##DESCRIPTION
#KEYWORDS('integrals', 'trigonometry','substitution')
## use trig identities, then substitute.
##ENDDESCRIPTION

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

## BenBush tagged and PAID on 2-20-2004

## DBsubject('Calculus')
## DBchapter('Techniques of Integration')
## DBsection('Trigonometric Integrals')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('7.2')
## Problem1('1,2,5,18')

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

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

$n = random(4,20,2);

$ans = 1/($n + 1) - 2/($n + 3) + 1/($n + 5);

TEXT(EV2(<<EOT));

Evaluate the definite integral.
$BR \[ \int_0^{\pi/2} \sin^5 x \cos^{$n} x \,dx \]
$BR $BR \{ans_rule( 40) \}
$BR
EOT

ANS(num_cmp($ans));

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

##DESCRIPTION
#KEYWORDS('integrals', 'trigonometry','substitution')
## use trig identities, then substitution.
##ENDDESCRIPTION

## BenBush tagged and PAID on 2-20-2004

## DBsubject('Calculus')
## DBchapter('Techniques of Integration')
## DBsection('Trigonometric Integrals')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('7.2')
## Problem1('4,6,7,8,9,10,16,47')


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(2,99,1);
$b = random(2,99,1);

$funct = "$a/$b*sin($b*x) - $a/(3*$b)*(sin($b*x))**3";

TEXT(EV2(<<EOT));

Evaluate the indefinite integral.
$BR \[ \int $a\cos^3 ($b x)\,dx \]
$BR $BR \{ans_rule( 45) \}
$BR
EOT

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

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

##KEYWORDS('integrals', 'trigonometric')
##DESCRIPTION
## Evaluate an integral
##ENDDESCRIPTION

## AmberHolden tagged
## Shotwell cleaned

## DBsubject('Calculus')
## DBchapter('Techniques of Integration')
## DBsection('Trigonometric Integrals')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('7.2')
## Problem1('1,2,5')

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

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

TEXT(beginproblem());

$showPartialCorrectAnswers = 1;

$l = random(3,10,1);
$a = random(2,20,1);
$l1 = $l+1;
$l3 = $l+3;
$funct = "-cos($a * x)^($l1)/($l1 *$a)+ cos($a * x)^($l3)/($l3 * $a)";

BEGIN_TEXT

Evaluate the indefinite integral.
$BR \[ \int \sin^3($a x)\cos^{$l}($a x) dx \]
$BR 
Answer: \{ans_rule( 60) \} \(+\) \(C\)
$BR
END_TEXT


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

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

##DESCRIPTION
##KEYWORDS('integrals', 'trig', 'half angle identity', 'substitution')
## Use half angle identity and substitution
##ENDDESCRIPTION

## DBsubject('Calculus')
## DBchapter('Techniques of Integration')
## DBsection('Trigonometric Integrals')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('7.2')
## Problem1('13,14,48')

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;

$c = random(2,9,1);

$a = random(0,8,1);
$len = random(3,10,1);
$b= $a + $len;

$ans = $len/8 - (sin(4*$b*$c) - sin(4*$a*$c))/(32*$c);

TEXT(EV2(<<EOT));

Evaluate the definite integral.
$BR \[ \int_{$a}^{$b} \sin^2( $c x ) \cos^2 ($c x) \,dx \]
$BR $BR \{ans_rule( 45) \}
$BR
EOT

ANS(num_cmp($ans));

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

##DESCRIPTION
##KEYWORDS('integrals', 'trigonometry', 'substitution')
##use trig identities, then substitute.
##ENDDESCRIPTION

## BenBush tagged and PAID on 2-20-2004

## DBsubject('Calculus')
## DBchapter('Techniques of Integration')
## DBsection('Trigonometric Integrals')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('7.2')
## Problem1('4,6,7,8,10,16,47')

DOCUMENT();

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

$showPartialCorrectAnswers = 1;

$a = random(2,20,1);
$b = random(2,20,1);
$c = $a*$b;
$ans = "($c*3*x)/8 +$b*(cos($a*x))^3*sin($a*x)/4+3*$b*cos($a*x)*sin($a*x)/8";

TEXT(beginproblem());

BEGIN_TEXT
Evaluate the indefinite integral.
$BR \[ \int $c \cos^4($a x) dx \]
$BR $BR \{ans_rule( 60) \}  \(+C\)
END_TEXT

##set $PG_environment{'textbook'} in webworkCourse.ph
if (defined($textbook)) {
   if ($textbook eq "EllisGulick5") {
BEGIN_TEXT
$PAR
This is similar to problems 7-10 in section 7.2 of the text.
END_TEXT
}
}

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

ENDDOCUMENT();
________________________________________________________________________________

##DESCRIPTION
##KEYWORDS('integrals', 'trigonometry', 'substitution')
## use trig identities, then substitute.
##ENDDESCRIPTION

## BenBush tagged and PAID on 2-20-2004

## DBsubject('Calculus')
## DBchapter('Techniques of Integration')
## DBsection('Trigonometric Integrals')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('7.2')
## Problem1('25,26,27,28,29,30,33,34,37,38,45')



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;

$l = random(2,15,1);
$a = random(2,20,1);
$a3 = 3*$a;
$soln = (2**$l -1)/($l*$a);

TEXT(EV2(<<EOT));

Evaluate the definite integral.
$BR \[ \int_{0}^{\frac{\pi}{$a3}} \frac{\sec^{$l}($a x)}{\cot($a x)} dx
\]
$BR $BR \{ans_rule( 60) \}
$BR
[NOTE:  Remeber to enter all necessary *, (, and )  !!      $BR
Enter arctan(x) for  \( \tan^{-1} x \) , sin(x) for \( \sin x \) . ]
EOT

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

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

##KEYWORDS('integrals', 'trigonometric')
##DESCRIPTION
## Evaluate an indefinite integral
##ENDDESCRIPTION

## AmberHolden tagged
## Shotwell cleaned

## DBsubject('Calculus')
## DBchapter('Techniques of Integration')
## DBsection('Trigonometric Integrals')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('7.2')
## Problem1('16')

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

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

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

$a = random(3,7);
$b = random(2,20);
$a1 = $a -1;
$c = $a1*$b;
$funct = "$c/$a*((tan(x^$a)^3)/3+tan(x^$a))";

BEGIN_TEXT

Evaluate the indefinite integral.
$BR \[ \int $c x^{$a1}\sec^4(x^{$a}) \, dx \]
$BR 
Answer: \{ans_rule( 40) \} \(+ C\)
END_TEXT


ANS(fun_cmp($funct, mode=>'antider', limits=>[[-2,2]]));

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

##KEYWORDS('integrals', 'trig')
##DESCRIPTION
## Evaluate the integral
##ENDDESCRIPTION

## AmberHolden tagged
## Shotwell cleaned

## DBsubject('Calculus')
## DBchapter('Techniques of Integration')
## DBsection('Trigonometric Integrals')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('7.2')
## Problem1('25,26')

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

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

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

$a = random(4,9,1);

$funct = "(tan(x))^($a+1)/($a+1)";

BEGIN_TEXT
Evaluate the indefinite integral.
$BR \[ \int \tan^{$a} x \sec^2 x \,dx \]
$BR 
Answer: \{ans_rule( 45) \} \(  + C \)
$BR
END_TEXT

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

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

##KEYWORDS('integrals', 'trig')
##DESCRIPTION
## Evaluate an indefinite integral
##ENDDESCRIPTION

## AmberHolden tagged
## Shotwell cleaned

## DBsubject('Calculus')
## DBchapter('Techniques of Integration')
## DBsection('Trigonometric Integrals')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('7.2')
## Problem1('27,28,29')

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

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

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

$a = random(2,9,1);

$funct = "-(-(cos($a*x))^(-3)/3+1/cos($a*x))/$a";

BEGIN_TEXT

Evaluate the indefinite integral.
$BR \[ \int \tan^3 ($a x) \sec ($a  x) \,dx \]
$BR 
Answer: \{ans_rule( 45) \} \(  + C \)
$BR
END_TEXT


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

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

##DESCRIPTION
#KEYWORDS('integrals', 'trigonometry','substitution')
## use trig identities, then substitution
##ENDDESCRIPTION

## BenBush tagged and PAID on 2-20-2004

## DBchapter('Techniques of Integration')
## DBsection('Trigonometric Integrals')
## Date('6/3/2002')
## Author('Tangan Gao')
## Institution('csulb')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('7.2')
## Problem1('26')

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(3,9,1);
$b = random(3,6,1);
$c = $a*$b;
$p = arccos(-1);
$t = tan($p/$b);
$soln = $t*$t*$t*$t*$t*$t*$t/($a*7)+$t*$t*$t*$t*$t/($a*5);

BEGIN_TEXT

Evaluate the definite integral.
$BR \[ \int_{0}^{\frac{\pi}{$c}} \tan^4($a x) \sec^4($a x) dx \]
$BR $BR \{ans_rule( 30) \}
$BR
END_TEXT

$ans = $soln;
&ANS(std_num_cmp($ans));

##set $PG_environment{'textbook'} in webworkCourse.ph

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