qu.1.topic=x approaches -inf approaches +inf@

qu.1.1.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><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&infin;</mi></mrow></mrow><mrow><mi>&infin;</mi></mrow></munderover></mrow></mstyle></math>$display<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math>. Your answer should be a finite number, infinity, -infinity, or enter DNE if it does not exist. Start this problem by rewriting the improper integral as a limit of a proper integral.</p>
<p>Hint: Remember BABA and the Chain Rule in Reverse:&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mfrac><mrow><mi>u</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo></mrow><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi>u</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>arctan</mi><mfenced open='(' close=')' separators=','><mrow><mi>u</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></mstyle></math></p>@
qu.1.1.maple=evalb(($ANSWER)=($RESPONSE));@
qu.1.1.allow2d=1@
qu.1.1.maple_answer=int(exp($b*x)/($a^2+exp($b*x)^2),x=-infinity..infinity);@
qu.1.1.type=formula@
qu.1.1.mode=Maple@
qu.1.1.name=int(exp(x)/(1+(exp(x)^2),-inf,inf)@
qu.1.1.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><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow><mrow><mi>&infin;</mi></mrow></munderover></mrow></mstyle></math>$display <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>m</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mi>&infin;</mi></mrow></munder><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>n</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></munder><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>n</mi></mrow><mrow><mi>m</mi></mrow></munderover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>$display <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>m</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mrow><mi>&infin;</mi></mrow></mrow></munder><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>n</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></munder><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$b</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>arctan</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><msup><mi>e</mi><mrow><mi>x</mi></mrow></msup></mrow></mfenced><msubsup><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mrow><mi>n</mi></mrow><mrow><mi>m</mi></mrow></msubsup><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$ab</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>m</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mi>&infin;</mi></mrow></munder><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>n</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mrow><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mrow></munder><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>arctan</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>e</mi><mrow><mi>m</mi></mrow></msup></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>arctan</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>e</mi><mrow><mi>n</mi></mrow></msup></mrow></mfenced></mrow></mfenced></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$ab</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0</mn></mrow></mfenced></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><mfrac><mi>&pi;</mi><mrow><mi mathvariant='normal'>$ab2</mi></mrow></mfrac></mrow></mrow></mstyle></math></p>@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=$a=rint(1,5);
$b=rint(1,5);
$ab=$a*$b;
$ab2=$ab*2;
condition:eq(gcd($a,$b),1);
$Q="exp($b*x)/($a^2+exp($b*x)^2)";
$display=maple("printf(MathML[ExportPresentation]($Q))");@
qu.1.1.uid=8f18cdd0-c398-445f-af45-a5b6dc1cb823@
qu.1.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Medium;
  Course=Introduction to Calculus II;
  Topic=Improper integrals;
  Sub-Topic=arctan;
@

qu.1.2.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><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&infin;</mi></mrow></mrow><mrow><mi>&infin;</mi></mrow></munderover></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>. Your answer should be a finite number, infinity, -infinity, or enter DNE if it does not exist. Start this problem by rewriting the improper integral as a limit of a proper integral.</p>
<p>Hint: Think arctan and remember BABA! Remember <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>t</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></munder><mi mathvariant='normal'>arctan</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><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></mrow><mrow><mfrac><mrow><mi>&pi;</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mstyle></math>and <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>t</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></munder><mi mathvariant='normal'>arctan</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.0em'>..</mo><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math></p>@
qu.1.2.maple=evalb(($ANSWER)=($RESPONSE));@
qu.1.2.allow2d=1@
qu.1.2.maple_answer=int(1/($a^2+$b^2*x^2),x=-infinity..infinity);@
qu.1.2.type=formula@
qu.1.2.mode=Maple@
qu.1.2.name=int(1/(a^2+b^x^2),-inf,inf)@
qu.1.2.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow><mrow><mi>&infin;</mi></mrow></munderover></mrow></mstyle></math>$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>= <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>m</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mrow><mi>&infin;</mi></mrow></mrow></munder><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>n</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mrow><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mrow></munder><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>n</mi></mrow><mrow><mi>m</mi></mrow></munderover></mrow></mstyle></math>$display <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math></p>
<p>= <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>m</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mrow><mi>&infin;</mi></mrow></mrow></munder><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>n</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mrow><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mrow></munder><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>$b</mi></mrow></mfrac><mi mathvariant='normal'>arctan</mi><msub><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac></mrow></mfenced><mi></mi></msub><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msubsup><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mrow><msup><mi>n</mi><mi></mi></msup></mrow><mrow><mi>m</mi></mrow></msubsup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfenced></mrow></mstyle></math></p>
<p>=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$ab</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>m</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mrow><mi>&infin;</mi></mrow></mrow></munder><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>n</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mrow><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mrow></munder><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>arctan</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>m</mi></mrow><mrow><mi mathvariant='normal'>$b</mi></mrow></mfrac></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>arctan</mi><mfenced open='(' close=')' separators=','><mrow><mfrac><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>n</mi></mrow><mrow><mi mathvariant='normal'>$b</mi></mrow></mfrac></mrow></mfenced></mrow></mfenced></mrow></mstyle></math></p>
<p>= <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$ab</mi></mrow></mfrac><mfenced open='(' close=')' separators=','><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi>&pi;</mi><mrow><mn>2</mn></mrow></mfrac></mrow></mrow></mfenced></mrow></mfenced></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mfrac><mi>&pi;</mi><mrow><mi mathvariant='normal'>$ab</mi></mrow></mfrac></mrow></mrow></mstyle></math></p>@
qu.1.2.editing=useHTML@
qu.1.2.hint.1=Think arctan for the integral!@
qu.1.2.hint.2=Remember BABA.@
qu.1.2.solution=@
qu.1.2.algorithm=$a=rint(1,5);
$b=rint(1,5);
$ab=$a*$b;
condition:eq(gcd($a,$b),1);
$Q="1/($a^2+$b^2*x^2)";
$display=maple("printf(MathML[ExportPresentation]($Q))");@
qu.1.2.uid=cdc74090-9d85-4401-952d-c2cc4ac2b60d@
qu.1.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Medium;
  Course=Introduction to Calculus II;
  Topic=Improper integrals;
  Sub-Topic=arctan;
@

qu.2.topic=x approaches a+/-Diverge@

qu.2.1.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><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi mathvariant='normal'>$b</mi></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></munderover></mrow></mstyle></math>$display<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math>. Your answer should be a finite number, infinity, -infinity, or enter DNE if it does not exist. Start this problem by rewriting the improper integral as a limit of a proper integral.</p>@
qu.2.1.maple=evalb(($ANSWER)=($RESPONSE));@
qu.2.1.allow2d=1@
qu.2.1.maple_answer=int($Q,x=$b..$a);@
qu.2.1.type=formula@
qu.2.1.mode=Maple@
qu.2.1.name=limx approaches a-1/(x-a)^n,n greater-than 1@
qu.2.1.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><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi mathvariant='normal'>$b</mi></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></munderover><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math>$display&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>a</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><msup><mi  mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></msup></mrow></munder><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi mathvariant='normal'>$b</mi></mrow><mrow><mi>a</mi></mrow></munderover><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$n1</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>a</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></msup></mrow></munder><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$n1</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msubsup><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mrow><mi mathvariant='normal'>$b</mi></mrow><mrow><mi>a</mi></mrow></msubsup></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$n1</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>a</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msup></mrow></munder><mfenced open='(' close=')' separators=','><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><mi>a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfrac></mrow></mfenced><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$b</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfrac></mrow></mfenced><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup></mrow></mfenced></mrow><mrow></mrow></mstyle></math></p>
<p>= $ANS</p>@
qu.2.1.editing=useHTML@
qu.2.1.solution=@
qu.2.1.algorithm=$a=rint(1,10);
$b=$a-1;
$n=rint(2,9);
$n1=$n-1;
$Q="1/(x-$a)^($n)";
$display=maple("printf(MathML[ExportPresentation]($Q))");
$ans=switch($n1-1, "infinity","-infinity","infinity","-infinity","infinity","-infinity","infinity");
$ANS=mathml("$ans");@
qu.2.1.uid=632f0f7a-5230-40ef-bcd4-39fdd9b79a71@
qu.2.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Medium;
  Course=Introduction to Calculus II;
  Topic=Improper integrals;
  Sub-Topic=to a left;
@

qu.2.2.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'>&#32;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi mathvariant='normal'>$a</mi></mrow><mrow><mi mathvariant='normal'>$c</mi></mrow></munderover></mrow></mstyle></math>$display<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math>. Your answer should be a finite number, infinity, -infinity, or enter DNE if it does not exist. Start this problem by rewriting the improper integral as a limit of a proper integral.</p>@
qu.2.2.maple=evalb(($ANSWER)=($RESPONSE));@
qu.2.2.allow2d=1@
qu.2.2.maple_answer=int($Q,x=$a..$c);@
qu.2.2.type=formula@
qu.2.2.mode=Maple@
qu.2.2.name=limx approaches a+1/(x-a)^n,n greater-than 1@
qu.2.2.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><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi mathvariant='normal'>$a</mi></mrow><mrow><mi mathvariant='normal'>$c</mi></mrow></munderover></mrow></mstyle></math>$display <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>a</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow></msup></mrow></munder><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>a</mi></mrow><mrow><mi mathvariant='normal'>$c</mi></mrow></munderover><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$n1</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>a</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow></msup></mrow></munder><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$n1</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msubsup><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mrow><mi>a</mi></mrow><mrow><mi mathvariant='normal'>$c</mi></mrow></msubsup></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$n1</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>a</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow></msup></mrow></munder><mfenced open='(' close=')' separators=','><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$c</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfrac></mrow></mfenced><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><mi>a</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfrac></mrow></mfenced><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup></mrow></mfenced></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$n1</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></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>&infin;</mi></mrow></mstyle></math></p>
<p>&nbsp;</p>@
qu.2.2.editing=useHTML@
qu.2.2.solution=@
qu.2.2.algorithm=$a=rint(1,10);
$c=$a+1;
$z=rint(11);
$n=switch($z, "5/4","3/2","2","5/2","7/3","3","7/2","4","5","6","8");
$n1=switch($z, "1/4", "1/2", "1", "3/2", "4/3", "2", "5/2", "3", "4", "5", "7");
$Q="1/(x-$a)^($n)";
$display=maple("printf(MathML[ExportPresentation]($Q))");@
qu.2.2.uid=f6b61d52-3bf2-490c-b771-e107cee02c3b@
qu.2.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Improper integrals;
  Sub-Topic=to a right;
@

qu.3.topic=x approaches a approaches inf@

qu.3.1.question=<p>Evaluate the improper integral<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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'>&#32;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi mathvariant='normal'>$c</mi></mrow><mrow><mi>&infin;</mi></mrow></munderover></mrow></mstyle></math>$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>. Your answer should be a finite number, infinity, -infinity, or enter DNE if it does not exist. Start this problem by rewriting the improper integral as the limit of a proper integral.</p>@
qu.3.1.maple=evalb(($ANSWER)=($RESPONSE));@
qu.3.1.allow2d=1@
qu.3.1.maple_answer=$n2@
qu.3.1.type=formula@
qu.3.1.mode=Maple@
qu.3.1.name=limx approaches a approaches inf converge@
qu.3.1.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi mathvariant='normal'>$c</mi></mrow><mrow><mi>&infin;</mi></mrow></munderover></mrow></mstyle></math>$display&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math></p>
<p>= <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>n</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mrow><mi>&infin;</mi></mrow></mrow></munder><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi mathvariant='normal'>$c</mi></mrow><mrow><mi>n</mi></mrow></munderover><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$n1</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>n</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mrow><mi>&infin;</mi></mrow></mrow></munder><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$n1</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msubsup><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mrow><mi mathvariant='normal'>$c</mi></mrow><mrow><mi>n</mi></mrow></msubsup></mrow><mrow><mi mathvariant='normal'></mi></mrow></mstyle></math></p>
<p>= <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$n2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>n</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mrow><mi>&infin;</mi></mrow></mrow></munder><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfrac></mrow></mfenced><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn></mrow></mfenced><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup></mrow></mfenced></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$n2</mi></mrow></mstyle></math></p>@
qu.3.1.editing=useHTML@
qu.3.1.solution=@
qu.3.1.algorithm=$a=rint(1,10);
$c=$a+1;
$z=rint(10);
$n=switch($z, "3", "4", "5", "3/2", "5/3", "7/3", "6", "10/9", "6/5", "8");
$n1=switch($z, "2", "3", "4", "1/2", "2/3", "4/3", "5", "1/9", "1/5", "7");
$n2=switch($z,  "1/2", "1/3", "1/4", "2", "3/2", "3/4", "1/5", "9", "5", "1/7");
$Q="1/(x-$a)^($n)";
$display=maple("printf(MathML[ExportPresentation]($Q))");@
qu.3.1.uid=43748f68-bfab-45ef-ac52-3800e6c9c14c@
qu.3.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Medium;
  Course=Introduction to Calculus II;
  Topic=Improper integrals;
  Sub-Topic=to infinity;
@

qu.3.2.question=<p>Evaluate the improper integral<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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'>&#32;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi mathvariant='normal'>$c</mi></mrow><mrow><mi>&infin;</mi></mrow></munderover></mrow></mstyle></math>$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>. Your answer should be a finite number, infinity, -infinity, or enter DNE if it does not exist. Start this problem by rewriting the improper integral as the limit of a proper integral.</p>@
qu.3.2.maple=evalb(($ANSWER)=($RESPONSE));@
qu.3.2.allow2d=1@
qu.3.2.maple_answer=infinity@
qu.3.2.type=formula@
qu.3.2.mode=Maple@
qu.3.2.name=limx approaches a approaches inf diverge@
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><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>$c</mi></mrow><mrow><mi>&infin;</mi></mrow></munderover></mrow></mstyle></math>$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><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><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>n</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mrow><mi>&infin;</mi></mrow></mrow></munder><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi mathvariant='normal'>$c</mi></mrow><mrow><mi>n</mi></mrow></munderover><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$n1</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>n</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mrow><mi>&infin;</mi></mrow></mrow></munder><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msubsup><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mrow><mi mathvariant='normal'>$c</mi></mrow><mrow><mi>n</mi></mrow></msubsup></mrow></mstyle></math></p>
<p>= <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$n2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>n</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mrow><mi>&infin;</mi></mrow></mrow></munder><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn></mrow></mfenced><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup></mrow></mfenced></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mi>&infin;</mi></mrow></mrow></mstyle></math></p>@
qu.3.2.editing=useHTML@
qu.3.2.solution=@
qu.3.2.algorithm=$a=rint(1,10);
$c=$a+1;
$z=rint(10);
$n=switch($z, "1/5", "3/7", "8/9", "2/3", "1/6", "3/8", "4/9", "3/5", "3/4", "4/5");
$n1=switch($z, "4/5", "4/7", "1/9", "1/3", "5/6", "5/8", "5/9", "2/5", "1/4", "1/5");
$n2=switch($z, "5/4", "7/4", "9", "3", "6/5", "8/5", "9/5", "5/2", "4", "5");

$Q="1/(x-$a)^($n)";
$display=maple("printf(MathML[ExportPresentation]($Q))");@
qu.3.2.uid=85b5a887-22ff-405f-a2d7-a15f8c25e403@
qu.3.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Medium;
  Course=Introduction to Calculus II;
  Topic=Improper integrals;
  Sub-Topic=to infinity;
@

qu.4.topic=x approaches aConverge@

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'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi mathvariant='normal'>$b</mi></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></munderover></mrow><mrow><mfrac><mn>1</mn><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mfrac><mi mathvariant='normal'>$m</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac></mrow></msup></mrow></mfrac><mi>dx</mi><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mi mathvariant='normal'></mi></mrow></mstyle></math>. Your answer should be a finite number, infinity, -infinity, or enter DNE if it does not exist. Start this problem by rewriting the improper integral as a limit of a proper integral.</p>@
qu.4.1.maple=evalb(($ANSWER)=($RESPONSE));@
qu.4.1.allow2d=1@
qu.4.1.maple_answer=int(1/surd((x-$a)^$m,$n),x=$b..$a);@
qu.4.1.type=formula@
qu.4.1.mode=Maple@
qu.4.1.name=limx approaches a- 1/(x-a)^(m/n),m/n less-than 1@
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><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi mathvariant='normal'>$b</mi></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></munderover><mfrac><mn>1</mn><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mfrac><mi mathvariant='normal'>$m</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac></mrow></msup></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow><mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>m</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></msup></mrow></munder><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi mathvariant='normal'>$b</mi></mrow><mrow><mi>m</mi></mrow></munderover><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi mathvariant='normal'>$m</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac></mrow></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$n2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>m</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></msup></mrow></munder><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msubsup><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mrow><mi mathvariant='normal'>$b</mi></mrow><mrow><mi>m</mi></mrow></msubsup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$n2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>m</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></msup></mrow></munder><mfenced open='(' close=')' separators=','><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>m</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup></mrow></mfenced></mrow></mstyle></math></p>
<p>= <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$n2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mn>0</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup></mrow></mfenced></mrow></mstyle></math></p>
<p>= <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$n3</mi></mrow></mstyle></math></p>@
qu.4.1.editing=useHTML@
qu.4.1.solution=@
qu.4.1.algorithm=$a=rint(1,10);
$b=$a-1;
$c=$a+1;
$m=rint(1,9);
$n=rint(3,11,2);
condition:eq(gcd($m,$n),1);
condition:lt($m,$n);
$Q="1/(x-$a)^($m/$n)";
$M=maple("
a:= (-1)^(-$m+$n):
ans:=(-a*(1/(-$m/$n+1))):
convert(-$m/$n+1,string),
convert((1/(-$m/$n+1)),string), 
convert(ans,string)
");
$n1=switch(0,$M);
$n2=switch(1,$M);
$n3=switch(3,$M);@
qu.4.1.uid=84e09014-9431-43d1-83b2-71595e69c35b@
qu.4.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Medium;
  Course=Introduction to Calculus II;
  Topic=Improper integrals;
  Sub-Topic=to a from left;
@

qu.4.2.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'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi mathvariant='normal'>$b</mi></mrow><mrow><mi mathvariant='normal'>$c</mi></mrow></munderover></mrow><mrow><mfrac><mn>1</mn><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mfrac><mi mathvariant='normal'>$m</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac></mrow></msup></mrow></mfrac><mi>dx</mi><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mi></mi></mrow></mstyle></math>. Your answer should be a finite number, infinity, -infinity, or enter DNE if it does not exist. Start this problem by rewriting the improper integral as a limit of a proper integral.</p>@
qu.4.2.maple=evalb(($ANSWER)=($RESPONSE));@
qu.4.2.allow2d=1@
qu.4.2.maple_answer=int(1/surd((x-$a)^$m,$n),x=$b..$c);@
qu.4.2.type=formula@
qu.4.2.mode=Maple@
qu.4.2.name=limx approaches a1/(x-a)^(m/n),m/n less-than 1@
qu.4.2.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><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi mathvariant='normal'>$b</mi></mrow><mrow><mi mathvariant='normal'>$c</mi></mrow></munderover><mfrac><mn>1</mn><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mfrac><mi mathvariant='normal'>$m</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac></mrow></msup></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>m</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></msup></mrow></munder><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi mathvariant='normal'>$b</mi></mrow><mrow><mi>m</mi></mrow></munderover><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi mathvariant='normal'>$m</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac></mrow></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><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><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>n</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow></msup></mrow></munder><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>n</mi></mrow><mrow><mi mathvariant='normal'>$c</mi></mrow></munderover><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi mathvariant='normal'>$m</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac></mrow></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$n2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>m</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></msup></mrow></munder><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msubsup><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mrow><mi mathvariant='normal'>$b</mi></mrow><mrow><mi>m</mi></mrow></msubsup><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><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>n</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow></msup></mrow></munder><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup><msubsup><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mrow><mi>n</mi></mrow><mrow><mi mathvariant='normal'>$c</mi></mrow></msubsup><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$n2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>m</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></msup></mrow></munder><mfenced open='(' close=')' separators=','><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>m</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>n</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow></msup></mrow></munder><mfenced open='(' close=')' separators=','><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn></mrow></mfenced><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup></mrow></mfenced></mrow></mfenced></mrow><mrow></mrow></mstyle></math></p>
<p>= <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$n2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mn>0</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>0</mn></mrow></mfenced></mrow><mrow></mrow></mstyle></math></p>
<p>= <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>$n3</mi></mrow></mstyle></math></p>@
qu.4.2.editing=useHTML@
qu.4.2.solution=@
qu.4.2.algorithm=$a=rint(1,10);
$b=$a-1;
$c=$a+1;
$m=rint(1,9);
$n=rint(3,11,2);
condition:eq(gcd($m,$n),1);
condition:lt($m,$n);
$Q="1/(x-$a)^($m/$n)";
$M=maple("
a:= (-1)^(-$m+$n)-1:
ans:=(a/(-$m/$n+1)):
convert(-$m/$n+1,string),
convert((1/(-$m/$n+1)),string), 
convert(ans,string)
");
$n1=switch(0,$M);
$n2=switch(1,$M);
$n3=switch(3,$M);@
qu.4.2.uid=1ce55fb1-3b9d-42b9-a967-6f5ccfecb32c@
qu.4.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Medium;
  Course=Introduction to Calculus II;
  Topic=Improper integrals;
  Sub-Topic=to a from left and right;
@

qu.4.3.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'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi mathvariant='normal'>$a</mi></mrow><mrow><mi mathvariant='normal'>$c</mi></mrow></munderover></mrow><mrow><mfrac><mn>1</mn><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mfrac><mi mathvariant='normal'>$m</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac></mrow></msup></mrow></mfrac><mi>dx</mi><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow><mrow><mi></mi></mrow></mstyle></math>. Your answer should be a finite number, infinity, -infinity, or enter DNE if it does not exist. Start this problem by rewriting the improper integral as a limit of a proper integral.</p>@
qu.4.3.maple=evalb(($ANSWER)=($RESPONSE));@
qu.4.3.allow2d=1@
qu.4.3.maple_answer=int(1/surd((x-$a)^$m,$n),x=$a..$c);@
qu.4.3.type=formula@
qu.4.3.mode=Maple@
qu.4.3.name=limx approaches a+   1/(x-a)^(m/n),m/n less-than 1@
qu.4.3.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><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi mathvariant='normal'>$a</mi></mrow><mrow><mi mathvariant='normal'>$c</mi></mrow></munderover><mfrac><mn>1</mn><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mfrac><mi mathvariant='normal'>$m</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac></mrow></msup></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow><mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>n</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow></msup></mrow></munder><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>n</mi></mrow><mrow><mi mathvariant='normal'>$c</mi></mrow></munderover><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mfrac><mi mathvariant='normal'>$m</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac></mrow></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$n2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>n</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow></msup></mrow></munder><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msubsup><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mrow><mi>n</mi></mrow><mrow><mi mathvariant='normal'>$c</mi></mrow></msubsup></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$n2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>n</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow></msup></mrow></munder><mfenced open='(' close=')' separators=','><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn></mrow></mfenced><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup></mrow></mfenced><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow><mrow><mi mathvariant='normal'></mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$n2</mi></mrow></mstyle></math></p>
<p>&nbsp;</p>@
qu.4.3.editing=useHTML@
qu.4.3.solution=@
qu.4.3.algorithm=$a=rint(1,10);
$b=$a-1;
$c=$a+1;
$m=rint(1,9);
$n=rint(3,11,2);
condition:eq(gcd($m,$n),1);
condition:lt($m,$n);
$Q="1/(x-$a)^($m/$n)";
$M=maple("
convert(-$m/$n+1,string),
convert((1/(-$m/$n+1)),string)
");
$n1=switch(0,$M);
$n2=switch(1,$M);@
qu.4.3.uid=2dd119c9-0b7d-44af-9b85-d30cc190831d@
qu.4.3.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Medium;
  Course=Introduction to Calculus II;
  Topic=Improper integrals;
  Sub-Topic=to a from right;
@

qu.5.topic=x approaches aDiverge@

qu.5.1.question=<p>Evaluate the <strong>improper </strong>integral: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi mathvariant='normal'>$b</mi></mrow><mrow><mi mathvariant='normal'>$c</mi></mrow></munderover></mrow></mstyle></math>$display<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math>. Your answer should be a finite number, infinity, -infinity, or enter DNE if it does not exist. Start this problem by rewriting the improper integral as the limit of a proper integral.</p>@
qu.5.1.maple=evalb(($ANSWER)=($RESPONSE));@
qu.5.1.allow2d=1@
qu.5.1.maple_answer=int($Q,x=$b..$c);@
qu.5.1.type=formula@
qu.5.1.mode=Maple@
qu.5.1.name=limx approaches a1/(x+a)^n,n greater-than or equal to 2, even@
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><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi mathvariant='normal'>$b</mi></mrow><mrow><mi mathvariant='normal'>$c</mi></mrow></munderover></mrow><mrow></mrow></mstyle></math>$display<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math></p>
<p>= <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>m</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></msup></mrow></munder><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi mathvariant='normal'>$b</mi></mrow><mrow><mi>m</mi></mrow></munderover><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>n</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow></msup></mrow></munder><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>n</mi></mrow><mrow><mi mathvariant='normal'>$c</mi></mrow></munderover><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow><mrow><mi mathvariant='normal'></mi></mrow><mrow></mrow></mstyle></math></p>
<p>= <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$n1</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>m</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></msup></mrow></munder><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$n1</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msubsup><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mrow><mi mathvariant='normal'>$b</mi></mrow><mrow><mi>m</mi></mrow></msubsup><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><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>n</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow></msup></mrow></munder><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$n1</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msubsup><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mrow><mi>n</mi></mrow><mrow><mi mathvariant='normal'>$c</mi></mrow></msubsup><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow></mstyle></math></p>
<p>= <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$n1</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>m</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></msup></mrow></munder><mfenced open='(' close=')' separators=','><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><mi>m</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfrac></mrow></mfenced><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>n</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow></msup></mrow></munder><msup><mfenced open='(' close=')' separators=','><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn></mrow></mfenced><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mfrac><mn>1</mn><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfrac></mrow></mfenced><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfenced></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$n1</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn mathvariant='italic'>1</mn><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mn mathvariant='italic'>1</mn><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></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>&infin;</mi></mrow></mstyle></math></p>@
qu.5.1.editing=useHTML@
qu.5.1.solution=@
qu.5.1.algorithm=$n=rint(2,10,2);
$n1=$n-1;
$a=rint(1,10);
$b=-$a-1;
$c=-$a+1;
$Q="1/(x+$a)^$n";
$display=maple("printf(MathML[ExportPresentation]($Q))");@
qu.5.1.uid=6f13424f-575b-4371-a444-103b27efb328@
qu.5.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Medium;
  Course=Introduction to Calculus II;
  Topic=Improper integrals;
  Sub-Topic=to a from left and right;
@

qu.5.2.question=<p>Evaluate the <strong>improper </strong>integral: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi mathvariant='normal'>$b</mi></mrow><mrow><mi mathvariant='normal'>$c</mi></mrow></munderover></mrow></mstyle></math>$display<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math>. Your answer should be a finite number, infinity, -infinity, or enter DNE if it does not exist. Start this problem by rewriting the improper integral as the limit of a proper integral.</p>@
qu.5.2.maple=evalb(($ANSWER)=($RESPONSE));@
qu.5.2.allow2d=1@
qu.5.2.maple_answer=int($Q,x=$b..$c);@
qu.5.2.type=formula@
qu.5.2.mode=Maple@
qu.5.2.name=limx approaches a1/(x-a)^n,n greater-than or equal to 2, even@
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><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi mathvariant='normal'>$b</mi></mrow><mrow><mi mathvariant='normal'>$c</mi></mrow></munderover></mrow><mrow></mrow></mstyle></math>$display<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math></p>
<p>= <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>m</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></msup></mrow></munder><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi mathvariant='normal'>$b</mi></mrow><mrow><mi>m</mi></mrow></munderover><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>n</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow></msup></mrow></munder><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>n</mi></mrow><mrow><mi mathvariant='normal'>$c</mi></mrow></munderover><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow><mrow></mrow></mstyle></math></p>
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<p>= <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$n1</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>m</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></msup></mrow></munder><mfenced open='(' close=')' separators=','><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><mi>m</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfrac></mrow></mfenced><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>n</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow></msup></mrow></munder><msup><mfenced open='(' close=')' separators=','><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn></mrow></mfenced><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mfrac><mn>1</mn><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfrac></mrow></mfenced><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfenced></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$n1</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn mathvariant='italic'>1</mn><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mn mathvariant='italic'>1</mn><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p>= <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>&infin;</mi></mrow></mstyle></math></p>@
qu.5.2.editing=useHTML@
qu.5.2.solution=@
qu.5.2.algorithm=$n=rint(2,10,2);
$n1=$n-1;
$a=rint(1,10);
$b=$a-1;
$c=$a+1;
$Q="1/(x-$a)^$n";
$display=maple("printf(MathML[ExportPresentation]($Q))");@
qu.5.2.uid=e0438c75-a9db-4bb4-a9ff-eaa406989144@
qu.5.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Medium;
  Course=Introduction to Calculus II;
  Topic=Improper integrals;
  Sub-Topic=to a from left and right;
@

qu.5.3.question=<p>Evaluate the <strong>improper </strong>integral: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi mathvariant='normal'>$b</mi></mrow><mrow><mi mathvariant='normal'>$c</mi></mrow></munderover></mrow></mstyle></math>$display<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math>. Your answer should be a finite number, infinity, -infinity, or enter DNE if it does not exist. Start this problem by rewriting the improper integral as the limit of a proper integral.</p>@
qu.5.3.maple=evalb(($ANSWER)=($RESPONSE));@
qu.5.3.allow2d=1@
qu.5.3.maple_answer=$ANS@
qu.5.3.type=formula@
qu.5.3.mode=Maple@
qu.5.3.name=limx approaches a1/(x-a)^n,n greater-than or equal to 3,odd@
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><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi mathvariant='normal'>$b</mi></mrow><mrow><mi mathvariant='normal'>$c</mi></mrow></munderover></mrow><mrow></mrow></mstyle></math>$display<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math></p>
<p>= <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>m</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></msup></mrow></munder><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi mathvariant='normal'>$b</mi></mrow><mrow><mi>m</mi></mrow></munderover><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>n</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow></msup></mrow></munder><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>n</mi></mrow><mrow><mi mathvariant='normal'>$c</mi></mrow></munderover><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow><mrow></mrow></mstyle></math></p>
<p>= <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$n1</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>m</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></msup></mrow></munder><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$n1</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msubsup><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mrow><mi mathvariant='normal'>$b</mi></mrow><mrow><mi>m</mi></mrow></msubsup><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><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>n</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow></msup></mrow></munder><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$n1</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msubsup><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mrow><mi>n</mi></mrow><mrow><mi mathvariant='normal'>$c</mi></mrow></msubsup><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow></mstyle></math></p>
<p>= <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$n1</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>m</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></msup></mrow></munder><mfenced open='(' close=')' separators=','><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><mi>m</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfrac></mrow></mfenced><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>n</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow></msup></mrow></munder><msup><mfenced open='(' close=')' separators=','><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn></mrow></mfenced><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mfrac><mn>1</mn><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfrac></mrow></mfenced><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfenced></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$n1</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn mathvariant='italic'>1</mn><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mn mathvariant='italic'>1</mn><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p>Therefore the integral does not exist.</p>@
qu.5.3.editing=useHTML@
qu.5.3.solution=@
qu.5.3.algorithm=$n=rint(3,9,2);
$n1=$n-1;
$a=rint(1,10);
$b=$a-1;
$c=$a+1;
$Q="1/(x-$a)^$n";
$display=maple("printf(MathML[ExportPresentation]($Q))");
$ANS=DNE;@
qu.5.3.uid=64c5c1d0-12eb-4d14-b806-fc600df892e7@
qu.5.3.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Medium;
  Course=Introduction to Calculus II;
  Topic=Improper integrals;
  Sub-Topic=to a from left and right;
@

qu.5.4.question=<p>Evaluate the <strong>improper </strong>integral: <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi mathvariant='normal'>$b</mi></mrow><mrow><mi mathvariant='normal'>$c</mi></mrow></munderover></mrow></mstyle></math>$display<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math>. Your answer should be a finite number, infinity, -infinity, or enter DNE if it does not exist. Start this problem by rewriting the improper integral as the limit of a proper integral.</p>@
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qu.5.4.maple_answer=$ANS@
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qu.5.4.mode=Maple@
qu.5.4.name=limx approaches a1/(x+ a)^n,n greater-than or equal to 3,odd@
qu.5.4.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi mathvariant='normal'>$b</mi></mrow><mrow><mi mathvariant='normal'>$c</mi></mrow></munderover></mrow><mrow></mrow></mstyle></math>$display<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dx</mi></mrow></mstyle></math></p>
<p>= <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>m</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></msup></mrow></munder><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi mathvariant='normal'>$b</mi></mrow><mrow><mi>m</mi></mrow></munderover><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>n</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow></msup></mrow></munder><munderover><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mi>n</mi></mrow><mrow><mi mathvariant='normal'>$c</mi></mrow></munderover><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow><mrow><mi mathvariant='normal'></mi></mrow></mstyle></math></p>
<p>= <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow><mrow><mi mathvariant='normal'>$n1</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>m</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></msup></mrow></munder><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$n1</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msubsup><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mrow><mi mathvariant='normal'>$b</mi></mrow><mrow><mi>m</mi></mrow></msubsup><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><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>n</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow></msup></mrow></munder><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$n1</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msubsup><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mrow><mi>n</mi></mrow><mrow><mi mathvariant='normal'>$c</mi></mrow></msubsup><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow></mstyle></math></p>
<p>= <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$n1</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>m</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow></msup></mrow></munder><mfenced open='(' close=')' separators=','><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>1</mn><mrow><mi>m</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfrac></mrow></mfenced><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><munder><mo lspace='0.0em' rspace='0.1666667em' movablelimits='true'>lim</mo><mrow><mi>n</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rightarrow;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi mathvariant='normal'>$a</mi><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow></msup></mrow></munder><msup><mfenced open='(' close=')' separators=','><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mn>1</mn></mrow></mfenced><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mfrac><mn>1</mn><mrow><mi>n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfrac></mrow></mfenced><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfenced></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$n1</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn mathvariant='italic'>1</mn><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mn mathvariant='italic'>1</mn><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mstyle></math></p>
<p>Therefore the integral does not exist.</p>@
qu.5.4.editing=useHTML@
qu.5.4.solution=@
qu.5.4.algorithm=$n=rint(3,9,2);
$n1=$n-1;
$a=rint(1,10);
$c=-($a-1);
$b=-($a+1);
$Q="1/(x+$a)^$n";
$display=maple("printf(MathML[ExportPresentation]($Q))");
$ANS=DNE;@
qu.5.4.uid=54c7ffca-3c8b-46c8-9a7a-0b840579acb8@
qu.5.4.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Medium;
  Course=Introduction to Calculus II;
  Topic=Improper integrals;
  Sub-Topic=to a from left and right;
@

