qu.1.topic=bvar=atrig@

qu.1.1.mode=Multiple Choice@
qu.1.1.name=bx=asin@
qu.1.1.comment=@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=$a=rint(2,5);
$b=rint(2,5);
condition:eq(gcd($a,$b),1);
$A=$a^2;
$B=$b^2;@
qu.1.1.uid=0ce21539-89d0-4b48-8635-d106f2d174a6@
qu.1.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by trig substitution;
  Sub-Topic=Choose substitution;
@
qu.1.1.question=<p>An integral involves<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$A</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$B</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mi>n</mi></msup><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math> If you solve it by trigonometric substitution, you set</p>@
qu.1.1.answer=1@
qu.1.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$b</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&sdot;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi mathvariant='normal'>sin</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>&theta;</mi><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo></mrow></mrow></mrow><mfenced open='[' close=']' separators=','><mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&minus;</mo></mrow><mrow><mfrac><mi>&pi;</mi><mn>2</mn></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow><mrow><mfrac><mi>&pi;</mi><mn>2</mn></mfrac></mrow></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>@
qu.1.1.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$b</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&sdot;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi mathvariant='normal'>sin</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>&theta;</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo></mrow></mrow><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi>&pi;</mi><mn>2</mn></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow><mfrac><mrow><mi>&pi;</mi></mrow><mn>2</mn></mfrac></mrow></mfenced><mrow><mi></mi></mrow></mrow></mstyle></math>@
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qu.1.1.fixed=@

qu.1.2.mode=Multiple Choice@
qu.1.2.name=bx=asec@
qu.1.2.comment=@
qu.1.2.editing=useHTML@
qu.1.2.solution=@
qu.1.2.algorithm=$a=rint(2,5);
$b=rint(2,5);
condition:eq(gcd($a,$b),1);
$A=$a^2;
$B=$b^2;@
qu.1.2.uid=65ae87a7-3623-4263-9b90-e016ce2b6a77@
qu.1.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by trig substitution;
  Sub-Topic=Choose substitution;
@
qu.1.2.question=<p>An integral involves<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$B</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mn>2</mn></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$A</mi></mrow></mfenced><mi>n</mi></msup><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math>If you solve it by trigonometric substitution, you set</p>@
qu.1.2.answer=1@
qu.1.2.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>&theta;</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo></mrow><mfenced open='[' close=')' separators=','><mrow><mn>0</mn><mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow><mfrac><mi>&pi;</mi><mn>2</mn></mfrac></mrow></mfenced><mrow><mi></mi></mrow><mrow><mo separator='true' lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo></mrow><mfenced open='(' close=']' separators=','><mrow><mrow><mfrac><mi>&pi;</mi><mn>2</mn></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow><mi>&pi;</mi></mrow></mfenced></mrow></mstyle></math>@
qu.1.2.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>&theta;</mi><mo lspace='0.0em' rspace='0.0em'>&isin;</mo><mfenced open='[' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mi>&pi;</mi><mn>2</mn></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mn>0</mn></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mfenced open='(' close=']' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mfrac><mi>&pi;</mi><mn>2</mn></mfrac></mrow></mfenced><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math>@
qu.1.2.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$B</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$A</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>&theta;</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo></mrow><mfenced open='[' close=')' separators=','><mrow><mn>0</mn><mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow><mfrac><mi>&pi;</mi><mn>2</mn></mfrac></mrow></mfenced><mrow><mi></mi></mrow><mrow><mo separator='true' lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo></mrow><mfenced open='(' close=']' separators=','><mrow><mrow><mfrac><mi>&pi;</mi><mn>2</mn></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow><mi>&pi;</mi></mrow></mfenced></mrow></mstyle></math>@
qu.1.2.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$A</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$B</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>&theta;</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo></mrow><mfenced open='[' close=')' separators=','><mrow><mn>0</mn><mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow><mfrac><mi>&pi;</mi><mn>2</mn></mfrac></mrow></mfenced><mrow><mi></mi></mrow><mrow><mo separator='true' lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo></mrow><mfenced open='(' close=']' separators=','><mrow><mrow><mfrac><mi>&pi;</mi><mn>2</mn></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow><mi>&pi;</mi></mrow></mfenced></mrow></mstyle></math>@
qu.1.2.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi>&theta;</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo></mrow><mfenced open='[' close=']' separators=','><mrow><mn>0</mn><mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow><mrow><mi></mi></mrow><mi>&pi;</mi></mrow></mfenced></mrow></mstyle></math>@
qu.1.2.fixed=@

qu.1.3.mode=Multiple Choice@
qu.1.3.name=bx=atan@
qu.1.3.comment=@
qu.1.3.editing=useHTML@
qu.1.3.solution=@
qu.1.3.algorithm=$a=rint(2,5);
$b=rint(2,5);
condition:eq(gcd($a,$b),1);
$A=$a^2;
$B=$b^2;@
qu.1.3.uid=bc13f961-f7b4-4477-b2a1-7e89ca2bd9d2@
qu.1.3.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by trig substitution;
  Sub-Topic=Choose substitution;
@
qu.1.3.question=<p>An integral involves<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$A</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$B</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mi>n</mi></msup><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math> If you solve it by trigonometric substitution, you set</p>@
qu.1.3.answer=1@
qu.1.3.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$b</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&sdot;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi mathvariant='normal'>tan</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>&theta;</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo></mrow></mrow><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi>&pi;</mi><mn>2</mn></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow><mfrac><mrow><mi>&pi;</mi></mrow><mn>2</mn></mfrac></mrow></mfenced><mrow><mi></mi></mrow></mrow></mstyle></math>@
qu.1.3.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$b</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&sdot;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi mathvariant='normal'>tan</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>&theta;</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo></mrow></mrow><mfenced open='[' close=']' separators=','><mrow><mrow><mfrac><mi>&pi;</mi><mn>2</mn></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow><mfrac><mrow><mi>&pi;</mi></mrow><mn>2</mn></mfrac></mrow></mfenced></mrow></mstyle></math>@
qu.1.3.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$A</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&sdot;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$B</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi mathvariant='normal'>tan</mi><mo lspace='0.0em' rspace='0.0em'>&ApplyFunction;</mo><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>&theta;</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo></mrow></mrow><mfenced open='[' close=']' separators=','><mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&minus;</mo></mrow><mrow><mfrac><mi>&pi;</mi><mn>2</mn></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow><mrow><mfrac><mi>&pi;</mi><mn>2</mn></mfrac></mrow></mrow></mfenced></mrow></mstyle></math>@
qu.1.3.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$a</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&sdot;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>&theta;</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo></mrow></mrow><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi>&pi;</mi><mn>2</mn></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow><mfrac><mi>&pi;</mi><mn>2</mn></mfrac></mrow></mfenced><mrow><mi></mi></mrow></mrow></mstyle></math>@
qu.1.3.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$B</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&sdot;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$A</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>&theta;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&isin;</mo></mrow><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi>&pi;</mi><mn>2</mn></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow><mfrac><mi>&pi;</mi><mn>2</mn></mfrac></mrow></mfenced><mrow><mi></mi></mrow></mrow></mstyle></math>@
qu.1.3.fixed=@

qu.2.topic=b(x+c)=atrig@

qu.2.1.mode=Multiple Choice@
qu.2.1.name=b(x-c)=atan@
qu.2.1.comment=<p>After completing the square we have a power of $CTS.</p>@
qu.2.1.editing=useHTML@
qu.2.1.solution=@
qu.2.1.algorithm=$a=rint(2,5);
$b=rint(2,5);
$c=rint(1,6);
condition:eq(gcd($a,$b),1);
$C=$a^2+$b^2*$c^2;
$f="($C)+$b^2*x^2-2*$b^2*$c*x";
$cts="($b^2)*(x-$c)^2+($C-$c^2*$b^2)";
$M=maple("
MathML[ExportPresentation]($f),
MathML[ExportPresentation]($cts)
");
$F=switch(0,$M);
$CTS=switch(1,$M);
$v=mathml(x+$c);@
qu.2.1.uid=474e267e-17dd-4b1e-a951-a1c932cdcbc6@
qu.2.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by trig subsitution;
  Sub-Topic=Choose substitution;
@
qu.2.1.question=<p>An integral involves a power of the expression $F. After completing the square, we would use the trigonometric substitution</p>@
qu.2.1.answer=1@
qu.2.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$b</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced><mo    mathvariant='normal'     lspace='thickmathspace' rspace='thickmathspace'       >&equals;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='normal' >tan</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&ApplyFunction;</mo><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced></mrow><mrow></mrow></mrow></math>@
qu.2.1.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$b</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced><mo    mathvariant='normal'     lspace='thickmathspace' rspace='thickmathspace'       >&equals;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='normal' >tan</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&ApplyFunction;</mo><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced></mrow><mrow></mrow></mrow></math>@
qu.2.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$b</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced><mo    mathvariant='normal'     lspace='thickmathspace' rspace='thickmathspace'       >&equals;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='normal' >sin</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&ApplyFunction;</mo><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced></mrow><mrow></mrow></mrow></math>@
qu.2.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$b</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced><mo    mathvariant='normal'     lspace='thickmathspace' rspace='thickmathspace'       >&equals;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='normal' >sec</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&ApplyFunction;</mo><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced></mrow><mrow></mrow></mrow></math>@
qu.2.1.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced><mo    mathvariant='normal'     lspace='thickmathspace' rspace='thickmathspace'       >&equals;</mo><mi    mathvariant='normal' >$b</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='normal' >tan</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&ApplyFunction;</mo><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced></mrow><mrow></mrow></mrow></math>@
qu.2.1.fixed=@

qu.2.2.mode=Multiple Choice@
qu.2.2.name=b(x-c)=asin@
qu.2.2.comment=<p>After completing the square we have a power of $CTS.</p>@
qu.2.2.editing=useHTML@
qu.2.2.solution=@
qu.2.2.algorithm=$a=rint(2,10);
$b=rint(2,10);
$c=rint(1,6);
condition:eq(gcd($a,$b),1);
$C=$a^2-$b^2*$c^2;
$f="($C)-$b^2*x^2+2*$b^2*$c*x";
$cts="($C)+($c^2*$b^2)-($b^2)*(x-$c)^2";
$M=maple("
MathML[ExportPresentation]($f),
MathML[ExportPresentation]($cts)
");
$F=switch(0,$M);
$CTS=switch(1,$M);

$v=mathml(x-$c);@
qu.2.2.uid=7b90ab30-5e10-43cc-90b8-b260477b5220@
qu.2.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by trig substitution;
  Sub-Topic=Choose substitution;
@
qu.2.2.question=<p>An integral involves a power of the expression $F. After completing the square, we would use the trigonometric substitution</p>@
qu.2.2.answer=1@
qu.2.2.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$b</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced><mo    mathvariant='normal'     lspace='thickmathspace' rspace='thickmathspace'       >&equals;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='normal' >sin</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&ApplyFunction;</mo><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced></mrow><mrow></mrow></mrow></math>@
qu.2.2.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$b</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced><mo    mathvariant='normal'     lspace='thickmathspace' rspace='thickmathspace'       >&equals;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='normal' >sin</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&ApplyFunction;</mo><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced></mrow><mrow></mrow></mrow></math>@
qu.2.2.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$b</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced><mo    mathvariant='normal'     lspace='thickmathspace' rspace='thickmathspace'       >&equals;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='normal' >sin</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&ApplyFunction;</mo><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced></mrow><mrow></mrow></mrow></math>@
qu.2.2.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$b</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced><mo    mathvariant='normal'     lspace='thickmathspace' rspace='thickmathspace'       >&equals;</mo><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='normal' >sin</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&ApplyFunction;</mo><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced></mrow><mrow></mrow></mrow></math>@
qu.2.2.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$b</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced><mo    mathvariant='normal'     lspace='thickmathspace' rspace='thickmathspace'       >&equals;</mo><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='normal' >sin</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&ApplyFunction;</mo><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced></mrow><mrow></mrow></mrow></math>@
qu.2.2.fixed=@

qu.2.3.mode=Multiple Choice@
qu.2.3.name=b(x+c)=asin@
qu.2.3.comment=<p>After completing the square we have a power of $CTS.</p>@
qu.2.3.editing=useHTML@
qu.2.3.solution=@
qu.2.3.algorithm=$a=rint(2,5);
$b=rint(2,5);
$c=rint(1,6);
condition:eq(gcd($a,$b),1);
$C=$a^2-$b^2*$c^2;
$f="($C)-$b^2*x^2-2*$b^2*$c*x";
$cts="($C)+($c^2*$b^2)-($b^2)*(x+$c)^2";
$M=maple("
MathML[ExportPresentation]($f),
MathML[ExportPresentation]($cts)
");
$F=switch(0,$M);
$CTS=switch(1,$M);
$v=mathml(x+$c);@
qu.2.3.uid=42ca3436-c328-407f-96d2-189dd094941f@
qu.2.3.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by trig subsititution;
  Sub-Topic=Choose substitution;
@
qu.2.3.question=<p>An integral involves a power of the expression $F. After completing the square, we would use the trigonometric substitution</p>@
qu.2.3.answer=1@
qu.2.3.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$b</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced><mo    mathvariant='normal'     lspace='thickmathspace' rspace='thickmathspace'       >&equals;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='normal' >sin</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&ApplyFunction;</mo><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced></mrow><mrow></mrow></mrow></math>@
qu.2.3.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$b</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced><mo    mathvariant='normal'     lspace='thickmathspace' rspace='thickmathspace'       >&equals;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='normal' >sin</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&ApplyFunction;</mo><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced></mrow><mrow></mrow></mrow></math>@
qu.2.3.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced><mo    mathvariant='normal'     lspace='thickmathspace' rspace='thickmathspace'       >&equals;</mo><mi    mathvariant='normal' >$b</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='normal' >sin</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&ApplyFunction;</mo><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced></mrow><mrow></mrow></mrow></math>@
qu.2.3.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$b</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced><mo    mathvariant='normal'     lspace='thickmathspace' rspace='thickmathspace'       >&equals;</mo><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='normal' >sin</mi><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced></mrow><mrow></mrow></mrow></math>@
qu.2.3.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo    mathvariant='normal'             >&minus;</mo><mi    mathvariant='normal' >$b</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced><mo    mathvariant='normal'     lspace='thickmathspace' rspace='thickmathspace'       >&equals;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='normal' >sin</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&ApplyFunction;</mo><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced></mrow><mrow></mrow></mrow></math>@
qu.2.3.fixed=@

qu.2.4.mode=Multiple Choice@
qu.2.4.name=b(x+c)=atan@
qu.2.4.comment=<p>After completing the square we have a power of $CTS.</p>@
qu.2.4.editing=useHTML@
qu.2.4.solution=@
qu.2.4.algorithm=$a=rint(2,5);
$b=rint(2,5);
$c=rint(1,6);
condition:eq(gcd($a,$b),1);
$C=$a^2+$b^2*$c^2;
$f="($C)+$b^2*x^2+2*$b^2*$c*x";
$cts="($b^2)*(x+$c)^2+($C-$c^2*$b^2)";
$M=maple("
MathML[ExportPresentation]($f),
MathML[ExportPresentation]($cts)
");
$F=switch(0,$M);
$CTS=switch(1,$M);
$v=mathml(x+$c);@
qu.2.4.uid=d4b3a412-50e9-4bbc-b169-d1c20dd2b2f9@
qu.2.4.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by trig substitutio;
  Sub-Topic=Choose substitution;
@
qu.2.4.question=<p>An integral involves a power of the expression $F. After completing the square, we would use the trigonometric substitution</p>@
qu.2.4.answer=1@
qu.2.4.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$b</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced><mo    mathvariant='normal'     lspace='thickmathspace' rspace='thickmathspace'       >&equals;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='normal' >tan</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&ApplyFunction;</mo><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced></mrow><mrow></mrow></mrow></math>@
qu.2.4.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$b</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced><mo    mathvariant='normal'     lspace='thickmathspace' rspace='thickmathspace'       >&equals;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='normal' >tan</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&ApplyFunction;</mo><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced></mrow><mrow></mrow></mrow></math>@
qu.2.4.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$b</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced><mo    mathvariant='normal'     lspace='thickmathspace' rspace='thickmathspace'       >&equals;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='normal' >sin</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&ApplyFunction;</mo><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced></mrow><mrow></mrow></mrow></math>@
qu.2.4.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$b</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced><mo    mathvariant='normal'     lspace='thickmathspace' rspace='thickmathspace'       >&equals;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='normal' >sec</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&ApplyFunction;</mo><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced></mrow><mrow></mrow></mrow></math>@
qu.2.4.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced><mo    mathvariant='normal'     lspace='thickmathspace' rspace='thickmathspace'       >&equals;</mo><mi    mathvariant='normal' >$b</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='normal' >tan</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&ApplyFunction;</mo><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced></mrow><mrow></mrow></mrow></math>@
qu.2.4.fixed=@

qu.2.5.mode=Multiple Choice@
qu.2.5.name=b(x+c)=asec@
qu.2.5.comment=<p>After completing the square we have a power of $CTS.</p>@
qu.2.5.editing=useHTML@
qu.2.5.solution=@
qu.2.5.algorithm=$a=rint(2,5);
$b=rint(2,5);
$c=rint(1,6);
condition:eq(gcd($a,$b),1);
$C=$b^2*$c^2-$a^2;
$f="$b^2*x^2+2*$b^2*$c*x+($C)";
$cts="($b^2)*(x+$c)^2+($C-$c^2*$b^2)";
$M=maple("
MathML[ExportPresentation]($f),
MathML[ExportPresentation]($cts)
");
$F=switch(0,$M);
$CTS=switch(1,$M);
$v=mathml(x+$c);@
qu.2.5.uid=71012402-fecf-4811-8437-3cb4c475cf98@
qu.2.5.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by trig substitution;
  Sub-Topic=Choose substitution;
@
qu.2.5.question=<p>An integral involves a power of the expression $F. After completing the square, we would use the trigonometric substitution</p>@
qu.2.5.answer=1@
qu.2.5.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$b</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced><mo    mathvariant='normal'     lspace='thickmathspace' rspace='thickmathspace'       >&equals;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='normal' >sec</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&ApplyFunction;</mo><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced></mrow><mrow></mrow></mrow></math>@
qu.2.5.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$b</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced><mo    mathvariant='normal'     lspace='thickmathspace' rspace='thickmathspace'       >&equals;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='normal' >sec</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&ApplyFunction;</mo><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced></mrow><mrow></mrow></mrow></math>@
qu.2.5.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$b</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced><mo    mathvariant='normal'     lspace='thickmathspace' rspace='thickmathspace'       >&equals;</mo><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='normal' >sin</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&ApplyFunction;</mo><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced></mrow><mrow></mrow></mrow></math>@
qu.2.5.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$b</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced><mo    mathvariant='normal'     lspace='thickmathspace' rspace='thickmathspace'       >&equals;</mo><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='normal' >tan</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&ApplyFunction;</mo><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced></mrow><mrow></mrow></mrow></math>@
qu.2.5.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced><mo    mathvariant='normal'     lspace='thickmathspace' rspace='thickmathspace'       >&equals;</mo><mi    mathvariant='normal' >$b</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mrow><mi    mathvariant='normal' >tan</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&ApplyFunction;</mo><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced></mrow><mrow></mrow></mrow></math>@
qu.2.5.fixed=@

qu.3.topic=ResultofSubsNoCTS@

qu.3.1.mode=Multiple Choice@
qu.3.1.name=bx=atanResult@
qu.3.1.comment=@
qu.3.1.editing=useHTML@
qu.3.1.solution=@
qu.3.1.algorithm=$a=rint(2,5);
$b=rint(2,5);
condition:eq(gcd($a,$b),1);
$A=$a^2;
$B=$b^2;@
qu.3.1.uid=da70b68a-a712-4b14-b0f9-c025a80dabe1@
qu.3.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by trig substitution;
  Sub-Topic=Result of substitution;
@
qu.3.1.question=<p>An integral involves <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$A</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$B</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mrow><mi>n</mi></mrow></msup></mrow></mstyle></math>. If you solve it by triginometric substitution,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$A</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$B</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mstyle></math> becomes</p>@
qu.3.1.answer=1@
qu.3.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$A</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><msup ><mi    mathvariant='normal' >sec</mi><mn    mathvariant='normal' >2</mn></msup><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.3.1.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$A</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><msup ><mi    mathvariant='normal' >tan</mi><mn    mathvariant='normal' >2</mn></msup><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.3.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$B</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><msup ><mi    mathvariant='normal' >sec</mi><mn    mathvariant='normal' >2</mn></msup><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.3.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$B</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><msup ><mi    mathvariant='normal' >tan</mi><mn    mathvariant='normal' >2</mn></msup><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.3.1.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup ><mi    mathvariant='normal' >sec</mi><mn    mathvariant='normal' >2</mn></msup><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.3.1.fixed=@

qu.3.2.mode=Multiple Choice@
qu.3.2.name=bx=asecResult@
qu.3.2.comment=@
qu.3.2.editing=useHTML@
qu.3.2.solution=@
qu.3.2.algorithm=$a=rint(2,5);
$b=rint(2,5);
condition:eq(gcd($a,$b),1);
$A=$a^2;
$B=$b^2;@
qu.3.2.uid=bd7e71c0-3c89-41f6-b6e3-016bacb87b41@
qu.3.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by trig substitution;
  Sub-Topic=Result of substitution;
@
qu.3.2.question=<p>An integral involves <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$B</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mn>2</mn></msup><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$A</mi></mrow></mfenced><mrow><mi>n</mi></mrow></msup></mrow></mstyle></math>. If you solve it by trigonometric substitution, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$B</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mn>2</mn></msup><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$A</mi></mrow></mstyle></math> becomes</p>@
qu.3.2.answer=1@
qu.3.2.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$A</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><msup ><mi    mathvariant='normal' >tan</mi><mn    mathvariant='normal' >2</mn></msup><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.3.2.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$A</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><msup ><mi    mathvariant='normal' >sec</mi><mn    mathvariant='normal' >2</mn></msup><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.3.2.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$B</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><msup ><mi    mathvariant='normal' >sec</mi><mn    mathvariant='normal' >2</mn></msup><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.3.2.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$B</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><msup ><mi    mathvariant='normal' >tan</mi><mn    mathvariant='normal' >2</mn></msup><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.3.2.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo    mathvariant='normal'     lspace='verythinmathspace' rspace='verythinmathspace' stretchy='true'      >&verbar;</mo><mi    mathvariant='normal' >tan</mi><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced><mrow></mrow><mo    mathvariant='normal'     lspace='verythinmathspace' rspace='verythinmathspace' stretchy='true'      >&verbar;</mo></mrow></math>@
qu.3.2.fixed=@

qu.3.3.mode=Multiple Choice@
qu.3.3.name=bx=asinResult@
qu.3.3.comment=@
qu.3.3.editing=useHTML@
qu.3.3.solution=@
qu.3.3.algorithm=$a=rint(2,5);
$b=rint(2,5);
condition:eq(gcd($a,$b),1);
$A=$a^2;
$B=$b^2;@
qu.3.3.uid=c03d1332-7723-4e9c-8cb7-cc820e2ea767@
qu.3.3.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by trig substitution;
  Sub-Topic=Result of substitution;
@
qu.3.3.question=<p>An integral involves <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$A</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$B</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mrow><mi>n</mi></mrow></msup></mrow></mstyle></math>. If you solve it by trigonometric substitution, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$A</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$B</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mn>2</mn></msup><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo></mrow></mstyle></math> becomes</p>@
qu.3.3.answer=1@
qu.3.3.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$A</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><msup ><mi    mathvariant='normal' >cos</mi><mn    mathvariant='normal' >2</mn></msup><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.3.3.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$A</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><msup ><mi    mathvariant='normal' >sin</mi><mn    mathvariant='normal' >2</mn></msup><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.3.3.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$B</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><msup ><mi    mathvariant='normal' >cos</mi><mn    mathvariant='normal' >2</mn></msup><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.3.3.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$B</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><msup ><mi    mathvariant='normal' >sin</mi><mn    mathvariant='normal' >2</mn></msup><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.3.3.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup ><mi    mathvariant='normal' >cos</mi><mn    mathvariant='normal' >2</mn></msup><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.3.3.fixed=@

qu.4.topic=bvar=atrig approaches CTS@

qu.4.1.mode=Multiple Choice@
qu.4.1.name=b(x-c)=asin@
qu.4.1.comment=@
qu.4.1.editing=useHTML@
qu.4.1.solution=@
qu.4.1.algorithm=$a=rint(2,5);
$b=rint(2,5);
$c=rint(1,6);
condition:eq(gcd($a,$b),1);
$C=$a^2-$b^2*$c^2;
$M=2*$b^2*$c;
$E=$b^2*$c^2;
$B=$b^2;
$A=$a^2;
$F=mathml($f);
$v=mathml(x+$c);@
qu.4.1.uid=1488bc3e-81b2-4812-b7ea-f7dab2d417d7@
qu.4.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by trig substitution;
  Sub-Topic=Result of substitution;
@
qu.4.1.question=<p>An integral involves a power of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$A</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$B</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$c</mi></mrow></mfenced><mn>2</mn></msup><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/></mrow></mstyle></math>. Using the substitution <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$c</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>, the expression <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$A</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$B</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$c</mi></mrow></mfenced><mn>2</mn></msup><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/></mrow></mstyle></math> becomes</p>@
qu.4.1.answer=1@
qu.4.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$A</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><msup ><mi    mathvariant='normal' >cos</mi><mn    mathvariant='normal' >2</mn></msup><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.4.1.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$A</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><msup ><mi    mathvariant='normal' >sin</mi><mn    mathvariant='normal' >2</mn></msup><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.4.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$B</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><msup ><mi    mathvariant='normal' >cos</mi><mn    mathvariant='normal' >2</mn></msup><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.4.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$B</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><msup ><mi    mathvariant='normal' >sin</mi><mn    mathvariant='normal' >2</mn></msup><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.4.1.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mi    mathvariant='normal' >cos</mi><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.4.1.fixed=@

qu.4.2.mode=Multiple Choice@
qu.4.2.name=b(x+c)=atan@
qu.4.2.comment=@
qu.4.2.editing=useHTML@
qu.4.2.solution=@
qu.4.2.algorithm=$a=rint(2,5);
$b=rint(2,5);
$c=rint(1,6);
condition:eq(gcd($a,$b),1);
$C=$a^2+$b^2*$c^2;
$M=2*$b^2*$c;
$E=$b^2*$c^2;
$B=$b^2;
$A=$a^2;
$F=mathml($f);
$v=mathml(x+$c);@
qu.4.2.uid=53719663-b094-4bef-b728-10b7672fe8d3@
qu.4.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by trig substitution;
  Sub-Topic=Result of substitution;
@
qu.4.2.question=<p>An integral involves a power of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$A</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$B</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$c</mi></mrow></mfenced><mn>2</mn></msup><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/></mrow></mstyle></math>. Using the substitution <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$c</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&sdot;</mo><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/></mrow></mstyle></math>, the expression <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$A</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$B</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$c</mi></mrow></mfenced><mn>2</mn></msup><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/></mrow></mstyle></math> becomes</p>@
qu.4.2.answer=1@
qu.4.2.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$A</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><msup ><mi    mathvariant='normal' >sec</mi><mn    mathvariant='normal' >2</mn></msup><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.4.2.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$A</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><msup ><mi    mathvariant='normal' >tan</mi><mn    mathvariant='normal' >2</mn></msup><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.4.2.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$B</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><msup ><mi    mathvariant='normal' >tan</mi><mn    mathvariant='normal' >2</mn></msup><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.4.2.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$B</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><msup ><mi    mathvariant='normal' >sec</mi><mn    mathvariant='normal' >2</mn></msup><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.4.2.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mi    mathvariant='normal' >sec</mi><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.4.2.fixed=@

qu.4.3.mode=Multiple Choice@
qu.4.3.name=b(x-c)=atan@
qu.4.3.comment=@
qu.4.3.editing=useHTML@
qu.4.3.solution=@
qu.4.3.algorithm=$a=rint(2,5);
$b=rint(2,5);
$c=rint(1,6);
condition:eq(gcd($a,$b),1);
$C=$a^2+$b^2*$c^2;
$M=2*$b^2*$c;
$E=$b^2*$c^2;
$B=$b^2;
$A=$a^2;
$F=mathml($f);
$v=mathml(x+$c);@
qu.4.3.uid=98836e49-25f7-4d3e-a2e6-bd85ca66ed2a@
qu.4.3.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by trig substitution;
  Sub-Topic=Result of substitution;
@
qu.4.3.question=<p>An integral involves a power of&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$A</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$B</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$c</mi></mrow></mfenced><mn>2</mn></msup><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/></mrow></mstyle></math>. Using the substitution <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$c</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&sdot;</mo><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/></mrow></mstyle></math>, the expression <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$A</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$B</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$c</mi></mrow></mfenced><mn>2</mn></msup><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/></mrow></mstyle></math> becomes</p>@
qu.4.3.answer=1@
qu.4.3.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$A</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><msup ><mi    mathvariant='normal' >sec</mi><mn    mathvariant='normal' >2</mn></msup><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.4.3.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$A</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><msup ><mi    mathvariant='normal' >tan</mi><mn    mathvariant='normal' >2</mn></msup><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.4.3.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$B</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><msup ><mi    mathvariant='normal' >tan</mi><mn    mathvariant='normal' >2</mn></msup><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.4.3.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$B</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><msup ><mi    mathvariant='normal' >sec</mi><mn    mathvariant='normal' >2</mn></msup><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.4.3.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mi    mathvariant='normal' >sec</mi><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.4.3.fixed=@

qu.4.4.mode=Multiple Choice@
qu.4.4.name=b(x+c)=asin@
qu.4.4.comment=@
qu.4.4.editing=useHTML@
qu.4.4.solution=@
qu.4.4.algorithm=$a=rint(2,5);
$b=rint(2,5);
$c=rint(1,6);
condition:eq(gcd($a,$b),1);
$C=$a^2-$b^2*$c^2;
$M=2*$b^2*$c;
$E=$b^2*$c^2;
$B=$b^2;
$A=$a^2;
$F=mathml($f);
$v=mathml(x+$c);@
qu.4.4.uid=c0da8de6-a8cb-4318-a883-7d2d4bb5d4ea@
qu.4.4.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by trig substitution;
  Sub-Topic=Result of substitution;
@
qu.4.4.question=<p>An integral involves a power of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$A</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$B</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$c</mi></mrow></mfenced><mn>2</mn></msup><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/></mrow></mstyle></math>. Using the substitution <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$c</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/></mrow></mstyle></math>, the expression <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$A</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$B</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$c</mi></mrow></mfenced><mn>2</mn></msup><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/></mrow></mstyle></math> becomes</p>@
qu.4.4.answer=1@
qu.4.4.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$A</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><msup ><mi    mathvariant='normal' >cos</mi><mn    mathvariant='normal' >2</mn></msup><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.4.4.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$A</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><msup ><mi    mathvariant='normal' >sin</mi><mn    mathvariant='normal' >2</mn></msup><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.4.4.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$B</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><msup ><mi    mathvariant='normal' >cos</mi><mn    mathvariant='normal' >2</mn></msup><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.4.4.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$B</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><msup ><mi    mathvariant='normal' >sin</mi><mn    mathvariant='normal' >2</mn></msup><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.4.4.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mi    mathvariant='normal' >cos</mi><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.4.4.fixed=@

qu.4.5.mode=Multiple Choice@
qu.4.5.name=b(x-c)=asec@
qu.4.5.comment=@
qu.4.5.editing=useHTML@
qu.4.5.solution=@
qu.4.5.algorithm=$a=rint(2,5);
$b=rint(2,5);
$c=rint(1,6);
condition:eq(gcd($a,$b),1);
$C=$b^2*$c^2-$a^2;
$M=2*$b^2*$c;
$E=$b^2*$c^2;
$B=$b^2;
$A=$a^2;
$F=mathml($f);
$v=mathml(x+$c);@
qu.4.5.uid=aa1cf9ae-ce69-4849-aa9a-a96b885beee4@
qu.4.5.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by trig substitution;
  Sub-Topic=Result of substitution;
@
qu.4.5.question=<p>An intgral involves a power of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$B</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$c</mi></mrow></mfenced><mn>2</mn></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$A</mi><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/></mrow></mstyle></math>. Using the substitution <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$c</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&sdot;</mo><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/></mrow></mstyle></math>, the expression <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$B</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$c</mi></mrow></mfenced><mn>2</mn></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$A</mi><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/></mrow></mstyle></math> becomes.</p>@
qu.4.5.answer=1@
qu.4.5.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$A</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><msup ><mi    mathvariant='normal' >tan</mi><mn    mathvariant='normal' >2</mn></msup><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.4.5.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$A</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><msup ><mi    mathvariant='normal' >sec</mi><mn    mathvariant='normal' >2</mn></msup><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.4.5.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$B</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><msup ><mi    mathvariant='normal' >tan</mi><mn    mathvariant='normal' >2</mn></msup><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.4.5.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$B</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><msup ><mi    mathvariant='normal' >sec</mi><mn    mathvariant='normal' >2</mn></msup><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.4.5.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi    mathvariant='normal' >$a</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mi    mathvariant='normal' >tan</mi><mfenced><mrow><mi    mathvariant='italic' >&theta;</mi></mrow></mfenced><mrow></mrow></mrow></math>@
qu.4.5.fixed=@

qu.5.topic=IntSinSubs@

qu.5.1.question=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math>Evaluate<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mo    mathvariant='normal'   fence='unset' separator='unset' lspace='' rspace='' stretchy='unset' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='unset' accent='unset'>&Integral;</mo></mrow></mrow></math>$display<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math>.</p>@
qu.5.1.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.5.1.allow2d=1@
qu.5.1.maple_answer=simplify(int((x+$c)/($a^2-$b^2*(x-$c)^2)^(3/2),x))+C;@
qu.5.1.type=formula@
qu.5.1.mode=Maple@
qu.5.1.name=(x+c)/(a-b(x-c)^2)^(3/2)@
qu.5.1.comment=<p>After completing the square we have ,</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo></mrow></mstyle></math>$cts <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math>.</p>
<p>Make the substitution</p>
<p>$sub = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&theta;</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced></mrow></mstyle></math>.</p>@
qu.5.1.editing=useHTML@
qu.5.1.hint.1=Complete the square in the denominator.@
qu.5.1.solution=@
qu.5.1.algorithm=$a=rint(1,5);
$b=rint(1,5);
$c=rint(1,3);
condition:eq(gcd($a,$b),1);
$f="(x+$c)/expand(($a^2-$b^2*(x-$c)^2))^(3/2)";
$M=maple("
MathML[ExportPresentation]($f),
MathML[ExportPresentation]((x+$c)/($a^2-$b^2*(x-$c)^2)^(3/2))
");
$display=switch(0,$M);
$cts=switch(1,$M);
$sub=mathml("$b*(x-$c)");@
qu.5.1.uid=f012012b-c595-4c9f-be65-f58065ce6747@
qu.5.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Hard;
  Course=Introduction to Calculus II;
  Topic=Integration by trig substitution;
  Sub-Topic=sin substitution;
@

qu.5.2.question=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math>Evaluate<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mo    mathvariant='normal'   fence='unset' separator='unset' lspace='' rspace='' stretchy='unset' symmetric='unset' maxsize='' minsize='' largeop='unset' movablelimits='unset' accent='unset'>&Integral;</mo></mrow></mrow></math>$display<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math>.</p>@
qu.5.2.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.5.2.allow2d=1@
qu.5.2.maple_answer=simplify(int((x-$c)/($a^2-$b^2*(x+$c)^2)^(3/2),x))+C;@
qu.5.2.type=formula@
qu.5.2.mode=Maple@
qu.5.2.name=(x-c)/(a-b(x+c)^2)^(3/2)@
qu.5.2.comment=<p>After completing the square we have,</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo></mrow></mstyle></math>$cts <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math>.</p>
<p>Make the substitution</p>
<p>$sub <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo mathvariant='italic' separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&theta;</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mfenced></mrow></mstyle></math></p>@
qu.5.2.editing=useHTML@
qu.5.2.hint.1=Complete the square in the denominator.@
qu.5.2.solution=@
qu.5.2.algorithm=$a=rint(1,5);
$b=rint(1,5);
$c=rint(1,3);
condition:eq(gcd($a,$b),1);
$f="(x-$c)/expand(($a^2-$b^2*(x+$c)^2))^(3/2)";
$M=maple("
MathML[ExportPresentation]($f),
MathML[ExportPresentation]((x-$c)/($a^2-$b^2*(x+$c)^2)^(3/2))
");
$display=switch(0,$M);
$cts=switch(1,$M);
$sub=mathml("$b*(x+$c)");@
qu.5.2.uid=4c73596f-1290-4800-b5c4-8b99a575702d@
qu.5.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Hard;
  Course=Introduction to Calculus II;
  Topic=Integration by trig substitution;
  Sub-Topic=sin substitution;
@

qu.5.3.question=<p>Evaluate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&Integral;</mo></mrow></mstyle></math>$display<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math>.</p>@
qu.5.3.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.5.3.allow2d=1@
qu.5.3.maple_answer=simplify(int(($a^2-$b^2*x^2)^(1/2)/x^2,x))+C;@
qu.5.3.type=formula@
qu.5.3.mode=Maple@
qu.5.3.name=(a-bx^2)^(1/2)/x^2@
qu.5.3.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo></mrow></mstyle></math>$display <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math></p>
<p>Let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&theta;</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='[' close=']' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced></mrow></mstyle></math>.</p>
<p>therefore<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mi mathvariant='normal'>$a</mi><mrow><mi mathvariant='normal'>$b</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mi>d</mi><mrow><mi>&theta;</mi></mrow></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi mathvariant='normal'>$as</mi><mrow><mi mathvariant='normal'>$bs</mi></mrow></mfrac><msup><mi mathvariant='normal'>sin</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mfrac><msqrt><mrow><mi mathvariant='normal'>$as</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$as</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>sin</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></msqrt><mrow><mfrac><mi mathvariant='normal'>$as</mi><mrow><mi mathvariant='normal'>$bs</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>sin</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mi mathvariant='normal'>$a</mi><mrow><mi mathvariant='normal'>$b</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mfenced><mi>d&theta;</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&Integral;</mo><mfrac><msqrt><mrow><msup><mi mathvariant='normal'>cos</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></msqrt><mrow><msup><mi mathvariant='normal'>sin</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mi>&theta;</mi></mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mfrac><mrow><msup><mi mathvariant='normal'>cos</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow><mrow><msup><mi mathvariant='normal'>sin</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mi>&theta;</mi></mrow></mrow></mstyle></math>,&nbsp;Note for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&theta;</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='[' close=']' separators=','><mrow><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>therefore <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msqrt><mrow><msup><mi mathvariant='normal'>cos</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><msup><mi mathvariant='normal'>cot</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mi>&theta;</mi></mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><msup><mi mathvariant='normal'>csc</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mi>&theta;</mi></mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cot</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&theta;</mi><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></mstyle></math></p>
<p>&nbsp;<img width="460" height="274" alt="" src="__BASE_URI__pictures/sin.png" /></p>
<p>Where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>a</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>b</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math></p>
<p>Therefore, in both cases</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><msqrt><mrow><mi mathvariant='normal'>$as</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$bs</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></msqrt><mrow><mi>x</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>arcsin</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></mstyle></math></p>@
qu.5.3.editing=useHTML@
qu.5.3.solution=@
qu.5.3.algorithm=$a=rint(1,5);
$b=rint(1,5);
$as=$a*$a;
$bs=$b*$b;
condition:eq(gcd($a,$b),1);
$f="sqrt($a^2-$b^2*x^2)/x^2";
$display=maple("printf(MathML[ExportPresentation]($f))");@
qu.5.3.uid=25e152aa-201a-4782-aee1-e4ef76458365@
qu.5.3.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by trig substitution;
  Sub-Topic=sin substitution;
@

qu.5.4.question=<p>Evaluate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&Integral;</mo></mrow></mstyle></math>$display<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math>.</p>@
qu.5.4.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.5.4.allow2d=1@
qu.5.4.maple_answer=simplify(int((x^2/($a^2-$b^2*x^2)^(3/2)),x))+C;;@
qu.5.4.type=formula@
qu.5.4.mode=Maple@
qu.5.4.name=x^2/(a-bx^2)^(3/2)@
qu.5.4.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo></mrow></mstyle></math>$display <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math></p>
<p>Let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&theta;</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mfenced></mrow></mstyle></math>.</p>
<p>therefore<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mi mathvariant='normal'>$a</mi><mrow><mi mathvariant='normal'>$b</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mi>d</mi><mrow><mi>&theta;</mi></mrow></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi mathvariant='normal'>$as</mi><mrow><mi mathvariant='normal'>$bs</mi></mrow></mfrac><msup><mi mathvariant='normal'>sin</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mfrac><mrow><mfrac><mi mathvariant='normal'>$as</mi><mrow><mi mathvariant='normal'>$bs</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>sin</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$as</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$as</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>sin</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mfenced><mrow><mfrac><mn>3</mn><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mi mathvariant='normal'>$a</mi><mrow><mi mathvariant='normal'>$b</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mfenced><mi>d&theta;</mi></mrow><mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$bc</mi></mrow></mfrac></mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mfrac><mrow><msup><mi mathvariant='normal'>sin</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi mathvariant='normal'>cos</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mfenced><mrow><mfrac><mn>3</mn><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>&theta;</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$bc</mi></mrow></mfrac></mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mfrac><mrow><msup><mi mathvariant='normal'>sin</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow><mrow><msup><mi mathvariant='normal'>cos</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mfrac><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>d&theta;</mi></mrow></mstyle></math>,&nbsp;Note for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&theta;</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='(' close=')' separators=','><mrow><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>therefore <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi mathvariant='normal'>cos</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mfenced><mrow><mfrac><mn>3</mn><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi mathvariant='normal'>cos</mi><mrow><mn>3</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$bc</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><msup><mi mathvariant='normal'>tan</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>&theta;</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$bc</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mi>&theta;</mi></mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$bc</mi></mrow></mfrac><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></mstyle></math></p>
<p>&nbsp;</p>
<p>&nbsp;<img width="460" height="274" src="__BASE_URI__pictures/sin.png" alt="" /></p>
<p>Where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>a</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>b</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math></p>
<p>Therefore, in both cases</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$bc</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow><mrow><msqrt><mrow><mi mathvariant='normal'>$as</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$bs</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>arcsin</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac></mrow></mfenced></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mi>x</mi><mrow><mi mathvariant='normal'>$bs</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msqrt><mrow><mi mathvariant='normal'>$as</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$bs</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></msqrt></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$bc</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>arcsin</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></mstyle></math></p>@
qu.5.4.editing=useHTML@
qu.5.4.solution=@
qu.5.4.algorithm=$a=rint(1,5);
$b=rint(1,5);
$as=$a*$a;
$bs=$b*$b;
$bs2=$bs*2;
$bc=$bs*$b;
$bc2=$bc*2;
condition:eq(gcd($a,$b),1);
$f="x^2/($a^2-$b^2*x^2)^(3/2)";
$display=maple("printf(MathML[ExportPresentation]($f))");@
qu.5.4.uid=6bcb850d-7a40-4d38-9afc-e19f1eeb2d69@
qu.5.4.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by trig substitution;
  Sub-Topic=sin substitution;
@

qu.5.5.question=<p>Evaluate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&Integral;</mo></mrow></mstyle></math>$display<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math>.</p>@
qu.5.5.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.5.5.allow2d=1@
qu.5.5.maple_answer=simplify(int(x^2/($a^2-$b^2*x^2)^(1/2),x))+C;@
qu.5.5.type=formula@
qu.5.5.mode=Maple@
qu.5.5.name=x^2/(a-bx^2)^(1/2)@
qu.5.5.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo></mrow></mstyle></math>$display <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math></p>
<p>Let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&theta;</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mfenced></mrow></mstyle></math>.</p>
<p>therefore<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mi mathvariant='normal'>$a</mi><mrow><mi mathvariant='normal'>$b</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mi>d</mi><mrow><mi>&theta;</mi></mrow></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi mathvariant='normal'>$as</mi><mrow><mi mathvariant='normal'>$bs</mi></mrow></mfrac><msup><mi mathvariant='normal'>sin</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mfrac><mrow><mfrac><mi mathvariant='normal'>$as</mi><mrow><mi mathvariant='normal'>$bs</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>sin</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow><mrow><msqrt><mrow><mi mathvariant='normal'>$as</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$as</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>sin</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></msqrt></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mi mathvariant='normal'>$a</mi><mrow><mi mathvariant='normal'>$b</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mfenced><mi>d&theta;</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mfrac><mi mathvariant='normal'>$as</mi><mrow><mi mathvariant='normal'>$bc</mi></mrow></mfrac></mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mfrac><mrow><msup><mi mathvariant='normal'>sin</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow><mrow><msqrt><mrow><msup><mi mathvariant='normal'>cos</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></msqrt></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>&theta;</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mfrac><mi mathvariant='normal'>$as</mi><mrow><mi mathvariant='normal'>$bc</mi></mrow></mfrac></mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><msup><mi mathvariant='normal'>sin</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>d&theta;</mi></mrow></mstyle></math>,&nbsp;Note for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&theta;</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='(' close=')' separators=','><mrow><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>therefore <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msqrt><mrow><msup><mi mathvariant='normal'>cos</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi mathvariant='normal'>$as</mi><mrow><mi mathvariant='normal'>$bc2</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mi>&theta;</mi></mrow></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>&theta;</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi mathvariant='normal'>$as</mi><mrow><mi mathvariant='normal'>$bc2</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mn>2</mn><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&theta;</mi></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi mathvariant='normal'>$as</mi><mrow><mi mathvariant='normal'>$bc2</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></mstyle></math></p>
<p>&nbsp;<img width="460" height="274" src="__BASE_URI__pictures/sin.png" alt="" /></p>
<p>Where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>a</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>b</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math></p>
<p>Therefore, in both cases</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mi mathvariant='normal'>$as</mi><mrow><mi mathvariant='normal'>$bc2</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>arcsin</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$bs2</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo></mrow><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msqrt><mrow><mi mathvariant='normal'>$as</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$bs</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></msqrt><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow><mrow></mrow></mstyle></math></p>@
qu.5.5.editing=useHTML@
qu.5.5.solution=@
qu.5.5.algorithm=$a=rint(1,5);
$b=rint(1,5);
$as=$a*$a;
$bs=$b*$b;
$bs2=$bs*2;
$bc=$bs*$b;
$bc2=$bc*2;
condition:eq(gcd($a,$b),1);
$f="x^2/($a^2-$b^2*x^2)^(1/2)";
$display=maple("printf(MathML[ExportPresentation]($f))");@
qu.5.5.uid=f83ce299-603e-4d73-adcd-44181066ea91@
qu.5.5.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by trig substitution;
  Sub-Topic=sin substitution;
@

qu.5.6.question=<p>Evaluate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&Integral;</mo></mrow></mstyle></math>$display<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math>.</p>@
qu.5.6.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.5.6.allow2d=1@
qu.5.6.maple_answer=simplify(int(($a^2-$b^2*x^2)^(1/2),x))+C;@
qu.5.6.type=formula@
qu.5.6.mode=Maple@
qu.5.6.name=(a-bx^2)^(1/2)@
qu.5.6.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo></mrow></mstyle></math>$display <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math></p>
<p>Let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&theta;</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='[' close=']' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced></mrow></mstyle></math>.</p>
<p>therefore<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mi mathvariant='normal'>$a</mi><mrow><mi mathvariant='normal'>$b</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mi>d</mi><mrow><mi>&theta;</mi></mrow></mrow></mstyle></math>&nbsp;</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><msqrt><mrow><mi mathvariant='normal'>$as</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$as</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>sin</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></msqrt><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mi mathvariant='normal'>$a</mi><mrow><mi mathvariant='normal'>$b</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mfenced><mi>d&theta;</mi></mrow><mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mfrac><mi mathvariant='normal'>$a</mi><mrow><mi mathvariant='normal'>$b</mi></mrow></mfrac></mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><msqrt><mrow><mi mathvariant='normal'>$as</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>cos</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></msqrt><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>&theta;</mi></mrow><mrow><mi mathvariant='normal'></mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mfrac><mrow><mi mathvariant='normal'>$as</mi></mrow><mrow><mi mathvariant='normal'>$b</mi></mrow></mfrac></mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><msup><mi mathvariant='normal'>cos</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>&theta;</mi></mrow></mstyle></math>,&nbsp;Note for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&theta;</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='[' close=']' separators=','><mrow><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>therefore <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msqrt><mrow><msup><mi mathvariant='normal'>cos</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi mathvariant='normal'>$as</mi></mrow><mrow><mi mathvariant='normal'>$b2</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>&theta;</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi mathvariant='normal'>$as</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&theta;</mi></mrow><mrow><mi mathvariant='normal'>$b2</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mrow><mi mathvariant='normal'>$as</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&theta;</mi></mrow></mfenced></mrow><mrow><mi mathvariant='normal'>$b4</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow><mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi mathvariant='normal'>$a</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&theta;</mi></mrow><mrow><mi mathvariant='normal'>$b2</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mrow><mi mathvariant='normal'>$as</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow><mrow><mi mathvariant='normal'>$b2</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow><mrow></mrow></mstyle></math></p>
<p>&nbsp;<img width="460" height="274" src="__BASE_URI__pictures/sin.png" alt="" /></p>
<p>Where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>a</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>b</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math></p>
<p>Therefore, in both cases</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mi mathvariant='normal'>$as</mi></mrow><mrow><mi mathvariant='normal'>$b2</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>arcsin</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mrow><mi mathvariant='normal'>$as</mi></mrow><mrow><mi mathvariant='normal'>$b2</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfrac><mrow><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><msqrt><mrow><mi mathvariant='normal'>$as</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$bs</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></msqrt><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow><mrow><mi mathvariant='normal'></mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi mathvariant='normal'>$as</mi></mrow><mrow><mi mathvariant='normal'>$b2</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>arcsin</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msqrt><mrow><mi mathvariant='normal'>$as</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi mathvariant='normal'>$bs</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></mrow></msqrt></mrow><mrow><mn>2</mn></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></mstyle></math></p>@
qu.5.6.editing=useHTML@
qu.5.6.solution=@
qu.5.6.algorithm=$a=rint(1,5);
$b=rint(1,5);
$as=$a*$a;
$bs=$b*$b;
$b2=$b*2;
$b4=$b*4;
condition:eq(gcd($a,$b),1);
$f="($a^2-$b^2*x^2)^(1/2)";
$display=maple("printf(MathML[ExportPresentation]($f))");@
qu.5.6.uid=561ad25d-7194-41af-bb9c-7960356ad683@
qu.5.6.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by trig substitution;
  Sub-Topic=sin substitution;
@

qu.6.topic=IntTanSubst@

qu.6.1.question=<p>Evaluate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&Integral;</mo></mrow></mstyle></math>$display<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math>.</p>@
qu.6.1.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.6.1.allow2d=1@
qu.6.1.maple_answer=simplify(int(1/($a^2+$b^2*x^2)^2,x))+C;@
qu.6.1.type=formula@
qu.6.1.mode=Maple@
qu.6.1.name=1/(a+bx^2)^2@
qu.6.1.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>$display <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math></p>
<p>Let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&theta;</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mi mathvariant='normal'>$a</mi><mrow><mi mathvariant='normal'>$b</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>d</mi><mrow><mi>&theta;</mi></mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mfrac><mn>1</mn><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$as</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$as</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>tan</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi mathvariant='normal'>$a</mi></mrow><mrow><mi mathvariant='normal'>$b</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mi>&theta;</mi></mrow></mrow><mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$acb</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mfrac><mn>1</mn><mrow><msup><mi mathvariant='normal'>sec</mi><mrow><mn>4</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mi>&theta;</mi></mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$acb</mi></mrow></mfrac></mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><msup><mi mathvariant='normal'>cos</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mi>&theta;</mi></mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$acb2</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mi>&theta;</mi></mrow></mrow><mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$acb2</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mrow><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>&theta;</mi></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>C</mi></mrow><mrow><mi></mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$acb2</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></mstyle></math></p>
<p><img width="460" height="274" src="__BASE_URI__pictures/tan.png" alt="" /></p>
<p>Where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>a</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>b</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math></p>
<p>In both cases,</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$acb2</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mi mathvariant='normal'>arctan</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow><mrow><mfrac><mrow><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow><mrow><msqrt><mrow><mi mathvariant='normal'>$bs</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$as</mi></mrow></msqrt></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfrac><mrow><mi mathvariant='normal'>$a</mi></mrow><mrow><msqrt><mrow><mi mathvariant='normal'>$bs</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$as</mi></mrow></msqrt></mrow></mfrac><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow><mrow><mi mathvariant='normal'></mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$acb2</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>arctan</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$as2</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfrac><mi>x</mi><mrow><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$bs</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$as</mi></mrow></mfenced></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow><mrow></mrow></mstyle></math></p>@
qu.6.1.editing=useHTML@
qu.6.1.solution=@
qu.6.1.algorithm=$a=rint(1,4);
$b=rint(1,4);
$as=$a*$a;
$as2=$as*2;
$bs=$b*$b;
$acb=(($a)^3)*$b;
$acb2=$acb*2;
condition:eq(gcd($a,$b),1);
$f="1/($a^2+$b^2*x^2)^2";
$display=maple("printf(MathML[ExportPresentation]($f))");@
qu.6.1.uid=ac262899-85db-4950-a15c-ddb93ffb564b@
qu.6.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Hard;
  Course=Introduction to Calculus II;
  Topic=Integration by trig sub;
  Sub-Topic=tan substitution;
@

qu.6.2.question=<p>Evaluate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&Integral;</mo></mrow></mstyle></math>$display<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math>.</p>@
qu.6.2.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.6.2.allow2d=1@
qu.6.2.maple_answer=expand(int(x^3*($a^2+$b^2*x^2)^(1/2),x))+C;@
qu.6.2.type=formula@
qu.6.2.mode=Maple@
qu.6.2.name=x^3*(a+bx^2)^(1/2)@
qu.6.2.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>$display <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math></p>
<p>Let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&theta;</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced></mrow></mstyle></math></p>
<p>Therefore<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mi mathvariant='normal'>$a</mi><mrow><mi mathvariant='normal'>$b</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>d</mi><mrow><mi>&theta;</mi></mrow></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>x</mi><mrow><mn>3</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi mathvariant='normal'>$ac</mi><mrow><mi mathvariant='normal'>$bc</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>tan</mi><mrow><mn>3</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mfrac><mi mathvariant='normal'>$ac</mi><mrow><mi mathvariant='normal'>$bc</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>tan</mi><mrow><mn>3</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><msqrt><mrow><mi mathvariant='normal'>$as</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$as</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>tan</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></msqrt></mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi mathvariant='normal'>$a</mi></mrow><mrow><mi mathvariant='normal'>$b</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>&theta;</mi></mrow><mrow><mi mathvariant='normal'></mi></mrow><mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mi mathvariant='normal'>$a5</mi></mrow><mrow><mi mathvariant='normal'>$b4</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><msup><mi mathvariant='normal'>tan</mi><mrow><mn>3</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><msqrt><mrow><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></msqrt></mrow><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>&theta;</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi mathvariant='normal'>$a5</mi><mrow><mi mathvariant='normal'>$b4</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><msup><mi mathvariant='normal'>tan</mi><mrow><mn>3</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>sec</mi><mrow><mn>3</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mi>&theta;</mi></mrow></mrow></mstyle></math>, Note for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&theta;</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced></mrow><mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>sec</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo></mrow><mrow><mi>&theta;</mi><mo mathvariant='italic' fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn mathvariant='italic'>0</mn><mo mathvariant='italic' separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow></mstyle></math>therefore&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mfenced><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi mathvariant='normal'>sec</mi><mrow><mi></mi></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>&nbsp;</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi mathvariant='normal'>$a5</mi><mrow><mi mathvariant='normal'>$b4</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><msup><mi mathvariant='normal'>tan</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mi>&theta;</mi></mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi mathvariant='normal'>$a5</mi><mrow><mi mathvariant='normal'>$b4</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mfenced open='(' close=')' separators=','><mrow><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>d</mi><mrow><mi>&theta;</mi></mrow></mrow></mstyle></math></p>
<p>=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mi mathvariant='normal'>$a5</mi><mrow><mi mathvariant='normal'>$b4</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><msup><mi mathvariant='normal'>sec</mi><mrow><mn>5</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow><mrow><mn>5</mn></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><msup><mi mathvariant='normal'>sec</mi><mrow><mn>3</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></mstyle></math></p>
<p><img alt="" width="460" height="274" src="__BASE_URI__pictures/tan.png" /></p>
<p>Where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>a</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>b</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math></p>
<p>In both cases,</p>
<p>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi mathvariant='normal'>$a5</mi><mrow><mi mathvariant='normal'>$b4</mi></mrow></mfrac></mrow><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mfrac><mn>1</mn><mrow><mn>5</mn></mrow></mfrac></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><msqrt><mrow><mi mathvariant='normal'>$bs</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$as</mi></mrow></msqrt><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac></mrow></mfenced><mrow><mn>5</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mn>1</mn><mrow><mn>3</mn></mrow></mfrac><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><msqrt><mrow><mi mathvariant='normal'>$bs</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$as</mi></mrow></msqrt><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac></mrow></mfenced><mrow><mn>3</mn></mrow></msup><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow><mrow></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></mstyle></math></p>
<p>&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$b45</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$bs</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$as</mi></mrow></mfenced><mrow><mfrac><mn>5</mn><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mi mathvariant='normal'>$as</mi><mrow><mi mathvariant='normal'>$b43</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$bs</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msub><mi mathvariant='normal'>$as</mi><mrow></mrow></msub></mrow></mfenced><msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mfrac><mn>3</mn><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></mstyle></math></p>@
qu.6.2.editing=useHTML@
qu.6.2.solution=@
qu.6.2.algorithm=$a=rint(1,4);
$b=rint(1,4);
$as=$a*$a;
$bs=$b*$b;
$ac=($a)^3;
$bc=($b)^3;
$a5=($a)^5;
$b4=($b)^4;
$b45=$b4*5;
$b43=$b4*3;
condition:eq(gcd($a,$b),1);
$f="x^3*($a^2+$b^2*x^2)^(1/2)";
$display=maple("printf(MathML[ExportPresentation]($f))");@
qu.6.2.uid=b8a8bf55-e5aa-4e9a-978f-c14ce7759b68@
qu.6.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Hard;
  Course=Introduction to Calculus II;
  Topic=Integration by trig sub;
  Sub-Topic=tan substitution;
@

qu.6.3.question=<p>Evaluate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&Integral;</mo></mrow></mstyle></math>$display<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math>.</p>@
qu.6.3.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.6.3.allow2d=1@
qu.6.3.maple_answer=simplify(int(1/(x^2*($a^2+$b^2*x^2)^(1/2)),x))+C;@
qu.6.3.type=formula@
qu.6.3.mode=Maple@
qu.6.3.name=1/(x^2*(a+bx^2)^(1/2)@
qu.6.3.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>$display <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math></p>
<p>Let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&theta;</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced></mrow></mstyle></math></p>
<p>Therefore<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mi mathvariant='normal'>$a</mi><mrow><mi mathvariant='normal'>$b</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>d</mi><mrow><mi>&theta;</mi></mrow></mrow></mstyle></math>&nbsp;and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi mathvariant='normal'>$as</mi><mrow><mi mathvariant='normal'>$bs</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>tan</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mfrac><mn>1</mn><mrow><mfrac><mi mathvariant='normal'>$as</mi><mrow><mi mathvariant='normal'>$bs</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>tan</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><msqrt><mrow><mi mathvariant='normal'>$as</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$as</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>tan</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></msqrt></mrow></mrow></mfrac><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi mathvariant='normal'>$a</mi></mrow><mrow><mi mathvariant='normal'>$b</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mi>&theta;</mi></mrow></mrow><mrow><mi mathvariant='normal'></mi></mrow><mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mi mathvariant='normal'>$b</mi></mrow><mrow><mi mathvariant='normal'>$as</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mfrac><mn>1</mn><mrow><msup><mi mathvariant='normal'>tan</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><msqrt><mrow><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></msqrt></mrow></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mi>&theta;</mi></mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi mathvariant='normal'>$b</mi></mrow><mrow><mi mathvariant='normal'>$as</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mfrac><mrow><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow><mrow><msup><mi mathvariant='normal'>tan</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>&theta;</mi></mrow></mstyle></math>, Note for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&theta;</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced></mrow><mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>sec</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo></mrow><mrow><mi>&theta;</mi><mo mathvariant='italic' fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn mathvariant='italic'>0</mn><mo mathvariant='italic' separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow></mstyle></math>therefore&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mfenced><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi mathvariant='normal'>sec</mi><mrow><mi></mi></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>&nbsp;</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi mathvariant='normal'>$b</mi><mrow><mi mathvariant='normal'>$as</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>sin</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mi>&theta;</mi></mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi mathvariant='normal'>$b</mi><mrow><mi mathvariant='normal'>$as</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi mathvariant='normal'>sin</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mi mathvariant='normal'>$b</mi><mrow><mi mathvariant='normal'>$as</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>csc</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></mstyle></math></p>
<p><img alt="" width="460" height="274" src="__BASE_URI__pictures/tan.png" /></p>
<p>Where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>a</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>b</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math></p>
<p>In both cases,</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mi mathvariant='normal'>$b</mi><mrow><mi mathvariant='normal'>$as</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><msqrt><mrow><mi mathvariant='normal'>$bs</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$as</mi></mrow></msqrt><mrow><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$as</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfrac><msqrt><mrow><mi mathvariant='normal'>$bs</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$as</mi></mrow></msqrt><mrow><mi>x</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></mstyle></math></p>@
qu.6.3.editing=useHTML@
qu.6.3.solution=@
qu.6.3.algorithm=$a=rint(1,4);
$b=rint(1,4);
$as=$a*$a;
$bs=$b*$b;

condition:eq(gcd($a,$b),1);
$f="1/(x^2*($a^2+$b^2*x^2)^(1/2))";
$display=maple("printf(MathML[ExportPresentation]($f))");@
qu.6.3.uid=94e34e6b-afaf-4442-b589-3d4de9dfe253@
qu.6.3.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Hard;
  Course=Introduction to Calculus II;
  Topic=Integration by trig sub;
  Sub-Topic=tan substitution;
@

qu.6.4.question=<p>Evaluate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&Integral;</mo></mrow></mstyle></math>$display<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math>.</p>@
qu.6.4.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.6.4.allow2d=1@
qu.6.4.maple_answer=simplify(int((x+$d)/($a^2+$b^2*(x-$c)^2)^(3/2),x))+C;@
qu.6.4.type=formula@
qu.6.4.mode=Maple@
qu.6.4.name=(x+d)//(a+b(x-c)^2)^(3/2)@
qu.6.4.comment=<p>After completing the square we have ,</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo></mrow></mstyle></math>$cts <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math>.</p>
<p>Make the substitution $sub = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&theta;</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced></mrow></mstyle></math>.</p>@
qu.6.4.editing=useHTML@
qu.6.4.hint.1=Complete the square in the denominator.@
qu.6.4.solution=@
qu.6.4.algorithm=$a=rint(1,5);
$b=rint(2,4);
$c=rint(1,3);
$d=rint(1,4);
condition:eq(gcd($a,$b),1);
$f="(x+$d)/(expand($a^2+$b^2*(x-$c)^2))^(3/2)";
$M=maple("
MathML[ExportPresentation]($f),
MathML[ExportPresentation]( ((x+$d)/($a^2+$b^2*(x-$c)^2))^(3/2) )
");
$display=switch(0,$M);
$cts=switch(1,$M);
$sub=mathml("$b*(x-$c)");@
qu.6.4.uid=27848dd0-0177-416f-b0f2-5ba40773ad95@
qu.6.4.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Hard;
  Course=Introduction to Calculus II;
  Topic=Integration by trig substitution;
  Sub-Topic=tan substitution;
@

qu.6.5.question=<p>Evaluate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&Integral;</mo></mrow></mstyle></math>$display<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math>.</p>@
qu.6.5.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.6.5.allow2d=1@
qu.6.5.maple_answer=simplify(int((x-$d)/($a^2+$b^2*(x+$c)^2)^(3/2),x))+C;@
qu.6.5.type=formula@
qu.6.5.mode=Maple@
qu.6.5.name=(x-d)//(a+b(x+c)^2)^(3/2)@
qu.6.5.comment=<p>After completing the square we have,</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo></mrow></mstyle></math>$cts <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math>.</p>
<p>Make the substitution $sub = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&theta;</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='(' close=')' separators=','><mrow><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced></mrow></mstyle></math>.</p>@
qu.6.5.editing=useHTML@
qu.6.5.hint.1=Complete the square in the denominator.@
qu.6.5.solution=@
qu.6.5.algorithm=$a=rint(1,5);
$b=rint(1,4);
$c=rint(1,3);
$d=rint(1,3);
condition:eq(gcd($a,$b),1);
$f="(x-$d)/(expand($a^2+$b^2*(x+$c)^2))^(3/2)";
$M=maple("
MathML[ExportPresentation]($f),
MathML[ExportPresentation]((x-$d)/($a^2+$b^2*(x+$c)^2)^(3/2))
");
$display=switch(0,$M);
$cts=switch(1,$M);
$sub=mathml("$b*(x+$c)");@
qu.6.5.uid=753b9c5e-8987-437f-9833-f2df0d824cad@
qu.6.5.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Hard;
  Course=Introduction to Calculus II;
  Topic=Integration by trig substitution;
  Sub-Topic=tan substitution;
@

qu.6.6.question=<p>Evaluate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&Integral;</mo></mrow></mstyle></math>$display<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math>.</p>@
qu.6.6.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.6.6.allow2d=1@
qu.6.6.maple_answer=simplify(int(1/($a^2+$b^2*(x+$c)^2)^(3/2),x))+C;@
qu.6.6.type=formula@
qu.6.6.mode=Maple@
qu.6.6.name=1/(a+b(x+c)^2)^(3/2)@
qu.6.6.comment=<p>After completing the square we have ,</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo></mrow></mstyle></math>$cts <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math>.</p>
<p>Make the substitution $sub = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&theta;</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced></mrow></mstyle></math></p>@
qu.6.6.editing=useHTML@
qu.6.6.hint.1=Complete the square in the denominator@
qu.6.6.solution=@
qu.6.6.algorithm=$a=rint(1,5);
$b=rint(1,4);
$c=rint(1,4);
condition:eq(gcd($a,$b),1);
$f="1/(expand($a^2+$b^2*(x+$c)^2))^(3/2)";
$display=maple("printf(MathML[ExportPresentation]($f))");
$cts=maple("printf(MathML[ExportPresentation](1/($a^2+$b^2*(x+$c)^2)^(3/2)))");
$sub=mathml("$b*(x+$c)");@
qu.6.6.uid=dc23a463-a8bb-435b-ad54-f5344edbdfe5@
qu.6.6.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Hard;
  Course=Introduction to Calculus II;
  Topic=Integration by trig substitution;
  Sub-Topic=tan substitution;
@

qu.6.7.question=<p>Evaluate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&Integral;</mo></mrow></mstyle></math>$display<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math>.</p>@
qu.6.7.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.6.7.allow2d=1@
qu.6.7.maple_answer=simplify(int(1/($a^2+$b^2*x^2)^(3/2),x))+C;@
qu.6.7.type=formula@
qu.6.7.mode=Maple@
qu.6.7.name=1/(a+bx^2)^(3/2)@
qu.6.7.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>$display <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math></p>
<p>Let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&theta;</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mi mathvariant='normal'>$a</mi><mrow><mi mathvariant='normal'>$b</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>d</mi><mrow><mi>&theta;</mi></mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mfrac><mn>1</mn><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$as</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$as</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>tan</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mfenced><mrow><mfrac><mn>3</mn><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></mfrac><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi mathvariant='normal'>$a</mi></mrow><mrow><mi mathvariant='normal'>$b</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mi>&theta;</mi></mrow></mrow><mrow></mrow><mrow><mi></mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$asb</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mfrac><mn>1</mn><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mfenced><mrow><mfrac><mn>3</mn><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mi>&theta;</mi></mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$asb</mi></mrow></mfrac></mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><msup><mi mathvariant='normal'>cos</mi><mrow><mi></mi></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mi>&theta;</mi></mrow></mrow></mstyle></math>, Note for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&theta;</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced></mrow><mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>sec</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo></mrow><mrow><mi>&theta;</mi><mo mathvariant='italic' fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn mathvariant='italic'>0</mn><mo mathvariant='italic' separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow></mstyle></math>therefore <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mfenced><mrow><mfrac><mn>3</mn><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi mathvariant='normal'>sec</mi><mrow><mn>3</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>&nbsp;</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$asb</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></mstyle></math></p>
<p><img alt="" width="460" height="274" src="__BASE_URI__pictures/tan.png" /></p>
<p>Where <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>a</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$a</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>b</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math></p>
<p>In both cases,</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$asb</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mrow><mfrac><mi mathvariant='normal'>$bx</mi><mrow><msqrt><mrow><mi mathvariant='normal'>$bs</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$as</mi></mrow></msqrt></mrow></mfrac></mrow></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$as</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfrac><mi>x</mi><mrow><msqrt><mrow><mi mathvariant='normal'>$bs</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$as</mi></mrow></msqrt></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></mstyle></math></p>@
qu.6.7.editing=useHTML@
qu.6.7.solution=@
qu.6.7.algorithm=$a=rint(1,4);
$b=rint(1,4);
$as=$a*$a;
$bs=$b*$b;
$asb=$as*$b;
condition:eq(gcd($a,$b),1);
$f="1/($a^2+$b^2*x^2)^(3/2)";
$display=maple("printf(MathML[ExportPresentation]($f))");@
qu.6.7.uid=0902440f-78cb-40f9-b255-3b010a6f5d15@
qu.6.7.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Hard;
  Course=Introduction to Calculus II;
  Topic=Integration by trig sub;
  Sub-Topic=tan substitution;
@

qu.7.topic=IntSecSubst@

qu.7.1.question=<p>Evaluate<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mfrac><mrow><msqrt><mrow><msup><mi>x</mi><mn>2</mn></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msqrt></mrow><mrow><mi>x</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow></mstyle></math>assuming <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1.</mn></mrow></mstyle></math></p>
<p><strong>Hint: </strong>Since <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow></mstyle></math>only use the&nbsp;second quadrant triangle. Don't use absolute value anywhere in your answer.</p>@
qu.7.1.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.7.1.allow2d=1@
qu.7.1.maple_answer=sqrt(x^2-1)+arcsec(x)+C;@
qu.7.1.type=formula@
qu.7.1.mode=Maple@
qu.7.1.name=sqrt(x^2-1)/x, x negative@
qu.7.1.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mfrac><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msqrt><mrow><mi>x</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow><mrow><mi></mi></mrow></mstyle></math></p>
<p>Let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&theta;</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='[' close=')' separators=','><mrow><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&pi;</mi></mrow></mfenced></mrow><mrow><mi></mi></mrow></mstyle></math>&nbsp;Note: since <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow></mrow></mstyle></math>we only use the&nbsp;second quadrant.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mi>d</mi><mrow><mi>&theta;</mi></mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mfrac><msqrt><mrow><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msqrt><mrow><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mi>d</mi><mi>&theta;</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><msqrt><mrow><msup><mi mathvariant='normal'>tan</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></msqrt><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mi>&theta;</mi></mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><msup><mi mathvariant='normal'>tan</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>&theta;</mi></mrow></mstyle></math>&nbsp;Note: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&theta;</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='[' close=')' separators=','><mrow><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&pi;</mi></mrow></mfenced></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>&theta;</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&theta;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>C</mi></mrow></mstyle></math></p>
<p><img width="332" height="226" src="__BASE_URI__pictures/sec2.png" alt="" /></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msqrt><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>arcsec</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></mstyle></math></p>@
qu.7.1.editing=useHTML@
qu.7.1.solution=@
qu.7.1.algorithm=@
qu.7.1.uid=bf440ded-b26b-41c1-ab58-767f6e4d7904@
qu.7.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Medium;
  Course=Introduction to Calculus II;
  Topic=Integration by trig substitution;
  Sub-Topic=sec substitution;
@

qu.7.2.question=<p>Evaluate<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mfrac><mrow><msqrt><mrow><msup><mi>x</mi><mn>2</mn></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msqrt></mrow><mrow><mi>x</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow></mstyle></math>assuming <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&ge;</mo><mn>1.</mn></mrow></mstyle></math>Don't forget + C.</p>
<p><strong>Hint: </strong>Since <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&ge;</mo><mn>1</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow></mstyle></math>only use the first quadrant triangle. Don't use absolute value anywhere in your answer.</p>@
qu.7.2.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.7.2.allow2d=1@
qu.7.2.maple_answer=sqrt(x^2-1)-arcsec(x)+C;@
qu.7.2.type=formula@
qu.7.2.mode=Maple@
qu.7.2.name=sqrt(x^2-1)/x@
qu.7.2.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mfrac><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msqrt><mrow><mi>x</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow><mrow><mi></mi></mrow></mstyle></math></p>
<p>Let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>, <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&theta;</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='[' close=')' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced></mrow></mstyle></math>&nbsp;Note: since <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mn>1</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mi></mi></mrow></mstyle></math>we only use the first quadrant.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mi>d</mi><mrow><mi>&theta;</mi></mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mfrac><msqrt><mrow><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msqrt><mrow><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mi>d</mi><mi>&theta;</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><msqrt><mrow><msup><mi mathvariant='normal'>tan</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></msqrt><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mi>&theta;</mi></mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><msup><mi mathvariant='normal'>tan</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mrow><mi>&theta;</mi></mrow></mrow></mstyle></math>&nbsp;Note: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&theta;</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='[' close=')' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>&theta;</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&theta;</mi><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>C</mi></mrow></mstyle></math></p>
<p><img width="227" height="178" src="__BASE_URI__pictures/sec1.png" alt="" /></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msqrt><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>arcsec</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></mstyle></math></p>@
qu.7.2.editing=useHTML@
qu.7.2.solution=@
qu.7.2.algorithm=@
qu.7.2.uid=7063f0ee-a74f-4a54-a7d0-b3f4922361f2@
qu.7.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Medium;
  Course=Introduction to Calculus II;
  Topic=Integration by trig substitution;
  Sub-Topic=sec substitution;
@

qu.7.3.question=<p>Evaluate<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msqrt></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow></mstyle></math>assuming <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn>1.</mn></mrow></mstyle></math></p>
<p><strong>Hint: </strong>Since <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn>1</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow></mstyle></math>only use the first quadrant triangle. Don't use absolute value anywhere in your answer.</p>@
qu.7.3.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.7.3.allow2d=1@
qu.7.3.maple_answer=ln(sqrt(x^2-1)+x)+C;@
qu.7.3.type=formula@
qu.7.3.mode=Maple@
qu.7.3.name=1/sqrt(x^2 -1)@
qu.7.3.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msqrt></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow><mrow></mrow><mrow><mi></mi></mrow></mstyle></math></p>
<p>Let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&theta;</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='(' close=')' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mfenced></mrow></mstyle></math>&nbsp;Note: since <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn>1</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow></mrow></mstyle></math>we only use the first quadrant.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mi>d</mi><mrow><mi>&theta;</mi></mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mfrac><mn>1</mn><mrow><msqrt><mrow><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msqrt></mrow></mfrac></mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mi>d</mi><mi>&theta;</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mfrac><mn>1</mn><mrow><msqrt><mrow><msup><mi mathvariant='normal'>tan</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>&theta;</mi></mrow></mfenced></mrow></msqrt></mrow></mfrac><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>&theta;</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>&theta;</mi></mrow></mstyle></math>&nbsp;Note: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&theta;</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='(' close=')' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mfenced></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>ln</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mfenced><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>C</mi></mrow></mstyle></math>&nbsp;Note both <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>and&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math> are positive for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&theta;</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='(' close=')' separators=','><mrow><mn>0</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced></mrow></mstyle></math></p>
<p><img width="227" height="178" src="__BASE_URI__pictures/sec1.png" alt="" /></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>ln</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msqrt></mrow><mrow><mi>x</mi></mrow><mrow><mi></mi></mrow></mfrac></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></mstyle></math></p>@
qu.7.3.editing=useHTML@
qu.7.3.solution=@
qu.7.3.algorithm=@
qu.7.3.uid=e6358ee7-28f5-441e-9cb1-7c3713f82869@
qu.7.3.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Medium;
  Course=Introduction to Calculus II;
  Topic=Integration by trig substitution;
  Sub-Topic=sec substitution;
@

qu.7.4.question=<p>Evaluate<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msqrt></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow></mstyle></math>assuming <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1.</mn></mrow></mstyle></math></p>
<p><strong>Hint: </strong>Since <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow></mstyle></math>only use the&nbsp;second quadrant triangle. Don't use absolute value anywhere in your answer. When <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msqrt><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mn>0</mn></mrow></mstyle></math> and so <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfenced open='|' close='|' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msqrt></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msqrt></mrow></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msqrt><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>x</mi></mrow></mstyle></math></p>@
qu.7.4.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.7.4.allow2d=1@
qu.7.4.maple_answer=-ln(sqrt(x^2-1)-x)+C;@
qu.7.4.type=formula@
qu.7.4.mode=Maple@
qu.7.4.name=1/sqrt(x^2 -1) x negative@
qu.7.4.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msqrt></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow><mrow></mrow><mrow><mi></mi></mrow></mstyle></math></p>
<p>Let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mstyle></math>,&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&theta;</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='(' close=')' separators=','><mrow><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&pi;</mi></mrow></mfenced></mrow></mstyle></math>&nbsp;Note: since <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mi mathvariant='normal'></mi></mrow></mstyle></math>we only use the&nbsp;second quadrant.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mi>d</mi><mrow><mi>&theta;</mi></mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mfrac><mn>1</mn><mrow><msqrt><mrow><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msqrt></mrow></mfrac></mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mi>d</mi><mi>&theta;</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mfrac><mn>1</mn><mrow><msqrt><mrow><msup><mi mathvariant='normal'>tan</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>&theta;</mi></mrow></mfenced></mrow></msqrt></mrow></mfrac><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>&theta;</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>&theta;</mi></mrow></mstyle></math>&nbsp;Note: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>for <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&theta;</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&isin;</mo><mfenced open='(' close=')' separators=','><mrow><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>&pi;</mi></mrow></mfenced></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>ln</mi><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='|' close='|' separators=','><mrow><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>&theta;</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfenced><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>C</mi></mrow></mstyle></math>&nbsp;</p>
<p><img width="332" height="226" src="__BASE_URI__pictures/sec2.png" alt="" /></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>ln</mi><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='|' close='|' separators=','><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msqrt></mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>ln</mi><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msqrt><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msqrt><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>C</mi></mrow></mstyle></math>&nbsp; from hint.</p>@
qu.7.4.editing=useHTML@
qu.7.4.solution=@
qu.7.4.algorithm=@
qu.7.4.uid=d4860c72-73f7-4417-ac20-2ab56ccaacf4@
qu.7.4.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Medium;
  Course=Introduction to Calculus II;
  Topic=Integration by trig substitution;
  Sub-Topic=sec substitution;
@

