qu.1.topic=ParFracDec@

qu.1.1.mode=Multiple Choice@
qu.1.1.name=quadPartialFractionRepeated@
qu.1.1.comment=@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=$a=rint(-9,9);
$b=rint(2,10);
$c=rint(1,9);
$d=rint(1,9);
$e=rint(1,9);
condition:eq(gcd($a,$b),1);
condition:ne($c,$d);
condition:ne($c,$e);
condition:ne($d,$e);
condition:eq(gcd($a,$b),1);
condtion:ne($a,1);
$f="($a*x+$b)/((x^2+$c)^2*(x^2+x+1)*(x-$e))";
$display=maple("printf(MathML[ExportPresentation]($f))");@
qu.1.1.uid=e6e05372-2d3f-49de-9e8d-140828b8741e@
qu.1.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by partial fractions;
  Sub-Topic=Setup;
@
qu.1.1.question=<p>As a partial fraction decomposition, we set $display equal to</p>@
qu.1.1.answer=1@
qu.1.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac    ><mrow><mi    mathvariant='italic' >A</mi><mo    mathvariant='normal'             >&sdot;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >B</mi></mrow><mrow><msup ><mfenced><mrow><msup ><mi    mathvariant='italic' >x</mi><mn    mathvariant='normal' >2</mn></msup><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced><mn    mathvariant='normal' >2</mn></msup></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mrow><mi    mathvariant='italic' >C</mi><mo    mathvariant='normal'             >&sdot;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >D</mi></mrow><mrow><msup ><mi    mathvariant='italic' >x</mi><mrow><mn    mathvariant='normal' >2</mn></mrow></msup><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mrow><mi    mathvariant='italic' >E</mi><mo    mathvariant='normal'             >&sdot;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >F</mi></mrow><mrow><msup ><mi    mathvariant='italic' >x</mi><mn    mathvariant='normal' >2</mn></msup><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mn    mathvariant='normal' >1</mn></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mi    mathvariant='italic' >G</mi><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$e</mi></mrow></mfrac></mrow></math>@
qu.1.1.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac    ><mrow><mi    mathvariant='italic' >A</mi><mo    mathvariant='normal'             >&sdot;</mo><msup ><mi    mathvariant='italic' >x</mi><mn    mathvariant='normal' >3</mn></msup><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >B</mi><mo    mathvariant='normal'             >&sdot;</mo><msup ><mi    mathvariant='italic' >x</mi><mn    mathvariant='normal' >2</mn></msup><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >C</mi><mo    mathvariant='normal'             >&sdot;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >D</mi></mrow><mrow><msup ><mfenced><mrow><msup ><mi    mathvariant='italic' >x</mi><mn    mathvariant='normal' >2</mn></msup><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced><mn    mathvariant='normal' >2</mn></msup></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mrow><mi    mathvariant='italic' >Ex</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >F</mi></mrow><mrow><msup ><mi    mathvariant='italic' >x</mi><mrow><mn    mathvariant='normal' >2</mn></mrow></msup><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mrow><mi    mathvariant='italic' >G</mi><mo    mathvariant='normal'             >&sdot;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >H</mi></mrow><mrow><msup ><mi    mathvariant='italic' >x</mi><mn    mathvariant='normal' >2</mn></msup><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mn    mathvariant='normal' >1</mn></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mi    mathvariant='italic' >I</mi><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$e</mi></mrow></mfrac></mrow></math>@
qu.1.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac    ><mrow><mi    mathvariant='italic' >A</mi><mo    mathvariant='normal'             >&sdot;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >B</mi></mrow><mrow><msup ><mfenced><mrow><msup ><mi    mathvariant='italic' >x</mi><mn    mathvariant='normal' >2</mn></msup><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced><mn    mathvariant='normal' >2</mn></msup></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mrow><mi    mathvariant='italic' >C</mi><mo    mathvariant='normal'             >&sdot;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >D</mi></mrow><mrow><msup ><mi    mathvariant='italic' >x</mi><mrow><mn    mathvariant='normal' >2</mn></mrow></msup><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mrow><mi    mathvariant='italic' >E</mi><mo    mathvariant='normal'             >&sdot;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >F</mi></mrow><mrow><msup ><mi    mathvariant='italic' >x</mi><mn    mathvariant='normal' >2</mn></msup><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mn    mathvariant='normal' >1</mn></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mrow><mi    mathvariant='italic' >G</mi><mo    mathvariant='normal'             >&sdot;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >H</mi></mrow><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$e</mi></mrow></mfrac></mrow></math>@
qu.1.1.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><mfrac><mrow><mi>A</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&sdot;</mo><msup><mi>x</mi><mn>3</mn></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>B</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&sdot;</mo><msup><mi>x</mi><mn>2</mn></msup></mrow><mrow><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mn>2</mn></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$c</mi></mrow></mfenced><mn>2</mn></msup></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mrow><mi>C</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&sdot;</mo><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>D</mi></mrow><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$c</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mrow><mi>E</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$F</mi></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mi>G</mi><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$e</mi></mrow></mfrac></mrow><mrow><mi></mi></mrow></mstyle></math>@
qu.1.1.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac    ><mrow><mi    mathvariant='italic' >A</mi></mrow><mrow><msup ><mfenced><mrow><msup ><mi    mathvariant='italic' >x</mi><mn    mathvariant='normal' >2</mn></msup><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced><mn    mathvariant='normal' >2</mn></msup></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mi    mathvariant='italic' >B</mi><mrow><msup ><mi    mathvariant='italic' >x</mi><mrow><mn    mathvariant='normal' >2</mn></mrow></msup><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mi    mathvariant='italic' >C</mi><mrow><msup ><mi    mathvariant='italic' >x</mi><mn    mathvariant='normal' >2</mn></msup><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mn    mathvariant='normal' >1</mn></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mi    mathvariant='italic' >D</mi><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&minus;</mo><mi    mathvariant='normal' >$e</mi></mrow></mfrac></mrow></math>@
qu.1.1.fixed=@

qu.1.2.mode=Multiple Choice@
qu.1.2.name=linearPartialFractionRepeated@
qu.1.2.comment=@
qu.1.2.editing=useHTML@
qu.1.2.solution=@
qu.1.2.algorithm=$a=rint(-9,9);
$b=rint(1,10);
$c=rint(1,9);
$d=rint(1,9);
$e=rint(1,9);
condition:ne($c,$d);
condition:ne($c,$e);
condition:ne($d,$e);
condition:eq(gcd($a,$b),1);
condtion:ne($a,1);
$f="($a*x+$b)/((x+$c)^3*(x+$d)^2*(x+$e))";
$display=maple("printf(MathML[ExportPresentation]($f))");@
qu.1.2.uid=ff9cc9d4-7dd9-463f-a702-b045b76f032d@
qu.1.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by partial fractions;
  Sub-Topic=Setup;
@
qu.1.2.question=<p>As a partial fraction decomposition, we set $display equal to</p>@
qu.1.2.answer=1@
qu.1.2.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac    ><mi    mathvariant='italic' >A</mi><mrow><msup ><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced><mn    mathvariant='normal' >3</mn></msup></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mi    mathvariant='italic' >B</mi><mrow><msup ><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced><mn    mathvariant='normal' >2</mn></msup></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mi    mathvariant='italic' >C</mi><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mi    mathvariant='italic' >D</mi><mrow><msup ><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$d</mi></mrow></mfenced><mn    mathvariant='normal' >2</mn></msup></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mi    mathvariant='italic' >E</mi><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$d</mi></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mi    mathvariant='italic' >F</mi><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$e</mi></mrow></mfrac></mrow></math>@
qu.1.2.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac    ><mrow><mi    mathvariant='italic' >A</mi><mo    mathvariant='normal'             >&sdot;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >B</mi></mrow><mrow><msup ><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced><mn    mathvariant='normal' >3</mn></msup></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mrow><mi    mathvariant='italic' >C</mi><mo    mathvariant='normal'             >&sdot;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >D</mi></mrow><mrow><msup ><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced><mn    mathvariant='normal' >2</mn></msup></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mrow><mi    mathvariant='italic' >E</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >F</mi></mrow><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mrow><mi    mathvariant='italic' >G</mi><mo    mathvariant='normal'             >&sdot;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >H</mi></mrow><mrow><msup ><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$d</mi></mrow></mfenced><mn    mathvariant='normal' >2</mn></msup></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'></mspace><mfrac    ><mrow><mi    mathvariant='italic' >J</mi><mo    mathvariant='normal'             >&sdot;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >K</mi></mrow><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$d</mi></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mrow><mi    mathvariant='italic' >M</mi><mo    mathvariant='normal'             >&sdot;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >N</mi></mrow><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$e</mi></mrow></mfrac></mrow></math>@
qu.1.2.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac    ><mrow><mi    mathvariant='italic' >A</mi><mo    mathvariant='normal'             >&sdot;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >B</mi></mrow><mrow><msup ><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced><mn    mathvariant='normal' >3</mn></msup></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mrow><mi    mathvariant='italic' >C</mi><mo    mathvariant='normal'             >&sdot;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >D</mi></mrow><mrow><msup ><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$d</mi></mrow></mfenced><mn    mathvariant='normal' >2</mn></msup></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mrow><mi    mathvariant='italic' >E</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >F</mi></mrow><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$e</mi></mrow></mfrac></mrow></math>@
qu.1.2.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac    ><mrow><mi    mathvariant='italic' >A</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><msup ><mi    mathvariant='italic' >x</mi><mn    mathvariant='normal' >2</mn></msup></mrow><mrow><msup ><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced><mn    mathvariant='normal' >3</mn></msup></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mrow><mi    mathvariant='italic' >B</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mi    mathvariant='italic' >x</mi></mrow><mrow><msup ><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$d</mi></mrow></mfenced><mn    mathvariant='normal' >2</mn></msup></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mi    mathvariant='italic' >C</mi><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$e</mi></mrow></mfrac></mrow></math>@
qu.1.2.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac    ><mi    mathvariant='normal' >$c</mi><mrow><msup ><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfenced><mn    mathvariant='normal' >3</mn></msup></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mrow><mi    mathvariant='normal' >$d</mi></mrow><mrow><msup ><mfenced><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$d</mi></mrow></mfenced><mn    mathvariant='normal' >2</mn></msup></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mi    mathvariant='normal' >$e</mi><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$e</mi></mrow></mfrac></mrow></math>@
qu.1.2.fixed=@

qu.1.3.mode=Multiple Choice@
qu.1.3.name=quadPartialFraction@
qu.1.3.comment=@
qu.1.3.editing=useHTML@
qu.1.3.solution=@
qu.1.3.algorithm=$a=rint(-9,9);
$b=rint(1,10);
$c=rint(1,9);
$d=rint(1,9);
$e=rint(1,9);
condition:ne($c,$d);
condition:ne($c,$e);
condition:ne($d,$e);
condition:eq(gcd($a,$b),1);
condtion:ne($a,1);
$f="($a*x+$b)/((x^2+$c)*(x^2+x+1)*(x+$e))";
$display=maple("printf(MathML[ExportPresentation]($f))");@
qu.1.3.uid=b007dda3-e422-45a4-9c80-735124dbb17d@
qu.1.3.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by partial fractions;
  Sub-Topic=Setup;
@
qu.1.3.question=<p>As a partial fraction decomposition, we set $display equal to</p>@
qu.1.3.answer=1@
qu.1.3.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac    ><mrow><mi    mathvariant='italic' >A</mi><mo    mathvariant='normal'             >&sdot;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >B</mi></mrow><mrow><msup ><mi    mathvariant='italic' >x</mi><mn    mathvariant='normal' >2</mn></msup><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mrow><mi    mathvariant='italic' >C</mi><mo    mathvariant='normal'             >&sdot;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >D</mi></mrow><mrow><msup ><mi    mathvariant='italic' >x</mi><mrow><mn    mathvariant='normal' >2</mn></mrow></msup><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mn    mathvariant='normal' >1</mn></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mi    mathvariant='italic' >E</mi><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$e</mi></mrow></mfrac></mrow></math>@
qu.1.3.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac    ><mrow><mi    mathvariant='italic' >A</mi></mrow><mrow><msup ><mi    mathvariant='italic' >x</mi><mn    mathvariant='normal' >2</mn></msup><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mi    mathvariant='italic' >B</mi><mrow><msup ><mi    mathvariant='italic' >x</mi><mrow><mn    mathvariant='normal' >2</mn></mrow></msup><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mn    mathvariant='normal' >1</mn></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mi    mathvariant='italic' >C</mi><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$e</mi></mrow></mfrac></mrow></math>@
qu.1.3.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac    ><mrow><mi    mathvariant='italic' >A</mi><mo    mathvariant='normal'             >&sdot;</mo><msup ><mi    mathvariant='italic' >x</mi><mn    mathvariant='normal' >2</mn></msup><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >B</mi><mo    mathvariant='normal'             >&sdot;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >C</mi></mrow><mrow><msup ><mi    mathvariant='italic' >x</mi><mn    mathvariant='normal' >2</mn></msup><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mrow><mi    mathvariant='italic' >D</mi><mo    mathvariant='normal'             >&sdot;</mo><msup ><mi    mathvariant='italic' >x</mi><mn    mathvariant='normal' >2</mn></msup><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >E</mi><mo    mathvariant='normal'             >&sdot;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >F</mi></mrow><mrow><msup ><mi    mathvariant='italic' >x</mi><mrow><mn    mathvariant='normal' >2</mn></mrow></msup><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mn    mathvariant='normal' >1</mn></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mrow><mi    mathvariant='italic' >G</mi><mo    mathvariant='normal'             >&sdot;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >H</mi></mrow><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$e</mi></mrow></mfrac></mrow></math>@
qu.1.3.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac    ><mrow><mi    mathvariant='italic' >A</mi><mo    mathvariant='normal'             >&sdot;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >B</mi></mrow><mrow><msup ><mi    mathvariant='italic' >x</mi><mn    mathvariant='normal' >2</mn></msup><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mrow><mi    mathvariant='italic' >C</mi><mo    mathvariant='normal'             >&sdot;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >D</mi></mrow><mrow><msup ><mi    mathvariant='italic' >x</mi><mrow><mn    mathvariant='normal' >2</mn></mrow></msup><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mn    mathvariant='normal' >1</mn></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mrow><mi    mathvariant='italic' >E</mi><mo    mathvariant='normal'             >&sdot;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >F</mi></mrow><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$e</mi></mrow></mfrac></mrow></math>@
qu.1.3.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac    ><mrow><mi    mathvariant='italic' >A</mi><mo    mathvariant='normal'             >&sdot;</mo><mi    mathvariant='italic' >x</mi></mrow><mrow><msup ><mi    mathvariant='italic' >x</mi><mn    mathvariant='normal' >2</mn></msup><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mrow><mi    mathvariant='italic' >B</mi><mo    mathvariant='normal'             >&sdot;</mo><mi    mathvariant='italic' >x</mi></mrow><mrow><msup ><mi    mathvariant='italic' >x</mi><mrow><mn    mathvariant='normal' >2</mn></mrow></msup><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mn    mathvariant='normal' >1</mn></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mi    mathvariant='italic' >C</mi><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$e</mi></mrow></mfrac></mrow></math>@
qu.1.3.fixed=@

qu.1.4.mode=Multiple Choice@
qu.1.4.name=linearPartialFraction@
qu.1.4.comment=@
qu.1.4.editing=useHTML@
qu.1.4.solution=@
qu.1.4.algorithm=$a=rint(-9,9);
$b=rint(1,10);
$c=rint(1,9);
$d=rint(1,9);
$e=rint(1,9);
condition:ne($c,$d);
condition:ne($c,$e);
condition:ne($d,$e);
condition:eq(gcd($a,$b),1);
condition:ne($a,1);
$f="($a*x+$b)/((x+$c)*(x+$d)*(x+$e))";
$display=maple("printf(MathML[ExportPresentation]($f))");@
qu.1.4.uid=284a3132-5641-4ff4-87b6-8890034cbd97@
qu.1.4.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by partial fractions;
  Sub-Topic=Setup;
@
qu.1.4.question=<p>As a partial fraction decomposition, we set $display equal to</p>@
qu.1.4.answer=1@
qu.1.4.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac    ><mi    mathvariant='italic' >A</mi><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mi    mathvariant='italic' >B</mi><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$d</mi></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mi    mathvariant='italic' >C</mi><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$e</mi></mrow></mfrac></mrow></math>@
qu.1.4.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac    ><mrow><mi    mathvariant='italic' >A</mi><mo    mathvariant='normal'             >&sdot;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >B</mi></mrow><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mrow><mi    mathvariant='italic' >C</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >D</mi></mrow><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$d</mi></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mrow><mi    mathvariant='italic' >E</mi><mo    mathvariant='normal'             >&sdot;</mo><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >F</mi></mrow><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$e</mi></mrow></mfrac></mrow></math>@
qu.1.4.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac    ><mrow><mi    mathvariant='normal' >$c</mi></mrow><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mrow><mi    mathvariant='normal' >$d</mi></mrow><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$d</mi></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mrow><mi    mathvariant='normal' >$e</mi></mrow><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$e</mi></mrow></mfrac></mrow></math>@
qu.1.4.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac    ><mn    mathvariant='normal' >1</mn><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mn    mathvariant='normal' >1</mn><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$d</mi></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mn    mathvariant='normal' >1</mn><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$e</mi></mrow></mfrac></mrow></math>@
qu.1.4.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac    ><mi    mathvariant='normal' >$a</mi><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$c</mi></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mrow><mi    mathvariant='normal' >$b</mi><mo    mathvariant='normal'             >&sdot;</mo><mi    mathvariant='italic' >x</mi></mrow><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$d</mi></mrow></mfrac><mo    mathvariant='normal'             >&plus;</mo><mfrac    ><mrow><mi    mathvariant='italic' >C</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='italic' >D</mi><mo    mathvariant='normal'     lspace='0em' rspace='0em'       >&sdot;</mo><mi    mathvariant='italic' >x</mi></mrow><mrow><mi    mathvariant='italic' >x</mi><mo    mathvariant='normal'             >&plus;</mo><mi    mathvariant='normal' >$e</mi></mrow></mfrac></mrow></math>@
qu.1.4.fixed=@

qu.2.topic=DivideFirst@

qu.2.1.mode=True False@
qu.2.1.name=do we divide?YES@
qu.2.1.comment=@
qu.2.1.editing=useHTML@
qu.2.1.solution=@
qu.2.1.algorithm=$m=rint(2,10);
$n=rint(2,10);
condition:le($n,$m);
$Q="(a*x^$m+b*x^($m-1)+c*x+d)/(A*x^$n+B*x^($n-1)+C*x+D)";
$display=maple("printf(MathML[ExportPresentation]($Q))");@
qu.2.1.uid=f0826519-20f7-4516-8b85-3104d3c081b3@
qu.2.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by partial fractions;
  Sub-Topic=divide first?;
@
qu.2.1.question=<p>True or False: When evaluating the following integral using Partial Fractions, you must first divide the bottom into the top.<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.2.1.answer=1@
qu.2.1.choice.1=True@
qu.2.1.choice.2=False@
qu.2.1.fixed=@

qu.2.2.mode=True False@
qu.2.2.name=do we divide?NO@
qu.2.2.comment=@
qu.2.2.editing=useHTML@
qu.2.2.solution=@
qu.2.2.algorithm=$m=rint(2,10);
$n=rint(2,10);
condition:lt($m,$n);
$Q="(a*x^$m+b*x^($m-1)+c*x+d)/(A*x^$n+B*x^($n-1)+C*x+D)";
$display=maple("printf(MathML[ExportPresentation]($Q))");@
qu.2.2.uid=bf83becb-5c38-4b6b-ad26-f0572f8f34d7@
qu.2.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by partial fractions;
  Sub-Topic=divide first?;
@
qu.2.2.question=<p>True or False: When evaluating the following integral using Partial Fractions, you must first divide the bottom into the top.<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.2.2.answer=2@
qu.2.2.choice.1=True@
qu.2.2.choice.2=False@
qu.2.2.fixed=@

qu.3.topic=(Ax+B)/(x-a)(x-b)@

qu.3.1.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></mrow></mstyle></math>$display<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math>.</p>@
qu.3.1.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.3.1.allow2d=1@
qu.3.1.maple_answer=(ln(abs(x+$a))*($A*$a+$B)+
ln(abs(x-$b))*($A*$b-$B))/($b+$a)+C@
qu.3.1.type=formula@
qu.3.1.mode=Maple@
qu.3.1.name=(Ax+B)/((x+a)(x-b))@
qu.3.1.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo></mrow><mrow><mi></mi></mrow></mstyle></math>$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</p>
<p>$display <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mi>A</mi><mrow><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mi>B</mi><mrow><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced></mrow></mfrac></mrow></mrow></mstyle></math>&nbsp;for all <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mrow><mo lspace='0.0em' rspace='0.0em'>&isin;</mo></mrow><mrow><mi>&reals;</mi></mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$A</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$B</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$b</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>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'>$a</mi></mrow></mfenced></mrow></mstyle></math></p>
<p>Set <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></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 mathvariant='normal'>$CT</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$CB</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><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><mi>A</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$C</p>
<p>Set <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi></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 mathvariant='normal'>$DT</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>B</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$DB</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><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$D</p>
<p>Now</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo></mrow></mstyle></math>$int <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>= $ans</p>@
qu.3.1.editing=useHTML@
qu.3.1.solution=@
qu.3.1.algorithm=$A=rint(1,5);
$B=rint(1,5);
$a=rint(1,5);
$b=rint(1,5);
condition:eq(gcd($A,$B),1);
condition:ne($b,$B);
$Q="($A*x-$B)/((x+$a)*(x-$b))";
$CT=-$A*($a)-$B;
$CB=-($a)-($b);
$DT=$A*($b)-$B;
$DB=($a)+($b);
$M=maple("
MathML[ExportPresentation]($Q),
MathML[ExportPresentation](($CT)/($CB)),
MathML[ExportPresentation](($DT)/($DB)),
MathML[ExportPresentation](($CT)/($CB)*(1/(x+$a))+($DT)/($DB)*(1/(x-$b))),
MathML[ExportPresentation]( ($CT)/($CB)*ln(abs(x+$a))+($DT)/($DB)*ln(abs(x-$b))+C )
");
$display=switch(0,$M);
$C=switch(1,$M);
$D=switch(2,$M);
$int=switch(3,$M);
$ans=switch(4,$M);@
qu.3.1.uid=fb75b466-3c11-40a4-9996-e3a78ddbd87b@
qu.3.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by partial fraction;
  Sub-Topic=Linear factors;
@

qu.3.2.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></mrow></mstyle></math>$display<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math>.</p>@
qu.3.2.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.3.2.allow2d=1@
qu.3.2.maple_answer=($A*$b-$B)/(($b+$a)*($c+$b))*ln(abs(x-$b))+(-$A*$a-$B)/( ($b+$a)*(-$c+$a))*ln(abs(x+$a))+($A*$c+$B)/((-$c+$a)* ($c+$b))*ln(abs(x+$c))+C@
qu.3.2.type=formula@
qu.3.2.mode=Maple@
qu.3.2.name=(Ax+B)/((x+a)(x-b)(x+c))@
qu.3.2.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo></mrow><mrow><mi></mi></mrow></mstyle></math>$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</p>
<p>$display&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mi>A</mi><mrow><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mi>B</mi><mrow><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced></mrow></mfrac></mrow></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mi>C</mi><mrow><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$c</mi></mrow></mfenced></mrow></mfrac></mrow></mrow></mstyle></math>&nbsp;for all <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mrow><mo lspace='0.0em' rspace='0.0em'>&isin;</mo></mrow><mrow><mi>&reals;</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 mathvariant='normal'>$A</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$B</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><mi>A</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'>$b</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$c</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>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'>$a</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$c</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</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'>$a</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced></mrow></mstyle></math></p>
<p>Set <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mstyle></math>:&nbsp;&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$CT</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$CB1</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$CB2</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><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><mi>A</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$C</p>
<p>Set <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi></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 mathvariant='normal'>$DT</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>B</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$DB1</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$DB2</mi></mrow></mfenced></mrow></mstyle></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$D</p>
<p>Set <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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'>&equals;</mo><mi mathvariant='normal'>$c</mi></mrow></mstyle></math>: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$ET</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>C</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$EB1</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$EB2</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><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><mi>C</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$E</p>
<p>Now</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo></mrow></mstyle></math>$int <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>= $ans <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>D</mi></mrow></mstyle></math></p>@
qu.3.2.editing=useHTML@
qu.3.2.solution=@
qu.3.2.algorithm=$A=rint(2,6,2);
$B=rint(1,5);
$a=rint(1,5);
$b=rint(1,5);
$c=rint(1,5);
condition:ne($a,$c);
condition:eq(gcd($A,$B),1);
condition:ne($b,$B);
$Q="($A*x-$B)/((x+$a)*(x-$b)*(x+$c))";
$CT=-$A*($a)-$B;
$DT=$A*($b)-$B;
$ET=-$A*($c)-$B;
$CB1=-($a)-($b);
$CB2=-($a)+($c);
$DB1=($a)+($b);
$DB2=($b)+($c);
$EB1=($a)-($c);
$EB2=-($c)-($b);
$M=maple("
MathML[ExportPresentation]($Q),
MathML[ExportPresentation](($CT)/(($CB1)*($CB2))),
MathML[ExportPresentation](($DT)/(($DB1)*($DB2))),
MathML[ExportPresentation](($ET)/(($EB1)*($EB2))),
MathML[ExportPresentation](($CT)/(($CB1)*($CB2))*(1/(x+$a))+($DT)/(($DB1)*($DB2))*(1/(x-$b))+($ET)/(($EB1)*($EB2))*(1/(x+$c))),
MathML[ExportPresentation](($CT)/(($CB1)*($CB2))*ln(abs(x+$a))+($DT)/(($DB1)*($DB2))*ln(abs(x-$b))+($ET)/(($EB1)*($EB2))*ln(abs(x+$c)))");
$display=switch(0,$M);
$C=switch(1,$M);
$D=switch(2,$M);
$E=switch(3,$M);
$int=switch(4,$M);
$ans=switch(5,$M);@
qu.3.2.uid=7df48e2f-0ff9-435b-ab5c-8eb8e6422ac1@
qu.3.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by partial fraction;
  Sub-Topic=Linear factors;
@

qu.4.topic=(Ax^2+B*x+c)/((x+a)(x-b))@

qu.4.1.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></mrow></mstyle></math>$display<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math>.</p>@
qu.4.1.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.4.1.allow2d=1@
qu.4.1.maple_answer=$A*x+ln(abs(x+$a))*(-$B*$a-$A*$a^2-$C)/($a+$b)+ln(abs(x-$b))*(-$B*$b+$C+$A*$b^2)/($a+$b)+C@
qu.4.1.type=formula@
qu.4.1.mode=Maple@
qu.4.1.name=(Ax^2+Bx+C)/((x+a)(x-b))@
qu.4.1.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='0.0em' 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></mrow></mstyle></math>$display <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math></p>
<p>After performing long division,</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo></mrow></mstyle></math>$divided <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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</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><mrow><mi mathvariant='normal'>$D</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$E</mi></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mi>A</mi><mrow><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mi>B</mi><mrow><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced></mrow></mfrac></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 mathvariant='normal'>$D</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$E</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>A</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'>$b</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>B</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow></mstyle></math></p>
<p>Set <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mstyle></math>: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$FT</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$FB</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><mi>A</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$F</p>
<p>Set <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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'>&equals;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$GT</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>B</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$GB</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$G</p>
<p>Now,</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></mrow></mstyle></math>$int <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$ans <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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'>&plus;</mo><mi>C</mi></mrow></mstyle></math></p>@
qu.4.1.editing=useHTML@
qu.4.1.solution=@
qu.4.1.algorithm=$A=1;
$B=rint(1,5);
$C=rint(1,5);
$a=rint(1,5);
$b=rint(1,5);
$D=(-$B-$a+$b);
$E=($C+$a*$b);
condition:lt($B^2-4*$A*$C,0);
condition:not(eq($D,0));
condition:not(eq($E,0));
$Q="($A*x^2-$B*x+$C)/((x+$a)*(x-$b))";
$FT=-$D*$a+$E;
$FB=-($a)-($b);
$GT=$D*$b+$E;
$GB=$b+$a;
$M=maple("
MathML[ExportPresentation]($Q),
MathML[ExportPresentation](1+ ((-$B-$a+$b)*x+($C+$a*$b))/((x+$a)*(x-$b))),
MathML[ExportPresentation](($FT)/($FB)),
MathML[ExportPresentation](($GT)/($GB)),
MathML[ExportPresentation](1+($FT)/($FB)*(1/(x+$a))+($GT)/($GB)*(1/($x-$b)) ),
MathML[ExportPresentation](x+($FT)/($FB)*ln(abs(x+$a))+($GT)/($GB)*ln(abs(x-$b)) )
");
$display=switch(0,$M);
$divided=switch(1,$M);
$F=switch(2,$M);
$G=switch(3,$M);
$int=switch(4,$M);
$ans=switch(5,$M);@
qu.4.1.uid=f2cc5b79-70cf-4bc4-89b5-168efea2ecbf@
qu.4.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Hard;
  Course=Introduction to Calculus II;
  Topic=Integration by partial fractions;
  Sub-Topic=divide first and integrate;
@

qu.5.topic=(Ax+B)/((x^2+a)(x-b))@

qu.5.1.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>$display<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math>.</p>@
qu.5.1.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.5.1.allow2d=1@
qu.5.1.maple_answer=(arctan(x/(sqrt($a)))*($A*$a*$b-$B*$b))/(($b^(2)+$a)*sqrt($a))+(ln(x^(2)+$a)*($A*$a-$B))/(2*($b^(2)+$a))+(ln(abs(x-$b))*($A*$b^(2)+$B))/(($b^(2)+$a))+C@
qu.5.1.type=formula@
qu.5.1.mode=Maple@
qu.5.1.name=(Ax^2+B)/((x^2+a)(x-b))@
qu.5.1.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></mrow></mstyle></math>$display <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math></p>
<p>Let</p>
<p>$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><mfrac><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>B</mi></mrow><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mrow><mi>C</mi></mrow><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfrac></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><mi mathvariant='normal'>$A</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$B</mi></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>B</mi></mrow></mfenced><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'>$b</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow><mrow><mi mathvariant='normal'></mi></mrow></mstyle></math></p>
<p>Set <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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'>&equals;</mo><mi mathvariant='normal'>$b</mi></mrow></mstyle></math>: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$ET</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>C</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$EB</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><mi>C</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$E</p>
<p>Coefficient 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><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo></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 mathvariant='normal'>$A</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>A</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>A</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$C</p>
<p>Coefficient 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><msup><mi>x</mi><mrow><mn>1</mn></mrow></msup></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><mn>0</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>A</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>B</mi></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$D</p>
<p>Therefore,</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></mrow></mstyle></math>$int <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>= $ans</p>@
qu.5.1.editing=useHTML@
qu.5.1.solution=@
qu.5.1.algorithm=$A=rint(2)+1;
$B=rint(1,5);
$a=rint(1,5);
$b=rint(1,5);
condition:eq(gcd($A,$B),1);
condition:ne($a,$B);
$Q="($A*x^2+$B)/((x^2+$a)*(x-$b))";
$ET=$A*($b)*($b)+$B;
$EB=($b)^2+$a;
$M=maple("
MathML[ExportPresentation]($Q),
MathML[ExportPresentation](($ET)/($EB)),
MathML[ExportPresentation]($A-($ET)/($EB)),
MathML[ExportPresentation]($b*($A-($ET)/($EB))),
MathML[ExportPresentation]( ( ($A-($ET)/($EB))*x+($b*($A-($ET)/($EB))) )/(x^2+$a)+(($ET)/($EB))*(1/(x-$b)) ),
MathML[ExportPresentation]((arctan(x/(sqrt($a)))*($A*$a*$b-$B*$b))/(($b^(2)+$a)*sqrt($a))+(ln(x^(2)+$a)*($A*$a-$B))/(2*($b^(2)+$a))+(ln(abs(x-$b))*($A*$b^(2)+$B))/(($b^(2)+$a))+C)
");
$display=switch(0,$M);
$E=switch(1,$M);
$C=switch(2,$M);
$D=switch(3,$M);
$int=switch(4,$M);
$ans=switch(5,$M);@
qu.5.1.uid=1988b0c5-9249-4e6c-828a-df83264f66d6@
qu.5.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Hard;
  Course=Integration by partial fractions;
  Sub-Topic=Quadratic factors;
  Topic=Integration by partial fractions;
@

qu.5.2.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>$display<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math>.</p>@
qu.5.2.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.5.2.allow2d=1@
qu.5.2.maple_answer=(($a*$A+$B*$b)*arctan(x/(sqrt($a))))/(($b^(2)+$a)*sqrt($a))+(ln(x^(2)+$a)*(-$A*$b+$B))/(2*($b^(2)+$a))+(ln(abs(x-$b))*($A*$b-$B))/($b^(2)+$a)+C@
qu.5.2.type=formula@
qu.5.2.mode=Maple@
qu.5.2.name=(Ax-B)/((x^2+a)(x-b))@
qu.5.2.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo></mrow></mstyle></math>$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</p>
<p>$display = <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>B</mi></mrow><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mrow><mi>C</mi></mrow><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfrac></mrow></mrow><mrow><mi mathvariant='normal'></mi></mrow><mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$A</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mrow></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$B</mi></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>B</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow><mrow></mrow></mstyle></math></p>
<p>Set <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$b</mi></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 mathvariant='normal'>$ET</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>C</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$EB</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><mi>C</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$E</p>
<p>Coefficient of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo></mrow></mstyle></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>0</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>A</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi><mo lspace='0.0em' rspace='0.0em'>&srarr;</mo><mi>A</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$C</p>
<p>Coefficient 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><msup><mi>x</mi><mrow><mn>1</mn></mrow></msup></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 mathvariant='normal'>$A</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>A</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>B</mi></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$D</p>
<p>Therefore,</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo></mrow></mstyle></math>$int <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>= $ans</p>@
qu.5.2.editing=useHTML@
qu.5.2.solution=@
qu.5.2.algorithm=$A=rint(4)+1;
$B=rint(1,5);
$a=rint(1,5);
$b=rint(1,5);
condition:eq(gcd($A,$B),1);
condition:ne($b,$B);
$Q="($A*x-$B)/((x^2+$a)*(x-$b))";
$ET=$A*($b)-$B;
$EB=($b)^2+$a;
$M=maple("
MathML[ExportPresentation]($Q),
MathML[ExportPresentation](($ET)/($EB)),
MathML[ExportPresentation](-($ET)/($EB)),
MathML[ExportPresentation]($A-$b*($ET)/($EB)),
MathML[ExportPresentation]( ( (-($ET)/($EB))*x+($A-$b*($ET)/($EB)) )/(x^2+$a)+(($ET)/($EB))*(1/(x-$b)) ),
MathML[ExportPresentation]((($a*$A+$B*$b)*arctan(x/(sqrt($a))))/(($b^(2)+$a)*sqrt($a))+(ln(x^(2)+$a)*(-$A*$b+$B))/(2*($b^(2)+$a))+(ln(abs(x-$b))*($A*$b-$B))/($b^(2)+$a)+C)
");
$display=switch(0,$M);
$E=switch(1,$M);
$C=switch(2,$M);
$D=switch(3,$M);
$int=switch(4,$M);
$ans=switch(5,$M);@
qu.5.2.uid=1fa52558-70a4-43da-87fd-ff08d4aa4ed1@
qu.5.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Hard;
  Course=Integration by partial fractions;
  Sub-Topic=Quadratic factors;
  Topic=Integration by partial fractions;
@

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)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.5.3.allow2d=1@
qu.5.3.maple_answer=(arctan(x+1)*($B+2*$A+$b*$B+$A*$b))/($b^(2)+2*$b+2)+(ln(abs(x-$b))*($A*$b-$B))/($b^(2)+2*$b+2)+(ln(x^(2)+2*x+2)*($B-$A*$b))/(2*($b^(2)+2*$b+2))+C@
qu.5.3.type=formula@
qu.5.3.mode=Maple@
qu.5.3.name=(Ax+B)/((x^2+2*x+2)(x-b)@
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><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo></mrow></mstyle></math>$display <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math></p>
<p>Let</p>
<p>$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><mfrac><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>B</mi></mrow><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>2</mn></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mrow><mfrac><mrow><mi>C</mi></mrow><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfrac></mrow></mrow><mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$A</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$B</mi></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='(' close=')' separators=','><mrow><mi>A</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>B</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>2</mn></mrow></mfenced></mrow><mrow></mrow></mstyle></math></p>
<p>Set <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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'>&equals;</mo><mi mathvariant='normal'>$b</mi></mrow></mstyle></math>: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$ET</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>C</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$EB</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><mi>C</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$E</p>
<p>Coefficient 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><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&colon;</mo></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><mn>0</mn><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>A</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><mi>A</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>C</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$C</p>
<p>Coefficient of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>x</mi><mrow><mn>0</mn></mrow></msup></mrow></mstyle></math>(constant): <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>B</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rarr;</mo><mi>B</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></mstyle></math>$D</p>
<p>Therefore,</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></mrow></mstyle></math>$int <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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>= $ans</p>@
qu.5.3.editing=useHTML@
qu.5.3.solution=@
qu.5.3.algorithm=$A=rint(2)+1;
$B=rint(1,5);
$a=rint(1,5);
$b=rint(1,5);
$c=rint(1,5);
condition:ne($a,$c);
condition:eq(gcd($A,$B),1);
condition:ne($b,$B);
$ET=$A*($b)-$B;
$EB=($b)^2+2*$b+2;
$Q="($A*x-$B)/((x^2+2*x+2)*(x-$b))";
$M=maple("
MathML[ExportPresentation]($Q),
MathML[ExportPresentation](($ET)/($EB)),
MathML[ExportPresentation](-($ET)/($EB)),
MathML[ExportPresentation](($B+2*($ET)/($EB))/($b)),
MathML[ExportPresentation]((-($ET)/($EB)*x+($B+2*($ET)/($EB))/($b))/(x^2+2*x+2)+(($ET)/($EB))*(1/(x-$b)) ),
MathML[ExportPresentation]((arctan(x+1)*($B+2*$A+$b*$B+$A*$b))/($b^(2)+2*$b+2)+(ln(abs(x-$b))*($A*$b-$B))/($b^(2)+2*$b+2)+(ln(x^(2)+2*x+2)*($B-$A*$b))/(2*($b^(2)+2*$b+2))+C)
");
$display=switch(0,$M);
$E=switch(1,$M);
$C=switch(2,$M);
$D=switch(3,$M);
$int=switch(4,$M);
$ans=switch(5,$M);@
qu.5.3.uid=ec132c57-bc7f-476b-ab54-5d3b9dc74670@
qu.5.3.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Hard;
  Course=Integration by partial fractions;
  Sub-Topic=Quadratic factors;
  Topic=Integration by partial fractions;
@

