qu.1.topic=IbyPtrigEXP@

qu.1.1.question=<p>Find<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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' stretchy='true' largeop='true'>&Integral;</mo><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&period;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo></mrow></mstyle></math></p>@
qu.1.1.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.1.1.allow2d=1@
qu.1.1.maple_answer=int(x*cos($a*x),x)+C;@
qu.1.1.type=formula@
qu.1.1.mode=Maple@
qu.1.1.name=IbyPx*cos($a*x)@
qu.1.1.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow><mrow><mi mathvariant='normal'></mi></mrow></mstyle></math></p>
<p>Let</p>
<p>
<table width="200" cellspacing="1" cellpadding="1" border="1">
    <tbody>
        <tr>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>x</mi></mrow></mstyle></math></td>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced><mi>dx</mi></mrow></mstyle></math></td>
        </tr>
        <tr>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>du</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>dx</mi></mrow></mstyle></math></td>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>v</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac></mrow></mstyle></math></td>
        </tr>
    </tbody>
</table>
</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$as</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></mstyle></math></p>@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=$a=rint(4)+2;

$as=$a*$a;@
qu.1.1.uid=f0cd1b35-e1aa-4ec3-a4fd-e85883317388@
qu.1.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by parts;
  Sub-Topic=Basic;
@

qu.1.2.question=<p>Find<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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' stretchy='true' largeop='true'>&Integral;</mo><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>e</mi><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&period;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo></mrow></mstyle></math></p>@
qu.1.2.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.1.2.allow2d=1@
qu.1.2.maple_answer=int(x*exp($a*x),x)+C;@
qu.1.2.type=formula@
qu.1.2.mode=Maple@
qu.1.2.name=IbyPx*e^x@
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><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>e</mi><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mstyle></math></p>
<p>Let</p>
<p>
<table width="200" cellspacing="1" cellpadding="1" border="1">
    <tbody>
        <tr>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>x</mi></mrow></mstyle></math></td>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>e</mi><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></msup><mi>dx</mi></mrow></mstyle></math></td>
        </tr>
        <tr>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>du</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>dx</mi></mrow></mstyle></math></td>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>v</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><msup><mi>e</mi><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></msup><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac></mrow></mstyle></math></td>
        </tr>
    </tbody>
</table>
</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>e</mi><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></msup></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><msup><mi>e</mi><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</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><mfrac><mrow><msup><mi>x</mi><mi></mi></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>e</mi><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></msup></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$as</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>e</mi><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></mstyle></math></p>@
qu.1.2.editing=useHTML@
qu.1.2.solution=@
qu.1.2.algorithm=$a=rint(4)+2;

$as=$a*$a;@
qu.1.2.uid=b6d137f6-5250-4458-b4ec-96f0a111fd32@
qu.1.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by parts;
  Sub-Topic=Basic;
@

qu.1.3.question=<p>Find<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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' stretchy='true' largeop='true'>&Integral;</mo><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&period;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo></mrow></mstyle></math></p>@
qu.1.3.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.1.3.allow2d=1@
qu.1.3.maple_answer=int(x*sin($a*x),x)+C;@
qu.1.3.type=formula@
qu.1.3.mode=Maple@
qu.1.3.name=IbyPx*sin($a*x)@
qu.1.3.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced><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>Let</p>
<p>
<table width="200" cellspacing="1" cellpadding="1" border="1">
    <tbody>
        <tr>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>x</mi></mrow></mstyle></math></td>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced><mi>dx</mi></mrow></mstyle></math></td>
        </tr>
        <tr>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>du</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>dx</mi></mrow></mstyle></math></td>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>v</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac></mrow></mstyle></math></td>
        </tr>
    </tbody>
</table>
</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$as</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></mstyle></math></p>@
qu.1.3.editing=useHTML@
qu.1.3.solution=@
qu.1.3.algorithm=$a=rint(4)+2;

$as=$a*$a;@
qu.1.3.uid=2fdbfe65-8629-4824-adec-fc80b1a9dc2d@
qu.1.3.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by parts;
  Sub-Topic=Basic;
@

qu.2.topic=IbyPf(sqrt(x))@

qu.2.1.question=<p>Find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mi mathvariant='normal'>sin</mi><msqrt><mrow><mi>x</mi></mrow></msqrt><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mstyle></math>.</p>
<p>Hint: Let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msqrt><mrow><mi>x</mi></mrow></msqrt></mrow></mrow></mstyle></math>so that <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>t</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math>Substitute and use Integration by Parts on the resulting integral.</p>@
qu.2.1.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.2.1.allow2d=1@
qu.2.1.maple_answer=int(sin(sqrt(x)),x)+C;@
qu.2.1.type=formula@
qu.2.1.mode=Maple@
qu.2.1.name=IbyPsin(sqrt($x))@
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><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><msqrt><mrow><mi>x</mi></mrow></msqrt></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow><mrow><mi></mi></mrow></mstyle></math></p>
<p>Let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msqrt><mrow><mi>x</mi></mrow></msqrt><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>so</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>t</mi><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msup><mtext>therefore</mtext><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>dx</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>t</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>dt</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mi>t</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>t</mi></mrow></mstyle></math>&nbsp;&nbsp;&nbsp;</p>
<p>Let</p>
<p>
<table border="1" cellspacing="1" cellpadding="1" width="200">
    <tbody>
        <tr>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>t</mi></mrow></mstyle></math></td>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mi>dt</mi></mrow></mstyle></math></td>
        </tr>
        <tr>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>du</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>dt</mi></mrow></mstyle></math></td>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>v</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced></mrow></mstyle></math></td>
        </tr>
    </tbody>
</table>
</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>t</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>t</mi></mrow><mrow><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='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>t</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>C</mi></mrow></mstyle></math></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><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msqrt><mrow><mi>x</mi></mrow></msqrt><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><msqrt><mrow><mi>x</mi></mrow></msqrt></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><msqrt><mrow><mi>x</mi></mrow></msqrt></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></mstyle></math></p>
<p>&nbsp;</p>@
qu.2.1.editing=useHTML@
qu.2.1.solution=@
qu.2.1.algorithm=@
qu.2.1.uid=e99fe48d-fa6d-4c53-96f9-4962af1c13a6@
qu.2.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by parts;
  Sub-Topic=Substitution first;
@

qu.2.2.question=<p>Find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mi mathvariant='normal'>cos</mi><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><msqrt><mrow><mi>x</mi></mrow></msqrt><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mstyle></math>.</p>
<p>Hint: Let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msqrt><mrow><mi>x</mi></mrow></msqrt></mrow></mrow></mstyle></math>so that <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>t</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math>Substitute and use Integration by Parts on the resulting integral.</p>@
qu.2.2.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.2.2.allow2d=1@
qu.2.2.maple_answer=int(cos(sqrt(x)),x)+C;@
qu.2.2.type=formula@
qu.2.2.mode=Maple@
qu.2.2.name=IbyPcos(sqrt($x))@
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><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><msqrt><mrow><mi>x</mi></mrow></msqrt></mrow></mfenced><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>Let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msqrt><mrow><mi>x</mi></mrow></msqrt><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>so</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>t</mi><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msup><mtext>therefore</mtext><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>dx</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>t</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>dt</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mi>t</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>t</mi></mrow></mstyle></math>&nbsp;&nbsp;&nbsp;</p>
<p>Let</p>
<p>
<table border="1" cellspacing="1" cellpadding="1" width="200">
    <tbody>
        <tr>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>t</mi></mrow></mstyle></math></td>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mi>dt</mi></mrow></mstyle></math></td>
        </tr>
        <tr>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>du</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>dt</mi></mrow></mstyle></math></td>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>v</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced></mrow></mstyle></math></td>
        </tr>
    </tbody>
</table>
</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>t</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>t</mi></mrow><mrow><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><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>t</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>t</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>C</mi></mrow></mstyle></math></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><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msqrt><mrow><mi>x</mi></mrow></msqrt><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><msqrt><mrow><mi>x</mi></mrow></msqrt></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><msqrt><mrow><mi>x</mi></mrow></msqrt></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></mstyle></math></p>
<p>&nbsp;</p>@
qu.2.2.editing=useHTML@
qu.2.2.solution=@
qu.2.2.algorithm=@
qu.2.2.uid=db044c88-62a9-4d74-9fdd-1080507808df@
qu.2.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by parts;
  Sub-Topic=Substitution first;
@

qu.2.3.question=<p>Find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><msup><mi>e</mi><mrow><msqrt><mrow><mi>x</mi></mrow></msqrt></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mstyle></math>.</p>
<p>Hint: Let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mrow><msqrt><mrow><mi>x</mi></mrow></msqrt></mrow></mrow></mstyle></math>so that <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi>t</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math>Substitute and use Integration by Parts on the resulting integral.</p>@
qu.2.3.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.2.3.allow2d=1@
qu.2.3.maple_answer=int(exp(sqrt(x)),x)+C;@
qu.2.3.type=formula@
qu.2.3.mode=Maple@
qu.2.3.name=IbyPexp(sqrt($x))@
qu.2.3.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><msup><mi>e</mi><mrow><msqrt><mrow><mi>x</mi></mrow></msqrt></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'>&DifferentialD;</mo><mi>x</mi></mrow><mrow><mi></mi></mrow></mstyle></math></p>
<p>Let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>t</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msqrt><mrow><mi>x</mi></mrow></msqrt><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>so</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>t</mi><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msup><mtext>therefore</mtext><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>dx</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>t</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>dt</mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mi>t</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>e</mi><mrow><mi>t</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>t</mi></mrow></mstyle></math>&nbsp;&nbsp;&nbsp;</p>
<p>Let</p>
<p>
<table border="1" cellspacing="1" cellpadding="1" width="200">
    <tbody>
        <tr>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>t</mi></mrow></mstyle></math></td>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>e</mi><mrow><mi>t</mi></mrow></msup><mi>dt</mi></mrow></mstyle></math></td>
        </tr>
        <tr>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>du</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>dt</mi></mrow></mstyle></math></td>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>v</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>e</mi><mrow><mi>t</mi></mrow></msup></mrow></mstyle></math></td>
        </tr>
    </tbody>
</table>
</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>t</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>e</mi><mrow><mi>t</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><msup><mi>e</mi><mrow><mi>t</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>t</mi></mrow><mrow><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><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>t</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>e</mi><mrow><mi>t</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>e</mi><mrow><mi>t</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>C</mi></mrow></mstyle></math></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><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msqrt><mrow><mi>x</mi></mrow></msqrt><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>e</mi><mrow><msqrt><mrow><mi>x</mi></mrow></msqrt></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>e</mi><mrow><msqrt><mrow><mi>x</mi></mrow></msqrt></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></mstyle></math></p>
<p>&nbsp;</p>@
qu.2.3.editing=useHTML@
qu.2.3.solution=@
qu.2.3.algorithm=@
qu.2.3.uid=62e5227a-fb2a-4511-bfbe-8389278c7b0b@
qu.2.3.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by parts;
  Sub-Topic=Substitution first;
@

qu.3.topic=IbyPxsec^2csc^2@

qu.3.1.question=<p>Find<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>csc</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math></p>
<p>Hint: Remember <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mi mathvariant='normal'>cot</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>ln</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mfenced open='|' close='|' separators=','><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfenced><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow><mrow><mi>C</mi></mrow></mstyle></math>.</p>@
qu.3.1.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.3.1.allow2d=1@
qu.3.1.maple_answer=-1/$a*x*cot($a*x)+1/$a^2*ln(abs(sin($a*x)))+C;@
qu.3.1.type=formula@
qu.3.1.mode=Maple@
qu.3.1.name=IbyPx*csc^2($a*x)@
qu.3.1.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>csc</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mstyle></math></p>
<p>Let</p>
<p>
<table width="200" cellspacing="1" cellpadding="1" border="1">
    <tbody>
        <tr>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>x</mi></mrow></mstyle></math></td>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi mathvariant='normal'>csc</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced><mi>dx</mi></mrow></mstyle></math></td>
        </tr>
        <tr>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>du</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>dx</mi></mrow></mstyle></math></td>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>v</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><mi mathvariant='normal'>cot</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac></mrow></mstyle></math></td>
        </tr>
    </tbody>
</table>
</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cot</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mi mathvariant='normal'>cot</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced><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='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cot</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$as</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>ln</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mfenced open='|' close='|' separators=','><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><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.3.1.editing=useHTML@
qu.3.1.solution=@
qu.3.1.algorithm=$a=rint(2,6);
$as=$a*$a;@
qu.3.1.uid=6a8f0c2e-fb50-47d6-add2-0eeb9c4930e8@
qu.3.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by Parts;
  Sub-Topic=Basic;
@

qu.3.2.question=<p>Find<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math></p>
<p>Hint: Remember <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>ln</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mfenced open='|' close='|' separators=','><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfenced><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow><mrow><mi>C</mi></mrow></mstyle></math>.</p>@
qu.3.2.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.3.2.allow2d=1@
qu.3.2.maple_answer=1/$a*x*tan($a*x)+1/$a^2*ln(abs(cos($a*x)))+C;@
qu.3.2.type=formula@
qu.3.2.mode=Maple@
qu.3.2.name=IbyPx*sec^2($a*x)@
qu.3.2.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mstyle></math></p>
<p>Let</p>
<p>
<table width="200" cellspacing="1" cellpadding="1" border="1">
    <tbody>
        <tr>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>x</mi></mrow></mstyle></math></td>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced><mi>dx</mi></mrow></mstyle></math></td>
        </tr>
        <tr>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>du</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>dx</mi></mrow></mstyle></math></td>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>v</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac></mrow></mstyle></math></td>
        </tr>
    </tbody>
</table>
</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced><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><mfrac><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$as</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>ln</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><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.3.2.editing=useHTML@
qu.3.2.solution=@
qu.3.2.algorithm=$a=rint(2,6);
$as=$a*$a;@
qu.3.2.uid=42d85d0d-629f-4b2f-a60e-f28b8a8f71bd@
qu.3.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by Parts;
  Sub-Topic=Basic;
@

qu.4.topic=IBYPf(a+x^n)x^(2n-1)@

qu.4.1.question=<p>Evaluate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><msup><mrow><mi>x</mi></mrow><mi mathvariant='normal'>$n</mi></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mi mathvariant='normal'>$m</mi></msup><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math></p>@
qu.4.1.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.4.1.allow2d=1@
qu.4.1.maple_answer=1/$n*x^$n*sin(x^$n+$a)+1/$n*cos(x^$n+$a)+C@
qu.4.1.type=formula@
qu.4.1.mode=Maple@
qu.4.1.name=PowerRulecos@
qu.4.1.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mrow><mi mathvariant='normal'>$m</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow><mrow><mi mathvariant='normal'></mi></mrow><mrow></mrow></mstyle></math></p>
<p>Let
<table width="200" cellspacing="1" cellpadding="1" border="1">
    <tbody>
        <tr>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup></mrow></mstyle></math></td>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow><mrow></mrow><mrow><mi>dx</mi></mrow></mstyle></math></td>
        </tr>
        <tr>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>du</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$n</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup><mi>dx</mi></mrow><mrow></mrow></mstyle></math></td>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>v</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac></mrow></mstyle></math></td>
        </tr>
    </tbody>
</table>
</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mi>dx</mi></mrow><mrow><mi mathvariant='normal'></mi></mrow></mstyle></math></p>
<p>&nbsp;= <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></mstyle></math></p>@
qu.4.1.editing=useHTML@
qu.4.1.solution=@
qu.4.1.algorithm=$a=range(2,8);
$n=range(2,10);
$m=2*$n-1;
$n1=$n-1;
$ns=$n*$n;
$n2=$ns*$n;@
qu.4.1.uid=22a9863b-c8a6-4aac-bc31-4db4476e50bc@
qu.4.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by parts;
  Sub-Topic=Power rule;
@

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'>&Integral;</mo><msup><mi>e</mi><mrow><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mi mathvariant='normal'>$m</mi></msup><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mstyle></math>.</p>@
qu.4.2.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.4.2.allow2d=1@
qu.4.2.maple_answer=int(exp(x^$n+$a)*x^$m,x)+C@
qu.4.2.type=formula@
qu.4.2.mode=Maple@
qu.4.2.name=PowerRuleexp@
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><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><msup><mi>e</mi><mrow><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mrow><mi mathvariant='normal'>$m</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mstyle></math></p>
<p>Let</p>
<p>
<table width="200" cellspacing="1" cellpadding="1" border="1">
    <tbody>
        <tr>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup></mrow></mstyle></math></td>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>e</mi><mrow><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></msup></mrow><mrow><mi>dx</mi></mrow></mstyle></math></td>
        </tr>
        <tr>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>du</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$n</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup></mrow><mrow><mi>dx</mi></mrow></mstyle></math></td>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>v</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><msup><mi>e</mi><mrow><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></msup><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac></mrow></mstyle></math></td>
        </tr>
    </tbody>
</table>
</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>e</mi><mrow><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></msup></mrow></msup></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>e</mi><mrow><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>e</mi><mrow><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></msup></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><msup><mi>e</mi><mrow><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></msup><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></mstyle></math></p>@
qu.4.2.editing=useHTML@
qu.4.2.solution=@
qu.4.2.algorithm=$a=range(2,8);
$b=range(2,8);
$n=range(2,10);
$m=2*$n-1;
$n1=$n-1;@
qu.4.2.uid=05d7df58-1a80-4696-aac5-c40a75ca2383@
qu.4.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by parts;
  Sub-Topic=Power rule;
@

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' stretchy='true' largeop='true'>&Integral;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><msup><mrow><mi>x</mi></mrow><mi mathvariant='normal'>$n</mi></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mi mathvariant='normal'>$m</mi></msup><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&period;</mo></mrow></mstyle></math></p>@
qu.4.3.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.4.3.allow2d=1@
qu.4.3.maple_answer=-1/$n*x^$n*cos(x^$n+$a)+1/$n*sin(x^$n+$a)+C@
qu.4.3.type=formula@
qu.4.3.mode=Maple@
qu.4.3.name=PowerRule sin@
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><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mrow><mi mathvariant='normal'>$m</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow><mrow></mrow><mrow><mi mathvariant='normal'></mi></mrow><mrow></mrow></mstyle></math></p>
<p>Let
<table width="200" cellspacing="1" cellpadding="1" border="1">
    <tbody>
        <tr>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup></mrow></mstyle></math></td>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow><mrow><mi mathvariant='normal'></mi></mrow><mrow><mi>dx</mi></mrow></mstyle></math></td>
        </tr>
        <tr>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>du</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$n</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup><mi>dx</mi></mrow><mrow></mrow></mstyle></math></td>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>v</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac></mrow></mstyle></math></td>
        </tr>
    </tbody>
</table>
</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mi>dx</mi></mrow><mrow><mi mathvariant='normal'></mi></mrow></mstyle></math></p>
<p>&nbsp;= <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mfrac><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$n</mi></mrow></mfrac><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>$a</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></mstyle></math></p>@
qu.4.3.editing=useHTML@
qu.4.3.solution=@
qu.4.3.algorithm=$a=range(2,8);
$n=range(2,10);
$m=2*$n-1;
$n1=$n-1;@
qu.4.3.uid=b796b540-5a84-479d-aa0d-20c22eef469f@
qu.4.3.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by parts;
  Sub-Topic=Power rule;
@

qu.4.4.question=<p>Evaluate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&Integral;</mo><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mi mathvariant='normal'>$n</mi></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mi mathvariant='normal'>$b</mi></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mi mathvariant='normal'>$m</mi></msup><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mrow></mstyle></math>.</p>@
qu.4.4.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.4.4.allow2d=1@
qu.4.4.maple_answer=1/($n*($b+1))*x^$n*(x^$n+$a)^($b+1)-1/($n*($b+1)*($b+2))*(x^$n+$a)^($b+2)+C@
qu.4.4.type=formula@
qu.4.4.mode=Maple@
qu.4.4.name=PowerRule(a+x^n)@
qu.4.4.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mi mathvariant='normal'>$b</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mrow><mi mathvariant='normal'>$m</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>Let</p>
<p>
<table width="200" cellspacing="1" cellpadding="1" border="1">
    <tbody>
        <tr>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup></mrow></mstyle></math></td>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mi mathvariant='normal'>$b</mi></mrow></msup></mrow><mrow><mi>dx</mi></mrow></mstyle></math></td>
        </tr>
        <tr>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>du</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>$n</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup></mrow><mrow><mi>dx</mi></mrow></mstyle></math></td>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>v</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mfrac><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mi mathvariant='normal'>$b1</mi></mrow></msup><mrow><mi mathvariant='normal'>$bn</mi></mrow></mfrac></mrow></mstyle></math></td>
        </tr>
    </tbody>
</table>
</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mi mathvariant='normal'>$b1</mi></mrow></msup></mrow><mrow><mi mathvariant='normal'>$bn</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mn>1</mn><mrow><mi mathvariant='normal'>$b1</mi></mrow></mfrac><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n1</mi></mrow></msup><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mi mathvariant='normal'>$b1</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow><mrow></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mrow><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mi mathvariant='normal'>$b1</mi></mrow></msup></mrow><mrow><mi mathvariant='normal'>$bn</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mi mathvariant='normal'>$b2</mi></mrow></msup><mrow><mi mathvariant='normal'>$bn1</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow><mrow><mi mathvariant='normal'></mi></mrow></mstyle></math></p>@
qu.4.4.editing=useHTML@
qu.4.4.solution=@
qu.4.4.algorithm=$a=range(1,9);
$b=range(2,5);
$n=range(2,5);
$m=2*$n-1;
$n1=$n-1;
$b1=$b+1;
$bn=($b+1)*$n;
$b2=$b+2;
$bn1=$b1*$b2*$n;@
qu.4.4.uid=55cbfb3e-d210-4e3f-992a-a58e38c200b6@
qu.4.4.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by parts;
  Sub-Topic=Power rule;
@

qu.5.topic=IbyPsecondstepXchoice@

qu.5.1.mode=Multiple Choice@
qu.5.1.name=e^xsin(x)u=e^x@
qu.5.1.comment=@
qu.5.1.editing=useHTML@
qu.5.1.solution=@
qu.5.1.algorithm=@
qu.5.1.uid=2b7faf67-c152-4d4a-a971-28e306720933@
qu.5.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by parts;
  Sub-Topic=Strategy;
@
qu.5.1.question=<p>After one application of integration by parts where we let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>e</mi><mrow><mi>x</mi></mrow></msup></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><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>dx</mi></mrow></mstyle></math>, we find <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><msup><mi>e</mi><mi>x</mi></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>dx</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi>e</mi><mi>x</mi></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&Integral;</mo><msup><mi>e</mi><mi>x</mi></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mi>dx</mi></mrow><mrow><mi></mi></mrow></mstyle></math>. Now the best strategy is to&nbsp;</p>@
qu.5.1.answer=1@
qu.5.1.choice.1=let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>e</mi><mrow><mi>x</mi></mrow></msup></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><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>dx</mi></mrow></mstyle></math> @
qu.5.1.choice.2=let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></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><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>e</mi><mrow><mi>x</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>dx</mi></mrow></mstyle></math> @
qu.5.1.choice.3=let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></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><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>e</mi><mrow><mi>x</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>dx</mi></mrow></mstyle></math> @
qu.5.1.choice.4=start over, letting <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>e</mi><mrow><mi>x</mi></mrow></msup></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><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>dx</mi></mrow></mstyle></math>@
qu.5.1.choice.5=let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>e</mi><mrow><mi>x</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></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><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>dx</mi></mrow></mstyle></math> @
qu.5.1.fixed=@

qu.5.2.mode=Multiple Choice@
qu.5.2.name=e^xsin(x)u=sinx@
qu.5.2.comment=@
qu.5.2.editing=useHTML@
qu.5.2.solution=@
qu.5.2.algorithm=@
qu.5.2.uid=23dc1631-0576-4397-b368-f70b3c9d2788@
qu.5.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by parts;
  Sub-Topic=Strategy;
@
qu.5.2.question=<p>After one application of integration by parts where we let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></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><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>e</mi><mrow><mi>x</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>dx</mi></mrow></mstyle></math>, we find</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><msup><mi>e</mi><mi>x</mi></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>dx</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>e</mi><mi>x</mi></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&Integral;</mo><msup><mi>e</mi><mi>x</mi></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mi>dx</mi></mrow></mstyle></math>. Now the best strategy is to</p>
<p>&nbsp;</p>@
qu.5.2.answer=2@
qu.5.2.choice.1=let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>e</mi><mrow><mi>x</mi></mrow></msup></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><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>dx</mi></mrow></mstyle></math>@
qu.5.2.choice.2=let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></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><mi>dv</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>e</mi><mrow><mi>x</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>dx</mi></mrow></mstyle></math>@
qu.5.2.choice.3=let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></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><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>e</mi><mrow><mi>x</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>dx</mi></mrow></mstyle></math> @
qu.5.2.choice.4=start over, letting <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>e</mi><mrow><mi>x</mi></mrow></msup></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><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>dx</mi></mrow></mstyle></math>@
qu.5.2.choice.5=let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>e</mi><mrow><mi>x</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></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><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>dx</mi></mrow></mstyle></math> @
qu.5.2.fixed=@

qu.5.3.mode=Multiple Choice@
qu.5.3.name=e^xcos(x)u=cosx@
qu.5.3.comment=@
qu.5.3.editing=useHTML@
qu.5.3.solution=@
qu.5.3.algorithm=@
qu.5.3.uid=682032f6-6dda-4c19-a49a-5f6028512174@
qu.5.3.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by parts;
  Sub-Topic=Strategy;
@
qu.5.3.question=<p>After one application of integration by parts where we let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></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><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>e</mi><mrow><mi>x</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>dx</mi></mrow></mstyle></math>, we find</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><msup><mi>e</mi><mi>x</mi></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>dx</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>e</mi><mi>x</mi></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&Integral;</mo><msup><mi>e</mi><mi>x</mi></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mi>dx</mi></mrow></mstyle></math>. Now the best strategy is to</p>@
qu.5.3.answer=2@
qu.5.3.choice.1=let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>e</mi><mrow><mi>x</mi></mrow></msup></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><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>dx</mi></mrow></mstyle></math>@
qu.5.3.choice.2=let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></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><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>e</mi><mrow><mi>x</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>dx</mi></mrow></mstyle></math>@
qu.5.3.choice.3=let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></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><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>e</mi><mrow><mi>x</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>dx</mi></mrow></mstyle></math>@
qu.5.3.choice.4=start over, letting <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>e</mi><mrow><mi>x</mi></mrow></msup></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><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>dx</mi></mrow></mstyle></math>@
qu.5.3.choice.5=let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>e</mi><mrow><mi>x</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></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><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>dx</mi></mrow></mstyle></math>@
qu.5.3.fixed=@

qu.5.4.mode=Multiple Choice@
qu.5.4.name=e^xcos(x)u=e^x@
qu.5.4.comment=@
qu.5.4.editing=useHTML@
qu.5.4.solution=@
qu.5.4.algorithm=@
qu.5.4.uid=09001d54-40a1-4e63-8048-438adafbd4d8@
qu.5.4.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by parts;
  Sub-Topic=Strategy;
@
qu.5.4.question=<p>After one application of integration by parts where we let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>e</mi><mrow><mi>x</mi></mrow></msup></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><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>dx</mi></mrow></mstyle></math>, we find</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><msup><mi>e</mi><mi>x</mi></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>dx</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>e</mi><mi>x</mi></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&Integral;</mo><msup><mi>e</mi><mi>x</mi></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mi>dx</mi></mrow></mstyle></math>. Now the best strategy is to</p>
<p>&nbsp;</p>@
qu.5.4.answer=1@
qu.5.4.choice.1=let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>e</mi><mrow><mi>x</mi></mrow></msup></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><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>dx</mi></mrow></mstyle></math>@
qu.5.4.choice.2=let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></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><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>e</mi><mrow><mi>x</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>dx</mi></mrow></mstyle></math>@
qu.5.4.choice.3=let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn></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><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>e</mi><mrow><mi>x</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>dx</mi></mrow></mstyle></math>@
qu.5.4.choice.4=start over, letting <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>e</mi><mrow><mi>x</mi></mrow></msup></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><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>dx</mi></mrow></mstyle></math>@
qu.5.4.choice.5=let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>e</mi><mrow><mi>x</mi></mrow></msup><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></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><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi>dx</mi></mrow></mstyle></math>@
qu.5.4.fixed=@

qu.6.topic=sec^3csc^3@

qu.6.1.question=<p>Evaluate<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><msup><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mi></mi></msup><msup><mi mathvariant='normal'>csc</mi><mrow><mn>3</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mstyle></math>. Don't forget absolute value where it is needed.</p>@
qu.6.1.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.6.1.allow2d=1@
qu.6.1.maple_answer=1/2*ln(abs(csc(x)-cot(x)))-1/2*csc(x)*cot(x)+C@
qu.6.1.type=formula@
qu.6.1.mode=Maple@
qu.6.1.name=intcsc^3@
qu.6.1.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><msup><mi mathvariant='normal'>csc</mi><mrow><mn>3</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><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>Let</p>
<p>
<table width="200" cellspacing="1" cellpadding="1" border="1">
    <tbody>
        <tr>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>csc</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math></td>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi mathvariant='normal'>csc</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mi>dx</mi></mrow></mstyle></math></td>
        </tr>
        <tr>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>du</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>csc</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>cot</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mi>dx</mi></mrow></mstyle></math></td>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>v</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>cot</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math></td>
        </tr>
    </tbody>
</table>
</p>
<p>&nbsp;</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>cot</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>csc</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mi mathvariant='normal'>csc</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>cot</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><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><mi>I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>cot</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>csc</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mi mathvariant='normal'>csc</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><msup><mi mathvariant='normal'>csc</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><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><mi>I</mi><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><mi mathvariant='normal'>cot</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>csc</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><msup><mi mathvariant='normal'>csc</mi><mrow><mn>3</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><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' stretchy='true' largeop='true'>&Integral;</mo><mi mathvariant='normal'>csc</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><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><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>cot</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>csc</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>I</mi><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>ln</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='|' close='|' separators=','><mrow><mi mathvariant='normal'>csc</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>cot</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></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><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>cot</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>csc</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>ln</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='|' close='|' separators=','><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>csc</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>cot</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mi></mi></mrow></mstyle></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac><mi mathvariant='normal'>cot</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>csc</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac><mi mathvariant='normal'>ln</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='|' close='|' separators=','><mrow><mi mathvariant='normal'>csc</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>cot</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><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.6.1.editing=useHTML@
qu.6.1.hint.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mi mathvariant='normal'>csc</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><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><mi mathvariant='normal'>ln</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='|' close='|' separators=','><mrow><mi mathvariant='normal'>csc</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>cot</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></mstyle></math>@
qu.6.1.solution=@
qu.6.1.algorithm=@
qu.6.1.uid=8aea1949-e0c9-4597-bcba-486356724052@
qu.6.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Hard;
  Course=Introduction to Calculus II;
  Topic=Integration by Parts;
  Sub-Topic=csc^3 sec^3;
@

qu.6.2.question=<p>Evaluate<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><msup><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mi></mi></msup><msup><mi mathvariant='normal'>sec</mi><mrow><mn>3</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mstyle></math>. Don't forget absolute value where it is needed.</p>@
qu.6.2.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.6.2.allow2d=1@
qu.6.2.maple_answer=1/2*sec(x)*tan(x)+1/2*ln(abs(sec(x)+tan(x)))+C@
qu.6.2.type=formula@
qu.6.2.mode=Maple@
qu.6.2.name=intsec^3@
qu.6.2.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><msup><mi mathvariant='normal'>sec</mi><mrow><mn>3</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow><mrow><mi></mi></mrow></mstyle></math></p>
<p>Let</p>
<p>
<table width="200" cellspacing="1" cellpadding="1" border="1">
    <tbody>
        <tr>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math></td>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mi>dx</mi></mrow></mstyle></math></td>
        </tr>
        <tr>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>du</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mi>dx</mi></mrow></mstyle></math></td>
            <td><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>v</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math></td>
        </tr>
    </tbody>
</table>
</p>
<p>&nbsp;</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>tan</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><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><mi>I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mfenced><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><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><msup><mi mathvariant='normal'>sec</mi><mrow><mn>3</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><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' stretchy='true' largeop='true'>&Integral;</mo><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><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><mi>I</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&#32;</mo><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>I</mi><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>ln</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</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><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>I</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>ln</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></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>I</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac><mi mathvariant='normal'>ln</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><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.6.2.editing=useHTML@
qu.6.2.hint.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><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><mi mathvariant='normal'>ln</mi><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lpar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></mstyle></math>@
qu.6.2.solution=@
qu.6.2.algorithm=@
qu.6.2.uid=5cefa4ae-6284-42c8-b230-eda1f1e85c01@
qu.6.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Hard;
  Course=Introduction to Calculus II;
  Topic=Integration by Parts;
  Sub-Topic=csc^3 sec^3;
@

qu.7.topic=sec^5csc^5@

qu.7.1.mode=Multiple Choice@
qu.7.1.name=Sec^5@
qu.7.1.comment=@
qu.7.1.editing=useHTML@
qu.7.1.solution=@
qu.7.1.algorithm=@
qu.7.1.uid=9316a692-515a-4ac1-b98e-1bdb21d615c4@
qu.7.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by parts;
  Sub-Topic=Strategy;
@
qu.7.1.question=<p>After one application of integrating by parts, we find</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 mathvariant='italic' lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><msup><mi>sec</mi><mn>5</mn></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><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'>&InvisibleTimes;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi>sec</mi><mn>3</mn></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mi>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>3</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&Integral;</mo><msup><mi>sec</mi><mn>3</mn></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>tan</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mi>dx</mi></mrow></mstyle></math>. Now the best strategy is to</p>
<p>&nbsp;</p>@
qu.7.1.answer=1@
qu.7.1.choice.1=let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi mathvariant='normal'>tan</mi><mn>2</mn></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi mathvariant='normal'>sec</mi><mn>2</mn></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mstyle></math>@
qu.7.1.choice.2=let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi mathvariant='normal'>tan</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></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><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi mathvariant='normal'>sec</mi><mrow><mn>3</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>dx</mi></mrow></mstyle></math>@
qu.7.1.choice.3=let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi mathvariant='normal'>sec</mi><mrow><mn>3</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></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><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi mathvariant='normal'>tan</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>dx</mi></mrow></mstyle></math>@
qu.7.1.choice.4=let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi mathvariant='normal'>tan</mi><mn>2</mn></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math>@
qu.7.1.choice.5=wring your integrating hands and give up!@
qu.7.1.fixed=@

qu.7.2.mode=Multiple Choice@
qu.7.2.name=Csc^5@
qu.7.2.comment=@
qu.7.2.editing=useHTML@
qu.7.2.solution=@
qu.7.2.algorithm=@
qu.7.2.uid=ccca1118-1226-4dc0-9d21-7af437690f84@
qu.7.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by parts;
  Sub-Topic=Strategy;
@
qu.7.2.question=<p>&nbsp;</p>
<p>After one application of integrating by parts, we find</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><msup><mi>csc</mi><mn>5</mn></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><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'>&InvisibleTimes;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><msup><mi>csc</mi><mn>3</mn></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mi>cot</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>3</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mo lspace='0.0em' rspace='0.0em'>&Integral;</mo><msup><mi>csc</mi><mn>3</mn></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>cot</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mi>dx</mi></mrow></mstyle></math>. Now the best strategy is to</p>@
qu.7.2.answer=1@
qu.7.2.choice.1=let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi mathvariant='normal'>cot</mi><mn>2</mn></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi mathvariant='normal'>csc</mi><mn>2</mn></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mstyle></math>@
qu.7.2.choice.2=let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi mathvariant='normal'>cot</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></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><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi mathvariant='normal'>csc</mi><mrow><mn>3</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>dx</mi></mrow></mstyle></math>@
qu.7.2.choice.3=let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi mathvariant='normal'>csc</mi><mrow><mn>3</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></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><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi mathvariant='normal'>cot</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>dx</mi></mrow></mstyle></math>@
qu.7.2.choice.4=let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi mathvariant='normal'>csc</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><msup><mi mathvariant='normal'>cot</mi><mn>2</mn></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math>@
qu.7.2.choice.5=wring your integrating hands and give up!@
qu.7.2.fixed=@

qu.8.topic=TrigProducts@

qu.8.1.mode=Multiple Choice@
qu.8.1.name=tanOdd@
qu.8.1.comment=@
qu.8.1.editing=useHTML@
qu.8.1.solution=@
qu.8.1.algorithm=$m=rint(3,15,2);
$n=rint(3,15,2);@
qu.8.1.uid=3967057a-6384-499a-abc0-604eb3aedc92@
qu.8.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Trig Products;
  Sub-Topic=Strategy;
@
qu.8.1.question=<p>To evaluate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo mathvariant='italic' lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><msup><mi mathvariant='normal'>tan</mi><mi mathvariant='normal'>$n</mi></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>sec</mi><mi mathvariant='normal'>$m</mi></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>dx</mi></mrow></mstyle></math> the best strategy is to</p>
<p>&nbsp;</p>
<p>&nbsp;</p>@
qu.8.1.answer=1@
qu.8.1.choice.1=replace <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi mathvariant='normal'>tan</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math> with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mstyle></math>@
qu.8.1.choice.2=replace <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math> with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow><mrow><msup><mi mathvariant='normal'>tan</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math>@
qu.8.1.choice.3=replace <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi mathvariant='normal'>tan</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math> with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow><mrow><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math>@
qu.8.1.choice.4=replace <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math> with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi mathvariant='normal'>tan</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mstyle></math>@
qu.8.1.choice.5=use integration by parts with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi mathvariant='normal'>tan</mi><mrow><mi mathvariant='normal'>$m</mi></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></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><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi mathvariant='normal'>sec</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>dx</mi></mrow></mstyle></math>@
qu.8.1.fixed=@

qu.8.2.mode=Multiple Choice@
qu.8.2.name=cos^oddsin^even@
qu.8.2.comment=@
qu.8.2.editing=useHTML@
qu.8.2.solution=@
qu.8.2.algorithm=$m=rint(3,15,2);
$n=rint(2,14,2);@
qu.8.2.uid=aeccda62-788c-4a7c-9567-8327ac37c72b@
qu.8.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Trig Products;
  Sub-Topic=Strategy;
@
qu.8.2.question=<p>To evaluate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo mathvariant='italic' lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><msup><mi mathvariant='normal'>sin</mi><mi mathvariant='normal'>$m</mi></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>cos</mi><mi mathvariant='normal'>$n</mi></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>dx</mi></mrow></mstyle></math> the best strategy is to</p>@
qu.8.2.answer=2@
qu.8.2.choice.1=replace <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi mathvariant='normal'>cos</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math> with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi mathvariant='normal'>sin</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math>@
qu.8.2.choice.2=replace <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi mathvariant='normal'>sin</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math> with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi mathvariant='normal'>cos</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math>@
qu.8.2.choice.3=replace <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi mathvariant='normal'>sin</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math> with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi mathvariant='normal'>cos</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></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><msup><mi mathvariant='normal'>cos</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math> with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi mathvariant='normal'>sin</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math>@
qu.8.2.choice.4=replace <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi mathvariant='normal'>cos</mi><mrow><mi></mi></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math> with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msqrt><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi mathvariant='normal'>sin</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></msqrt></mrow></mstyle></math>@
qu.8.2.choice.5=replace <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='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'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math> with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msqrt><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi mathvariant='normal'>cos</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></msqrt></mrow></mstyle></math>@
qu.8.2.fixed=@

qu.8.3.mode=Multiple Choice@
qu.8.3.name=tanEsecO@
qu.8.3.comment=<p>The question asked for the BEST strategy. If you let <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi mathvariant='normal'>tan</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mstyle></math>, you will end up with a question that requires <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' 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><msup><mi mathvariant='normal'>sec</mi><mrow><mn>3</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mstyle></math>. You know the integral of <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math>and you can now use I by P to integrate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi mathvariant='normal'>sec</mi><mrow><mn>3</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math>. This works but is two steps longer than the BEST strategy, which is to use I by P right away.</p>@
qu.8.3.editing=useHTML@
qu.8.3.solution=@
qu.8.3.algorithm=$m=rint(3,15,2);
$n=rint(2,14,2);@
qu.8.3.uid=458e6b19-577a-41b5-9f5a-7f0f2d612f26@
qu.8.3.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Trig Products;
  Sub-Topic=Strategy;
@
qu.8.3.question=<p>To evaluate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><msup><mi mathvariant='normal'>tan</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>dx</mi></mrow></mstyle></math> the best strategy is to</p>@
qu.8.3.answer=5@
qu.8.3.choice.1=group one <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math> with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math> and replace the other <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math> with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msqrt><mrow><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></msqrt></mrow></mstyle></math>@
qu.8.3.choice.2=replace <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi mathvariant='normal'>tan</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math> with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mstyle></math>@
qu.8.3.choice.3=replace <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi mathvariant='normal'>sec</mi><mrow><mi></mi></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math> with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msqrt><mrow><msup><mi mathvariant='normal'>tan</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></msqrt></mrow></mstyle></math>@
qu.8.3.choice.4=replace <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math> with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi mathvariant='normal'>tan</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mstyle></math>@
qu.8.3.choice.5=use integration by parts with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></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><mi>dv</mi><mo mathvariant='italic' lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mi mathvariant='normal'>sec</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi mathvariant='normal'>tan</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>dx</mi></mrow></mstyle></math>@
qu.8.3.fixed=@

qu.8.4.mode=Multiple Choice@
qu.8.4.name=sin^oddcos^even@
qu.8.4.comment=@
qu.8.4.editing=useHTML@
qu.8.4.solution=@
qu.8.4.algorithm=$m=rint(3,15,2);
$n=rint(2,14,2);@
qu.8.4.uid=ba59b5be-d8c7-4638-a2b8-0b85128ac1c5@
qu.8.4.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Trig Products;
  Sub-Topic=Strategy;
@
qu.8.4.question=<p>To evaluate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mo mathvariant='italic' lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><msup><mi mathvariant='normal'>sin</mi><mi mathvariant='normal'>$n</mi></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>cos</mi><mi mathvariant='normal'>$m</mi></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>dx</mi></mrow></mstyle></math> the best strategy is to</p>
<p>&nbsp;</p>@
qu.8.4.answer=1@
qu.8.4.choice.1=replace <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi mathvariant='normal'>cos</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math> with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi mathvariant='normal'>sin</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math>@
qu.8.4.choice.2=replace <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi mathvariant='normal'>sin</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math> with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi mathvariant='normal'>cos</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math>@
qu.8.4.choice.3=replace <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi mathvariant='normal'>sin</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math> with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi mathvariant='normal'>cos</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></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><msup><mi mathvariant='normal'>cos</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math> with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow><mrow><msup><mi mathvariant='normal'>sin</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math>@
qu.8.4.choice.4=replace <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi mathvariant='normal'>cos</mi><mrow><mi></mi></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math> with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msqrt><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi mathvariant='normal'>sin</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></msqrt></mrow></mstyle></math>@
qu.8.4.choice.5=replace <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='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'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math> with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msqrt><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi mathvariant='normal'>cos</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></msqrt></mrow></mstyle></math>@
qu.8.4.fixed=@

qu.8.5.mode=Multiple Choice@
qu.8.5.name=secEven@
qu.8.5.comment=@
qu.8.5.editing=useHTML@
qu.8.5.solution=@
qu.8.5.algorithm=$m=rint(2,14,2);
$n=rint(4,14,2);@
qu.8.5.uid=cb2d5c73-33c4-4db8-81d4-6bb4f6b832fd@
qu.8.5.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Trig Products;
  Sub-Topic=Strategy;
@
qu.8.5.question=<p>To evaluate <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle fontfamily='Times New Roman' mathsize='12' mathcolor='#000000'  veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mo mathvariant='italic' lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></msup><msup><mi mathvariant='normal'>tan</mi><mrow><mi mathvariant='normal'>$m</mi></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi mathvariant='normal'>sec</mi><mi mathvariant='normal'>$n</mi></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mi>dx</mi></mrow><mrow><mi></mi></mrow></mstyle></math> the best strategy is to</p>
<p>&nbsp;</p>@
qu.8.5.answer=1@
qu.8.5.choice.1=replace <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math> with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo></mrow><mrow><msup><mi mathvariant='normal'>tan</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow><mrow><mi></mi></mrow></mstyle></math>@
qu.8.5.choice.2=replace <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi mathvariant='normal'>tan</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math> with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mstyle></math>@
qu.8.5.choice.3=replace <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi mathvariant='normal'>tan</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math> with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow><mrow><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow><mrow><mi></mi></mrow></mstyle></math>@
qu.8.5.choice.4=replace <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi mathvariant='normal'>sec</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mstyle></math> with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi mathvariant='normal'>tan</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>1</mn></mrow></mstyle></math>@
qu.8.5.choice.5=use integration by parts with <math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi mathvariant='normal'>tan</mi><mrow><mi mathvariant='normal'>$m</mi></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></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><mi>dv</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><msup><mi mathvariant='normal'>sec</mi><mrow><mi mathvariant='normal'>$n</mi></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>dx</mi></mrow></mstyle></math>@
qu.8.5.fixed=@

qu.9.topic=evenSinCos@

qu.9.1.question=<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'>&Integral;</mo><mrow><msup><mi mathvariant='normal'>cos</mi><mn>2</mn></msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced></mrow><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mrow></mstyle></math></p>@
qu.9.1.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.9.1.allow2d=1@
qu.9.1.maple_answer=1/2*x+1/(4*$a)*sin(2*$a*x)+C@
qu.9.1.type=formula@
qu.9.1.mode=Maple@
qu.9.1.name=cos($ax)^2@
qu.9.1.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><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><msup><mi mathvariant='normal'>cos</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></mstyle></math></p>
<p>=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced><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>=&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><mfrac><mi>x</mi><mrow><mn>2</mn></mrow></mfrac><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><mfrac><mrow><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced></mrow><mrow><mi mathvariant='normal'>$a4</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></mstyle></math></p>@
qu.9.1.editing=useHTML@
qu.9.1.hint.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi mathvariant='normal'>cos</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mstyle></math>&nbsp@
qu.9.1.solution=@
qu.9.1.algorithm=$a=rint(2,9);
$a2=$a*2;
$a4=$a*4;
$v=mathml($a*x);@
qu.9.1.uid=c56b26a5-b2cf-497d-a36e-45b0f8df0d85@
qu.9.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by parts;
  Sub-Topic=Trig products;
@

qu.9.2.question=<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'>&Integral;</mo><mrow><msup><mi mathvariant='normal'>sin</mi><mn>2</mn></msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced></mrow><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mrow><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mrow></mstyle></math></p>@
qu.9.2.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.9.2.allow2d=1@
qu.9.2.maple_answer=1/2*x-1/(4*$a)*sin(2*$a*x)+C@
qu.9.2.type=formula@
qu.9.2.mode=Maple@
qu.9.2.name=sin($ax)^2@
qu.9.2.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><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><msup><mi mathvariant='normal'>sin</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&InvisibleTimes;</mo><mo lspace='0.0em' rspace='0.0em'>&DifferentialD;</mo><mi>x</mi></mrow></mstyle></math></p>
<p>=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle     veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><mrow><mfrac><mn>1</mn><mrow><mn>2</mn></mrow></mfrac></mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced><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><mfrac><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mrow><mi mathvariant='normal'>sin</mi><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a2</mi><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mi>x</mi></mrow></mfenced></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$a4</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></mstyle></math></p>@
qu.9.2.editing=useHTML@
qu.9.2.hint.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mstyle    veryverythinmathspace='0.0555556em' verythinmathspace='0.111111em' thinmathspace='0.166667em' mediummathspace='0.222222em' thickmathspace='0.277778em' verythickmathspace='0.333333em' veryverythickmathspace='0.388889em' scriptlevel='0' scriptsizemultiplier='0.71' scriptminsize='8.0pt'><mrow><msup><mi mathvariant='normal'>sin</mi><mrow><mn>2</mn></mrow></msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfrac><mrow><mn>1</mn><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>cos</mi><mfenced open='(' close=')' separators=','><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mstyle></math>@
qu.9.2.solution=@
qu.9.2.algorithm=$a=rint(2,9);
$a2=2*$a;
$a4=4*$a;
$v=mathml($a*x);@
qu.9.2.uid=5ea98312-bb1d-4b5c-a125-525bc47510e5@
qu.9.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus II;
  Topic=Integration by parts;
  Sub-Topic=Trig products;
@

