qu.1.topic=BasicIntFormula@

qu.1.1.question=<p>Find $F.</p>@
qu.1.1.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.1.1.allow2d=1@
qu.1.1.maple_answer=$ANS+C@
qu.1.1.type=formula@
qu.1.1.mode=Maple@
qu.1.1.name=expand@
qu.1.1.comment=<p>Expand the function.</p>
<p>$F</p>
<p>= $STEP</p>
<p>= $ANS1</p>@
qu.1.1.editing=useHTML@
qu.1.1.hint.1=Expand the function.@
qu.1.1.solution=@
qu.1.1.algorithm=$a=rint(-5,5);
$b=rint(-5,5);
condition:ne(($a)*($b),0);
$n=rint(2,10);
$m=rint(2,10);
$f=switch(rint(2),"(x^($n)+($a))^2","(x^($n)+($a))*(x^($m)+($b))");
$ANS="int(expand($f),x)";
$M=maple("
MathML[ExportPresentation](Int($f,x)),
MathML[ExportPresentation](Int(expand($f),x)),
MathML[ExportPresentation]($ANS+C)
");
$F=switch(0,$M);
$STEP=switch(1,$M);
$ANS1=switch(2,$M);@
qu.1.1.uid=0009b2ba-60f9-475c-9735-c356a468bc25@
qu.1.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus I;
  Topic=Basic Integration;
  Sub-Topic=Techniques;
@

qu.1.2.question=<p>Find $F.</p>
<p>&nbsp;</p>@
qu.1.2.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.1.2.allow2d=1@
qu.1.2.maple_answer=$ANS+C@
qu.1.2.type=formula@
qu.1.2.mode=Maple@
qu.1.2.name=make separate fractions@
qu.1.2.comment=<p>Make separate fractions.</p>
<p>&nbsp;</p>
<p>$F</p>
<p>= $STEP</p>
<p>= $ANS1</p>@
qu.1.2.editing=useHTML@
qu.1.2.hint.1=Make separate fractions.@
qu.1.2.solution=@
qu.1.2.algorithm=$a=rint(-5,5);
$b=rint(-5,5);
$z=rint(3);
condition:ne(($a)*($b),0);
$n=rint(2,5);
$m=rint(6,10);
$f=switch($z,"(x^($n)+($a)*x^($m)+($b)*x)/x",
"(x^($n)+($a)*x^($m)+($b)*x)/(3*x)","(x^($n)+($a)*x^($m)+($b))/x^(3/4)");
$step=switch($z, "(x^(($n)-1) + ($a)*x^(($m)-1)+($b))", "((1/3)*x^($n-1)+($a/3)*x^($m-1)+($b/3))", "(x^($n-(3/4))+($a)*x^($m-(3/4))+($b)*x^(-3/4))");
$ANS=switch($z,"int($f,x)","int($f,x)",
"x^($n+1/4)/($n+1/4)+($a)*x^($m+1/4)/($m+1/4)+4*($b)*x^(1/4)");
$M=maple("
MathML[ExportPresentation](Int($f,x)),
MathML[ExportPresentation](Int($step,x)),
MathML[ExportPresentation]($ANS+C)
");
$F=switch(0,$M);
$STEP=switch(1,$M);
$ANS1=switch(2,$M);@
qu.1.2.uid=4617cd65-5fcf-4e9a-86b0-203b548c5d96@
qu.1.2.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Difficulty=Easy;
  Topic=Basic Integration;
  Sub-Topic=Techniques;
@

qu.1.3.question=<p>Find $F.</p>@
qu.1.3.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.1.3.allow2d=1@
qu.1.3.maple_answer=$ANS+C@
qu.1.3.type=formula@
qu.1.3.mode=Maple@
qu.1.3.name=BasicIntegrals@
qu.1.3.comment=@
qu.1.3.editing=useHTML@
qu.1.3.solution=@
qu.1.3.algorithm=$a=rint(-5,5);
$b=rint(-5,5);
condition:ne($b,0);
condition:ne($a,0);
$p=rint(-10,10);
$q=rint(-10,10);
condition:ne($q,0);
condition:ne(($p)/($q),-1);
condition:eq(gcd($p,$q),1);
$f=switch(rint(10),"($b)*x^(($p)/($q))","($b)*x^(($p)/($q))","($b)*x^(($p)/($q))",
"($b)*x^(($p)/($q))","($b)*x^(($p)/($q))","($b)*sin(x)","($b)*cos(x)",
"($b)*sec(x)^2","($b)*csc(x)^2");
$F=maple("printf(MathML[ExportPresentation](Int($f,x)))");
$ANS="int($f,x)";@
qu.1.3.uid=b04b56a9-2abe-46a5-b0dd-3551e220aaf4@
qu.1.3.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Difficulty=Easy;
  Topic=Basic Integration;
  Sub-Topic=Techniques;
@

qu.2.topic=ChainRuleInReverse@

qu.2.1.question=<p>Find $F.</p>
<p>&nbsp;</p>@
qu.2.1.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.2.1.allow2d=1@
qu.2.1.maple_answer=$ANS+C@
qu.2.1.type=formula@
qu.2.1.mode=Maple@
qu.2.1.name=CRIR NoAdjustments@
qu.2.1.comment=@
qu.2.1.editing=useHTML@
qu.2.1.hint.1=This question is an example of the chain rule in reverse, with no "adjustments" needed.    <p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mi>g</mi><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced><mi>f</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.1111111em' rspace='0.0em'>&apos;</mo><mfenced open='(' close=')' separators=','><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></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>f</mi><mfenced open='(' close=')' separators=','><mrow><mi>g</mi><mfenced open='(' close=')' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow><mrow><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></math>.</p>@
qu.2.1.solution=@
qu.2.1.algorithm=$a=rint(-5,5);
condition:ne($a,0);
$m=rint(2,10);
$n=rint(2,10);
$z=rint(7);
$f=switch($z,"x^($m)","x^($m)","x^($m)","sin(x)","cos(x)",
"sec(x)^2","csc(x)^2");
$g=switch(rint(5),"sin(x)+($a)","cos(x)+($a)",
"x^($n)+($a)","sqrt(x)+($a)","1/x^($n)+($a)");
$fg="subs(x=($g),$f)";
$gp="diff($g,x)";
$F=maple("printf(MathML[ExportPresentation](Int($gp*$fg,x)))");
$ANS=switch($z,"($g)^(($m)+1)/(($m)+1)","($g)^(($m)+1)/(($m)+1)","($g)^(($m)+1)/(($m)+1)","-cos($g)","sin($g)",
"tan($g)","-cot($g)");@
qu.2.1.uid=1e23adce-3ef3-4a35-8ade-b003b90d4471@
qu.2.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus I;
  Topic=Basic Integration;
  Sub-Topic=Chain Rule in Reverse;
@

qu.2.2.question=<p>Find $F.</p>
<p>&nbsp;</p>@
qu.2.2.maple=if not(simplify(diff(($ANSWER)-($RESPONSE),x))=0) then
grade:=0:
else
if (type(simplify(($ANSWER)-($RESPONSE)-C), numeric)) then
grade:=0.75:
else
grade:=1:
end if: 
end if:
grade@
qu.2.2.allow2d=1@
qu.2.2.maple_answer=$ANS+C@
qu.2.2.type=formula@
qu.2.2.mode=Maple@
qu.2.2.name=CRIR Adjustments@
qu.2.2.comment=@
qu.2.2.editing=useHTML@
qu.2.2.hint.1=This question is an example of the chain rule in reverse, but you will need to adjust the integrand with a multiplicative constant. For example,    <p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>3</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mn>1</mn></mrow></mfenced><mrow><mfrac><mn>3</mn><mrow><mn>4</mn></mrow></mfrac></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><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mrow><mfrac><mn>1</mn><mrow><mn>3</mn></mrow></mfrac></mrow></mrow><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo><mfenced open='(' close=')' separators=','><mrow><mn>3</mn><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow></mfenced><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>3</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo><mn>1</mn></mrow></mfenced><mrow><mfrac><mn>3</mn><mrow><mn>4</mn></mrow></mfrac></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='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mfrac><mn>1</mn><mrow><mn>3</mn></mrow></mfrac><mfrac><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>3</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo><mn>1</mn></mrow></mfenced><mrow><mfrac><mn>7</mn><mrow><mn>4</mn></mrow></mfrac></mrow></msup><mrow><mfenced open='(' close=')' separators=','><mrow><mfrac><mn>7</mn><mrow><mn>4</mn></mrow></mfrac></mrow></mfenced></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo><mi>C</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mfrac><mn>4</mn><mrow><mn>21</mn></mrow></mfrac><msup><mfenced open='(' close=')' separators=','><mrow><msup><mi>x</mi><mrow><mn>3</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo><mn>1</mn></mrow></mfenced><mrow><mfrac><mn>7</mn><mrow><mn>4</mn></mrow></mfrac></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&amp;plus;</mo><mi>C</mi></mrow></math>.</p>@
qu.2.2.solution=@
qu.2.2.algorithm=$a=rint(-5,5);
condition:ne($a,0);
$m=rint(2,10);
$n=rint(2,10);
$z=rint(5);
$f=switch($z,"x^($m)","sin(x)","cos(x)","sec(x)^2","csc(x)^2");
$g=switch(rint(6),"($n)*sin(x)+($a)","($n)*cos(x)+($a)","x^($n)+($a)",
"($n)*sqrt(x)+($a)","1/x^($n)+($a)","sqrt(x)+($a)");
$fg="subs(x=($g),$f)";
$gp="diff($g,x)";
$F=maple("printf(MathML[ExportPresentation](Int(($gp)/($n)*($fg),x)))");
$Q="($gp)/($n)*($fg)";
$ANS=switch($z,"(1/($n))*($g)^(($m)+1)/(($m)+1)","-1/($n)*cos($g)",
"1/($n)*sin($g)","(1/($n))*tan(($g))","-(1/($n))*cot($g)");@
qu.2.2.uid=bca67fd7-ea19-49b1-b8b0-ad990ac431e7@
qu.2.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Medium;
  Course=Introduction to Calculus I;
  Topic=Basic Integration;
  Sub-Topic=Chain Rule in Reverse;
@

qu.3.topic=substitution@

qu.3.1.mode=Inline@
qu.3.1.name=substitution/(x+a)^n@
qu.3.1.comment=@
qu.3.1.editing=useHTML@
qu.3.1.solution=@
qu.3.1.algorithm=$a=rint(1,5);
$b=rint(1,5);
$n=switch(rint(4),"1/2","2/3","3/2",3,4);
$f="(x-($a))/(x+($b))^($n)";
$u="x+$b";
$g="(u-($a)-($b))/u^($n)";
$M=maple("
MathML[ExportPresentation](Int($f,x)),
MathML[ExportPresentation](Int($g,u))
");
$F=switch(0,$M);
$G=switch(1,$M);@
qu.3.1.uid=6eed2552-c961-4fee-8d89-a4044679114f@
qu.3.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus I;
  Topic=Basic Integration;
  Sub-Topic=Substitution;
@
qu.3.1.weighting=1,1,1,1@
qu.3.1.numbering=alpha@
qu.3.1.part.1.editing=useHTML@
qu.3.1.part.1.question=(Unset)@
qu.3.1.part.1.name=sro_id_1@
qu.3.1.part.1.answer=$u@
qu.3.1.part.1.mode=Formula@
qu.3.1.part.2.editing=useHTML@
qu.3.1.part.2.question=(Unset)@
qu.3.1.part.2.name=sro_id_2@
qu.3.1.part.2.answer=dx@
qu.3.1.part.2.mode=Formula@
qu.3.1.part.3.name=sro_id_3@
qu.3.1.part.3.maple_answer=$g@
qu.3.1.part.3.editing=useHTML@
qu.3.1.part.3.question=(Unset)@
qu.3.1.part.3.libname=@
qu.3.1.part.3.mode=Maple@
qu.3.1.part.3.allow2d=1@
qu.3.1.part.3.plot=@
qu.3.1.part.3.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.3.1.part.3.type=formula@
qu.3.1.part.4.name=sro_id_4@
qu.3.1.part.4.maple_answer=(x+($b))^(2-($n))/(2-($n))-(($a)+($b))*(x+($b))^(1-($n))/(1-($n)) + C@
qu.3.1.part.4.editing=useHTML@
qu.3.1.part.4.question=(Unset)@
qu.3.1.part.4.libname=@
qu.3.1.part.4.mode=Maple@
qu.3.1.part.4.allow2d=1@
qu.3.1.part.4.plot=@
qu.3.1.part.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.3.1.part.4.type=formula@
qu.3.1.question=<p>In order to find $F, we should use the substitution method.</p><p>&nbsp;</p><p>(a) Let <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math><span> </span><1><span> </span>so that</p><p>&nbsp;</p><p>(b) <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>du</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math><span> </span><2></p><p>&nbsp;</p><p><span>(c) The transformed integral is</span><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo></mrow></math><span> </span><3><span> </span><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>du</mi></mrow></math></span>.</p><p><br /><span>(Just enter the integrand without </span><span><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>du</mi></mrow></math></span>. As you can see, </span><span><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>du</mi></mrow></math></span> is already entered by TA!)</span></p><p>&nbsp;</p><p>(d) Finally, after resubstituting for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>u</mi></mrow></math>in terms of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math>and adding <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></math>, the answer is</p><p>$F=<span> </span><4></p><p><span> </span></p><p><span>Avoid frustration. Work out your answer to (d) carefully on paper and only then type it in. If you are using text entry, use the PREVIEW button to check that all your *'s, ^'s, and (,) 's are in place. </span></p>@

qu.3.2.mode=Inline@
qu.3.2.name=substitution(x+a)^n@
qu.3.2.comment=@
qu.3.2.editing=useHTML@
qu.3.2.solution=@
qu.3.2.algorithm=$a=rint(1,5);
$b=rint(1,5);
$n=switch(rint(4),"1/2","2/3","3/4",10);
$f="(x-($a))*(x+($b))^($n)";
$u="x+$b";
$g="(u-($a)-($b))*u^($n)";
$M=maple("
MathML[ExportPresentation](Int($f,x)),
MathML[ExportPresentation](Int($g,u))
");
$F=switch(0,$M);
$G=switch(1,$M);@
qu.3.2.uid=8b8357f3-854f-418e-b51c-49ddd9830461@
qu.3.2.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Easy;
  Course=Introduction to Calculus I;
  Topic=Basic Integration;
  Sub-Topic=Substitution;
@
qu.3.2.weighting=1,1,1,1@
qu.3.2.numbering=alpha@
qu.3.2.part.1.editing=useHTML@
qu.3.2.part.1.question=(Unset)@
qu.3.2.part.1.name=sro_id_1@
qu.3.2.part.1.answer=$u@
qu.3.2.part.1.mode=Formula@
qu.3.2.part.2.editing=useHTML@
qu.3.2.part.2.question=(Unset)@
qu.3.2.part.2.name=sro_id_2@
qu.3.2.part.2.answer=dx@
qu.3.2.part.2.mode=Formula@
qu.3.2.part.3.name=sro_id_3@
qu.3.2.part.3.maple_answer=$g@
qu.3.2.part.3.editing=useHTML@
qu.3.2.part.3.question=(Unset)@
qu.3.2.part.3.libname=@
qu.3.2.part.3.mode=Maple@
qu.3.2.part.3.allow2d=1@
qu.3.2.part.3.plot=@
qu.3.2.part.3.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.3.2.part.3.type=formula@
qu.3.2.part.4.name=sro_id_4@
qu.3.2.part.4.maple_answer=(x+($b))^(($n)+2)/(($n)+2)-(($a)+($b))*(x+($b))^(($n)+1)/(($n)+1)+C@
qu.3.2.part.4.editing=useHTML@
qu.3.2.part.4.question=(Unset)@
qu.3.2.part.4.libname=@
qu.3.2.part.4.mode=Maple@
qu.3.2.part.4.allow2d=1@
qu.3.2.part.4.plot=@
qu.3.2.part.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.3.2.part.4.type=formula@
qu.3.2.question=<p>In order to find $F, we should use the substitution method.</p><p>&nbsp;</p><p>(a) Let <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math><span> </span><1><span> </span>so that</p><p>&nbsp;</p><p>(b) <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>du</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math><span> </span><2></p><p><span> </span></p><p><span>(c) The transformed integral is</span><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo></mrow></math><span> </span><3><span> </span><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>du</mi></mrow></math>.</span></p><p>&nbsp;</p><p><span>(Just enter the integrand without </span><span><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>du</mi></mrow></math></span>. As you can see, </span><span><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>du</mi></mrow></math></span> is already entered by TA!)</span></p><p>&nbsp;</p><p>(d) Finally, after resubstituting for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>u</mi></mrow></math>in terms of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math>and adding <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></math>, the answer is</p><p>$F=<span> </span><4>.</p><p><span> </span></p><p><span>Avoid frustration. Work out your answer to (d) carefully on paper and only then type it in. If you are using text entry, use the PREVIEW button to check that all your *'s, ^'s, and (,) 's are in place. </span></p>@

qu.3.3.mode=Inline@
qu.3.3.name=substitution (ax+b)^n@
qu.3.3.comment=@
qu.3.3.editing=useHTML@
qu.3.3.solution=@
qu.3.3.algorithm=$a=rint(1,5);
$b=rint(1,5);
$c=rint(2,5);
condition:eq(gcd($b,$c),1);
$n=switch(rint(4),"1/2","2/3","3/4");
$f="(x+($a))*(($c)*x-($b))^($n)";
$u="$c*x-$b";
$g="(u+($b)+($c)*($a))*u^($n)/($c)^2";
$M=maple("
MathML[ExportPresentation](Int($f,x)),
MathML[ExportPresentation](Int($g,u))
");
$F=switch(0,$M);
$G=switch(1,$M);@
qu.3.3.uid=fa684ae4-f8ba-46f8-859a-6fdf88cd00a8@
qu.3.3.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Difficulty=Easy;
  Topic=Basic Integration;
  Sub-Topic=Substitution;
@
qu.3.3.weighting=1,1,1,1@
qu.3.3.numbering=alpha@
qu.3.3.part.1.editing=useHTML@
qu.3.3.part.1.question=(Unset)@
qu.3.3.part.1.name=sro_id_1@
qu.3.3.part.1.answer=$u@
qu.3.3.part.1.mode=Formula@
qu.3.3.part.2.editing=useHTML@
qu.3.3.part.2.question=(Unset)@
qu.3.3.part.2.name=sro_id_2@
qu.3.3.part.2.answer=($c)*dx@
qu.3.3.part.2.mode=Formula@
qu.3.3.part.3.name=sro_id_3@
qu.3.3.part.3.maple_answer=$g@
qu.3.3.part.3.editing=useHTML@
qu.3.3.part.3.question=(Unset)@
qu.3.3.part.3.libname=@
qu.3.3.part.3.mode=Maple@
qu.3.3.part.3.allow2d=1@
qu.3.3.part.3.plot=@
qu.3.3.part.3.maple=is(($ANSWER)-($RESPONSE) = 0);@
qu.3.3.part.3.type=formula@
qu.3.3.part.4.name=sro_id_4@
qu.3.3.part.4.maple_answer=1/($c)^2*((($c)*x-($b))^(($n)+2)/(($n)+2)+(($a)*($c)+($b))*(($c)*x-($b))^(($n)+1)/(($n)+1))+C@
qu.3.3.part.4.editing=useHTML@
qu.3.3.part.4.question=(Unset)@
qu.3.3.part.4.libname=@
qu.3.3.part.4.mode=Maple@
qu.3.3.part.4.allow2d=1@
qu.3.3.part.4.plot=@
qu.3.3.part.4.maple=if not(simplify(diff(($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.3.3.part.4.type=formula@
qu.3.3.question=<p>In order to find $F, we should use the substitution method.</p><p>&nbsp;</p><p>(a) Let <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>u</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math><span> </span><1><span> </span>so that</p><p>&nbsp;</p><p>(b) <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>du</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math><span> </span><2></p><p><span> </span></p><p><span>(c) The transformed integral is</span><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' largeop='true'>&Integral;</mo></mrow></math><span> </span><3><span> <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>du</mi></mrow></math>. (Don't forget to substitute for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>dx</mi></mrow></math>.)</span></span></p><p>&nbsp;</p><p><span>(Just enter the integrand without </span><span><span><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>du</mi></mrow></math></span></span>. As you can see, </span><span><span><span><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>du</mi></mrow></math> </span></span>is already entered by TA!)</span></p><p>&nbsp;</p><p>(d) Finally, after resubstituting for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>u</mi></mrow></math>in terms of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math>and adding <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi>C</mi></mrow></math>, the answer is</p><p>$F=<span> </span><4></p><p><span> </span></p><p><span>Avoid frustration. Work out your answer to (d) carefully on paper and only then type it in. If you are using text entry, use the PREVIEW button to check that all your *'s, ^'s, and (,) 's are in place. </span></p>@

