## DESCRIPTION
## Calculus: The Substitution Rule
## ENDDESCRIPTION

## KEYWORDS('calculus', 'integrals', 'substitution rule')
## Tagged by XW

## DBsubject('Calculus')
## DBchapter('Integrals')
## DBsection('The Substitution Rule')
## Date('6/5/2005')
## Author('Jeff Holt')
## Institution('UVA')
## TitleText1('Calculus')
## EditionText1('5e')
## AuthorText1('Stewart')
## Section1('5.5')
## Problem1('23')

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a = random(2,5,1);
$power = $a - 1;
$coeff = 1 / $a;
$funct = "$coeff * e^(x^$a)";

TEXT(EV2(<<EOT));

Evaluate the indefinite integral.
$BR \[ \int x^$power e^{x^$a} dx \]
$BR $BR Integral = \{ans_rule( 30) \} + C
$BR
EOT

$ans = $funct;
ANS(fun_cmp($ans, mode=>"antider"));

ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________

## DESCRIPTION
## Calculus: The Substitution Rule
## ENDDESCRIPTION

## KEYWORDS('calculus', 'integrals', 'substitution rule')
## Tagged by XW

## DBsubject('Calculus')
## DBchapter('Integrals')
## DBsection('The Substitution Rule')
## Date('6/5/2005')
## Author('Jeff Holt')
## Institution('UVA')
## TitleText1('Calculus')
## EditionText1('5e')
## AuthorText1('Stewart')
## Section1('5.5')
## Problem1('35')

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a = random(1,5,1);
$b = 2 * $a;
$c = 1 + $a**2;
$funct = "1/2 * ln(x^2 + $b * x + $c)";

TEXT(EV2(<<EOT));

Evaluate the indefinite integral.
$BR \[ \int \frac{x + $a}{x^2 + $b x +$c} \, dx \]
$BR $BR Integral =  \{ans_rule( 30) \} + C
$BR
EOT

$ans = $funct;
ANS(fun_cmp($ans, mode=>"antider"));

ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________

## DESCRIPTION
## Calculus: The Substitution Rule
## ENDDESCRIPTION

## KEYWORDS('calculus', 'integrals', 'substitution rule')
## Tagged by XW

## DBsubject('Calculus')
## DBchapter('Integrals')
## DBsection('The Substitution Rule')
## Date('6/5/2005')
## Author('Jeff Holt')
## Institution('UVA')
## TitleText1('Calculus')
## EditionText1('5e')
## AuthorText1('Stewart')
## Section1('5.5')
## Problem1('25')

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a = random(1,5,2);
$b = random(1,3,1);

TEXT(EV2(<<EOT));
Evaluate the indefinite integral.
$BR
\[ \int \sec x \tan x ($b + \sec x)^{$a/2} \, dx \]
$BR $BR
Integral = \{ ans_rule(40) \} + C
$BR
[NOTE:  Remember to enter all necessary *, (, and )  !!      $BR
Enter arctan(x) for  \( \tan^{-1} x \) , sin(x) for \( \sin x \) . ]
EOT

$ans = "2/($a+2) * ( $b + sec(x) )^( ($a+2)/2 )";
ANS(fun_cmp($ans, mode=>"antider"));

ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________

## DESCRIPTION
## Calculus: The Substitution Rule
## ENDDESCRIPTION

## KEYWORDS('calculus', 'integrals', 'substitution rule')
## Tagged by XW

## DBsubject('Calculus')
## DBchapter('Integrals')
## DBsection('The Substitution Rule')
## Date('6/5/2005')
## Author('Jeff Holt')
## Institution('UVA')
## TitleText1('Calculus')
## EditionText1('5e')
## AuthorText1('Stewart')
## Section1('5.5')
## Problem1('5')

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a = random(2,8,1);
$b = random(2,4,1);
$c = 1 - $b;
$d = random(1,15,1);
$power = $c;
$coeff1 = 1 / $a;
$coeff2 = 1 / $c;
$funct = "$coeff1 * $coeff2 * ($a * x + $d)^$power";

TEXT(EV2(<<EOT));

Evaluate the integral by making the given substitution.
$BR \[ \int \frac {dx}{($a x + $d)^$b} \]$BR \( u = $a x + $d \)
$BR $BR \{ans_rule( 30) \} +C
$BR
EOT

$ans = $funct;
ANS(fun_cmp($ans, mode=>"antider"));

ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________

## DESCRIPTION
## Calculus: The Substitution Rule
## ENDDESCRIPTION

## KEYWORDS('calculus', 'integrals', 'substitution rule')
## Tagged by XW

## DBsubject('Calculus')
## DBchapter('Integrals')
## DBsection('The Substitution Rule')
## Date('6/5/2005')
## Author('Jeff Holt')
## Institution('UVA')
## TitleText1('Calculus')
## EditionText1('5e')
## AuthorText1('Stewart')
## Section1('5.5')
## Problem1('12')

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a = random(2,6,2);
$b = random(1,7,1);
$c = $a - 1;
$coeff = 1 / $a;
$funct = "$coeff * ln(x^$a + $b)";

TEXT(EV2(<<EOT));

Evaluate the indefinite integral.
$BR \[ \int \frac{x^$c}{x^$a + $b} \, dx \]
$BR $BR \{ans_rule( 30) \} + C
$BR
[NOTE:  Remember to enter all necessary *, (, and )  !!      $BR
Enter arctan(x) for  \( \tan^{-1} x \) , sin(x) for \( \sin x \) . ]
EOT

$ans = $funct;
ANS(fun_cmp($ans, mode=>"antider"));

ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________

## DESCRIPTION
## Calculus: The Substitution Rule
## ENDDESCRIPTION

## KEYWORDS('calculus', 'integrals', 'substitution rule')
## Tagged by XW

## DBsubject('Calculus')
## DBchapter('Integrals')
## DBsection('The Substitution Rule')
## Date('6/5/2005')
## Author('Jeff Holt')
## Institution('UVA')
## TitleText1('Calculus')
## EditionText1('5e')
## AuthorText1('Stewart')
## Section1('5.5')
## Problem1('21')

## Before doing anything, we must import the macro definitions on the next few lines.

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a = random(2,9,1);
$power = $a + 1;
$coeff = 1 / $power;
$funct = "(ln(x)^$power)/$power ";

TEXT(EV2(<<EOT));

Evaluate the indefinite integral.
$BR \[ \int \frac{(\ln(x))^$a}{x} dx \]
$BR $BR Integral = \{ans_rule( 30) \} + C
$BR
EOT

$ans = $funct;
ANS(fun_cmp($ans, limits=>[1,2], mode=>"antider", vars=>'x'));

ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________

## DESCRIPTION
## Calculus: The Substitution Rule
## ENDDESCRIPTION

## KEYWORDS('calculus', 'integrals', 'substitution rule')
## Tagged by XW

## DBsubject('Calculus')
## DBchapter('Integrals')
## DBsection('The Substitution Rule')
## Date('6/5/2005')
## Author('Jeff Holt')
## Institution('UVA')
## TitleText1('Calculus')
## EditionText1('5e')
## AuthorText1('Stewart')
## Section1('5.5')
## Problem1('29')

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a = random(2,6,1);
$b = random(1,9,1);
$c = $a + 1;
$funct = "$b / $c * (sin(x))^$c";

TEXT(EV2(<<EOT));

Evaluate the indefinite integral.
$BR \[ \int $b \sin ^$a x \cos x \; dx \]
$BR $BR Integral = \{ans_rule( 30) \} + C
$BR
EOT

$ans = $funct;
ANS(fun_cmp($ans, mode=>"antider"));

ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________

## DESCRIPTION
## Calculus: The Substitution Rule
## ENDDESCRIPTION

## KEYWORDS('calculus', 'integrals', 'substitution rule')
## Tagged by XW

## DBsubject('Calculus')
## DBchapter('Integrals')
## DBsection('The Substitution Rule')
## Date('6/5/2005')
## Author('Jeff Holt')
## Institution('UVA')
## TitleText1('Calculus')
## EditionText1('5e')
## AuthorText1('Stewart')
## Section1('5.5')
## Problem1('34')

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a = random(2,6,1);
$b = $a-1;

TEXT(EV2(<<EOT));
Evaluate the indefinite integral.
$BR
\[ \int \frac{ \cos (\pi/x^{$b}) }{ x^{$a} } \, dx \]
$BR $BR
Integral = \{ ans_rule(30) \} + C
$BR
[NOTE:  Remember to enter all necessary *, (, and )  !!      $BR
Enter arctan(x) for  \( \tan^{-1} x \) , sin(x) for \( \sin x \) . ]
EOT

$ans = "-1/(pi*$b) * sin(pi/x^$b)";
ANS(fun_cmp($ans, mode=>"antider"));

ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________

## DESCRIPTION
## Calculus: The Substitution Rule
## ENDDESCRIPTION

## KEYWORDS('calculus', 'integrals', 'substitution rule')
## Tagged by XW

## DBsubject('Calculus')
## DBchapter('Integrals')
## DBsection('The Substitution Rule')
## Date('6/5/2005')
## Author('Jeff Holt')
## Institution('UVA')
## TitleText1('Calculus')
## EditionText1('5e')
## AuthorText1('Stewart')
## Section1('5.5')
## Problem1('38')

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a = random(2,4,1);
$b = 2*$a-1;

TEXT(EV2(<<EOT));
Evaluate the indefinite integral.
$BR
\[ \int x^{$b} \sqrt[3]{x^$a + 1} \;  \, dx \]
$BR $BR
Integral = \{ ans_rule(55) \} + C
$BR
[NOTE:  Remember to enter all necessary *, (, and )  !!      $BR
Enter arctan(x) for  \( \tan^{-1} x \) , sin(x) for \( \sin x \) . ]
EOT

$ans = "3/(7*$a) * (x^{$a} + 1)**(7/3) - 3/(4*$a) * (x^{$a} + 1)**(4/3)";
ANS(fun_cmp($ans, mode=>"antider"));

ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________

## DESCRIPTION
## Calculus: The Substitution Rule
## ENDDESCRIPTION

## KEYWORDS('calculus', 'integrals', 'substitution rule')
## Tagged by XW

## DBsubject('Calculus')
## DBchapter('Integrals')
## DBsection('The Substitution Rule')
## Date('6/5/2005')
## Author('Jeff Holt')
## Institution('UVA')
## TitleText1('Calculus')
## EditionText1('5e')
## AuthorText1('Stewart')
## Section1('5.5')
## Problem1('50')

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a = random(3,7,4);
$u = 21/$a;

TEXT(EV2(<<EOT));
Evaluate
$BR
\[ \int_{0}^{$u} \sqrt{ 4 + $a x } \; dx . \]
$BR $BR
Definite Integral = \{ ans_rule(35) \}
$BR
EOT

$ans = 78/$a;
ANS(num_cmp($ans));

ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________

## DESCRIPTION
## Calculus: The Substitution Rule
## ENDDESCRIPTION

## KEYWORDS('calculus', 'integrals', 'substitution rule')
## Tagged by XW

## DBsubject('Calculus')
## DBchapter('Integrals')
## DBsection('The Substitution Rule')
## Date('6/5/2005')
## Author('Jeff Holt')
## Institution('UVA')
## TitleText1('Calculus')
## EditionText1('5e')
## AuthorText1('Stewart')
## Section1('5.5')
## Problem1('52')

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a = random(2,4,1);
$b = $a-1;

TEXT(EV2(<<EOT));
Evaluate
$BR
\[ \int_{0}^{ \sqrt[$a]{\pi} } x^{$b} \cos( x^{$a} ) \, dx . \]
$BR $BR
Definite Integral = \{ ans_rule(35) \}
$BR
EOT

$ans = 0;
ANS(num_cmp($ans));

ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________

## DESCRIPTION
## Calculus: The Substitution Rule
## ENDDESCRIPTION

## KEYWORDS('calculus', 'integrals', 'substitution rule')
## Tagged by XW

## DBsubject('Calculus')
## DBchapter('Integrals')
## DBsection('The Substitution Rule')
## Date('6/5/2005')
## Author('Jeff Holt')
## Institution('UVA')
## TitleText1('Calculus')
## EditionText1('5e')
## AuthorText1('Stewart')
## Section1('5.5')
## Problem1('55')

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a = random(3,7,1);
$b = $a-1;

TEXT(EV2(<<EOT));
Evaluate
$BR
\[ \int_{1}^{4} \frac{ 1 }{ x^{$a} }
\sqrt{ 1 + \frac{ 1 }{ x^{$b} } } \, dx . \]
$BR
If the integral does not exist, enter DNE.
$BR
Definite Integral = \{ ans_rule(35) \}
$BR
EOT

$ans = 2/($b*3) * ( 2**(3/2) - ( 1 + 1/(4**$b) )**(3/2) );
ANS(num_cmp($ans));

ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________

## DESCRIPTION
## Calculus: The Substitution Rule
## ENDDESCRIPTION

## KEYWORDS('calculus', 'integrals', 'substitution rule')
## Tagged by XW

## DBsubject('Calculus')
## DBchapter('Integrals')
## DBsection('The Substitution Rule')
## Date('6/5/2005')
## Author('Jeff Holt')
## Institution('UVA')
## TitleText1('Calculus')
## EditionText1('5e')
## AuthorText1('Stewart')
## Section1('5.5')
## Problem1('58')

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a = random(2,4,1);
$b = $a-1;
$u = random(1,2,1);

TEXT(EV2(<<EOT));
Evaluate
$BR
\[ \int_{0}^{$u} x^{$b} e^{ -x^{$a} } \, dx . \]
$BR
If the integral does not exist, enter DNE.
$BR
Definite Integral = \{ ans_rule(35) \}
$BR
EOT

$ans = 1/$a * ( 1 - ( exp( -($u**$a) ) ) );
ANS(num_cmp($ans));

ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________

## DESCRIPTION
## Calculus: The Substitution Rule
## ENDDESCRIPTION

## KEYWORDS('calculus', 'integrals', 'substitution rule')
## Tagged by XW

## DBsubject('Calculus')
## DBchapter('Integrals')
## DBsection('The Substitution Rule')
## Date('6/5/2005')
## Author('Jeff Holt')
## Institution('UVA')
## TitleText1('Calculus')
## EditionText1('5e')
## AuthorText1('Stewart')
## Section1('5.5')
## Problem1('63')

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a = random(1,2,1);
$u = random(3,4,1);

TEXT(EV2(<<EOT));
Evaluate
$BR
\[ \int_{$a}^{$u} x \sqrt{ x - $a} \; dx . \]
$BR
If the integral does not exist, enter DNE.
$BR
Definite Integral = \{ ans_rule(45) \}
$BR
EOT

$ans = 2/5 * ($u-$a)**(5/2) + 2*$a/3 * ($u-$a)**(3/2);
ANS(num_cmp($ans));

ENDDOCUMENT();        # This should be the last executable line in the problem.
________________________________________________________________________________

## DESCRIPTION
## Calculus: The Substitution Rule
## ENDDESCRIPTION

## KEYWORDS('calculus', 'integrals', 'substitution rule')
## Tagged by XW

## DBsubject('Calculus')
## DBchapter('Integrals')
## DBsection('The Substitution Rule')
## Date('6/5/2005')
## Author('Jeff Holt')
## Institution('UVA')
## TitleText1('Calculus')
## EditionText1('5e')
## AuthorText1('Stewart')
## Section1('5.5')
## Problem1('66')

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a = random(2,6,1);
$pi = arccos(-1);

TEXT(EV2(<<EOT));
Evaluate
$BR
\[ \int_{0}^{1/2} \frac{ $a \sin^{-1} x }{ \sqrt{ 1 - x^2 } } \, dx . \]
$BR
If the integral does not exist, enter DNE.
$BR
Definite Integral = \{ ans_rule(35) \}
$BR
EOT

$ans = $a/2 * ($pi/6)**2;
ANS(num_cmp($ans));

ENDDOCUMENT();        # This should be the last executable line in the problem.