##KEYWORDS('integrals', 'partial fractions')
##DESCRIPTION
## Evaluate an indefinite integral.
##ENDDESCRIPTION

## AmberHolden tagged
## Shotwell cleaned

## DBsubject('Calculus')
## DBchapter('Techniques of Integration')
## DBsection('Integration by Partial Fractions')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('7.4')
## Problem1('15,16,18')

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

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

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

$a=random(-3,0,1);
$b=random(1,4,1);

BEGIN_TEXT
Evaluate the following integral.
$BR
$BR \[ \int \frac{1}{(x+$a)(x + $b)} dx \]
$BR Answer: \{ans_rule( 60) \} \(+C\)
END_TEXT
$d1= $a-$b;
$soln = FEQ("(ln(x + $b))/($d1) - (ln(x+$a))/($d1) ");

ANS(fun_cmp($soln,mode=>'antider',limits=>[10,100]));

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

##KEYWORDS('integrals', 'partial fractions')
##DESCRIPTION
## Use partial fractions to integrate.
##ENDDESCRIPTION

## AmberHolden tagged
## Shotwell cleaned

## DBsubject('Calculus')
## DBchapter('Techniques of Integration')
## DBsection('Integration by Partial Fractions')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('7.4')
## Problem1('17')

DOCUMENT();

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

$showPartialCorrectAnswers = 1;

$aa = random(3,6,1);
$bb = random(2,5,1);
$cc = random(1,5,1);
$dd = random(2,4,1);
$uu = random(1, 5,1);
$lin0 = $cc + $dd - $aa*$uu**2;
$const0 = $cc*$uu -$bb*$uu**2-$dd*$uu;
$u2 = $uu**2;
$num = nicestring([$aa, $bb, $lin0, $const0], ['x^3', 'x^2', 'x', '']);
TEXT(beginproblem());

BEGIN_TEXT

Consider the following indefinite integral. $BR
\[ \int \frac{$num}{x^2-$u2}\, dx\] $BR
$BR

The integrand decomposes into the form: 
\[ a x + b + \frac{c}{x-$uu} +\frac{d}{x+$uu}\] 
Compute the coefficients:
$BR
\(a =\) \{ ans_rule()\}
$BR
\(b =\)  \{ ans_rule()\}
$BR
\(c =\)  \{ ans_rule()\}
$BR
\(d =\)  \{ ans_rule()\}
$BR$BR
Now integrate term by term to evaluate the integral. $BR
Answer: \{ ans_rule(60)\} \(+\) \(C\)

END_TEXT

ANS(num_cmp($aa));
ANS(num_cmp($bb));
ANS(num_cmp($cc));
ANS(num_cmp($dd));
ANS(fun_cmp("$aa*x^2/2+$bb*x+$cc*ln(x-$uu)+$dd*ln(x+$uu)",mode=>'antider',limits=>[$uu+1,$uu+2]));

ENDDOCUMENT();
________________________________________________________________________________

##KEYWORDS('integrals', 'partial fractions')
##DESCRIPTION
## Evaluate a definite integral.
##ENDDESCRIPTION

## AmberHolden tagged
## Shotwell cleaned

## DBsubject('Calculus')
## DBchapter('Techniques of Integration')
## DBsection('Integration by Partial Fractions')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('7.4')
## Problem1('18,19,20')

DOCUMENT();

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


TEXT(beginproblem());

$m=random(1,3,1);
$n=random(1,6,1);
$a=random(-3,3,1);
$c=random(1,5,1);
$b=$a+$c;
$p=$m+$n;
$q=$n*$a+$m*$b;
$l=$a+$b;
$k=$a*$b;
$soln="$m*ln(1.5)+$n*ln(($c+3)/($c+2))";
$r=2-$a;
$s=3-$a;
$num=nicestring([$p, $q], ['x', '']);
$den=nicestring([1, $l, $k], ['x^2', 'x', '']);
BEGIN_TEXT

Evaluate the integral. 
$BR
$BR \[ \int_{$r}^{$s} \frac{$num}{$den} dx \]
$BR Answer: \{ans_rule(60) \}
END_TEXT
ANS(num_cmp($soln));

ENDDOCUMENT();
________________________________________________________________________________

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

## AmberHolden tagged
## Shotwell cleaned

## DBsubject('Calculus')
## DBchapter('Techniques of Integration')
## DBsection('Integration by Partial Fractions')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('7.4')
## Problem1('18,19,20')

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

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

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

$a = non_zero_random(-6,6,1);
$b = non_zero_random(-4,4,1);

$coef1 = 2 * $a;
$coef2 = $a * $a;
$coef3 = -$b;
$pol = nicestring([1, $coef1, $coef2], ['x^2', 'x', '']);
$soln = "$coef3/(x+$a)";

BEGIN_TEXT

Evaluate the indefinite integral.
$BR \[ \int \frac {$b}{$pol} dx \]$BR
$BR Answer: \{ans_rule(30) \} \(+\) \(C\)
$BR
END_TEXT

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

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

##KEYWORDS('integrals', 'partial fractions')
##DESCRIPTION
## Evaluate the indefinite integral.
##ENDDESCRIPTION

## AmberHolden tagged
## Shotwell cleaned

## DBsubject('Calculus')
## DBchapter('Techniques of Integration')
## DBsection('Integration by Partial Fractions')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('7.4')
## Problem1('25')

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

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

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

$a = random(-4,4,1);
$A = 0;
$B = non_zero_random(-5,5,1);
$C = -$B*$a;
$D = non_zero_random(-5,5,1);

$const = $a*$C;
$coef1 = -$B;
$coef2 = -$C;

$soln = "$coef1/x + $coef2/(2*x^2) + $D*ln(abs(x+$a))";
$num = nicestring([$D, $B, $const], ['x^3', 'x^2', '']);
$den = nicestring([1, $a], ['x^4', 'x^3']);
BEGIN_TEXT
Evaluate the indefinite integral.
$BR
$BR \[ \int \frac{$num}{$den} dx \]
$BR Answer: \{ ans_rule(50) \} \(+C\)
END_TEXT

ANS(fun_cmp($soln,mode=>'antider',limits=>[-10,6]));

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

##KEYWORDS('integrals','partial fractions', 'substitution')
##DESCRIPTION
## Evaluate an indefinite integral
##ENDDESCRIPTION

## AmberHolden tagged
## Shotwell cleaned

## DBsubject('Calculus')
## DBchapter('Techniques of Integration')
## DBsection('Integration by Partial Fractions')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('7.4')
## Problem1('43,44')


DOCUMENT();

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


$showPartialCorrectAnswers = 1;
$a = random(3,7,1);
$b = random(4,8,1);
if ($a == $b) {$b++;}

TEXT(beginproblem());


BEGIN_TEXT
Evaluate the indefinite integral. $BR
\[ \int \frac{1}{$a x +$b \sqrt{x}}\, dx\]
$BR
Answer: \{ ans_rule(40)\} \(+\) \(C\)
$BR
END_TEXT

ANS(fun_cmp("2*log($a*sqrt(x)+$b)/$a",mode=>'antider'));
ENDDOCUMENT();
________________________________________________________________________________

##KEYWORDS('integrals', 'partial fractions')
##DESCRIPTION
## Evaluate a definite integral.
##ENDDESCRIPTION

## AmberHolden tagged
## Shotwell cleaned

## DBsubject('Calculus')
## DBchapter('Techniques of Integration')
## DBsection('Integration by Partial Fractions')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('7.4')
## Problem1('15,16,18')



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

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

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

$a=random(-3,0,1);
$b=random(1,4,1);
$c=random(-4,4,1);
$d=random(5,9,1);
$e=random(5,9,1);
$function = sub { my $x=shift;
                  ($x**3 + $c)/( ($x+$d) * ($x + $e) );
                };
$soln=romberg($function, $a, $b);

BEGIN_TEXT
Evaluate the integral.
$BR
$BR \[ \int_{$a}^{$b} \frac{x^3 + $c}{(x+$d)(x+$e)} dx \]
$BR Answer:  \{ans_rule( 60) \}
END_TEXT

ANS(num_cmp($soln));

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

##KEYWORDS('integrals', 'partial fractions')
##DESCRIPTION
## Use partial fraction decomposition to evaluate an integral
##ENDDESCRIPTION

## AmberHolden tagged
## Shotwell cleaned

## DBsubject('Calculus')
## DBchapter('Techniques of Integration')
## DBsection('Integration by Partial Fractions')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('7.4')
## Problem1('31,32')


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

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

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

$a = non_zero_random(-5,5,1);
$av = abs($a);
$bs = random(2,3,1);
$b = $bs * $bs;
while ($b==$av) {
   $bs = random(2,3,1);
   $b = $bs * $bs;
   };

$generator = random(0,1,1);

if ($generator == 0) {
   $A = non_zero_random(-5,6,1);
   $B = non_zero_random(-5,6,1);
   $C = 0;
   while($A == -$B) {$B = non_zero_random(-5,5,1)};

   $soln = "$A*ln(abs(x+$a))+$B*ln(x^2+$b)/2";

   $x2_coeff = $A+$B;
   $x_coeff  = $a * $B;
   $const    = $A * $b;
    }

if ($generator == 1) {
   $A = non_zero_random(-5,5,1);
   $B = 0;
   $C = non_zero_random(-5,5,1);
   $temp1 = -$A*$b;
   $temp2 = $a*$C;
   while($temp1 == $temp2) {
      $C = non_zero_random(-5,5,1);
      $temp2 = $a*$C;
      };

   $soln = "$A*ln(abs(x+$a))+$C*arctan(x/$bs)/$bs";

   $x2_coeff = $A;
   $x_coeff  = $C;
   $const    = $A*$b + $a*$C;
   }
$num = nicestring([$x2_coeff, $x_coeff, $const], ['x^2', 'x', '']);

BEGIN_TEXT
The form of the partial fraction decomposition of a rational function is given below.
$BR
$BR \[ \frac{$num}{(x+$a)(x^2+$b)} 
        = \frac{A}{x+$a} + \frac{B x + C}{x^2+$b} \]
$BR
\( A = \) \{ ans_rule(10) \}
\( B = \) \{ ans_rule(10) \}
\( C = \) \{ ans_rule(10) \}
$BR$BR
Now evaluate the indefinite integral.
$BR
$BR \[ \int \frac{$num}{(x+$a)(x^2+$b)} dx \]
$BR Answer: \{ ans_rule(40) \} \(+C\)
END_TEXT
ANS(num_cmp($A));
ANS(num_cmp($B));
ANS(num_cmp($C));
ANS(fun_cmp($soln,mode=>'antider',limits=>[-10,6]));

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

##KEYWORDS('integrals', 'partial fractions')
##DESCRIPTION
## Use partial fraction decomp. to help evaluate an integral.
##ENDDESCRIPTION

## AmberHolden tagged
## Shotwell cleaned

## DBsubject('Calculus')
## DBchapter('Techniques of Integration')
## DBsection('Integration by Partial Fractions')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('7.4')
## Problem1('29,39')

DOCUMENT();

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

$showPartialCorrectAnswers = 1;

$aa = random(3,6,1);
$bb = random(2,5,1);
$cc = random(1,5,1);
$dd = random(2,4,1);
$uu = random(1, 5,1);
$cub0 = $cc+$bb;
$quad0 = $aa+$dd;
$u2 = $uu**2;
$lin0 = $bb*$u2;
$const0 = $aa*$u2;

TEXT(beginproblem());
$pol =   nicestring([$cub0, $quad0, $lin0, $const0], ['x^3', 'x^2', 'x', '']);

BEGIN_TEXT
$BR
Consider the following indefinite integral.
\[ \int \frac{$pol}{x^4+$u2 x^2}\, dx\]
$BR$BR
The integrand has partial fractions decomposition: $BR
\[\frac{a}{x^2} + \frac{b}{x} + \frac{cx + d}{x^2+$u2}\]
where
$BR
\(a =\) \{ ans_rule()\}
$BR
\(b =\) \{ ans_rule()\}
$BR
\(c =\) \{ ans_rule()\}
$BR
\(d =\) \{ ans_rule()\}
$BR$BR
Now integrate term by term to evaluate the integral.
$BR
Answer: \{ ans_rule(60)\} \(+C\)
END_TEXT

ANS(num_cmp($aa));
ANS(num_cmp($bb));
ANS(num_cmp($cc));
ANS(num_cmp($dd));
ANS(fun_cmp(
"-$aa/x+$bb*ln(x)+($cc/2)*ln(x^2+$u2)+($dd/$uu)*arctan(x/$uu)",mode=>'antider',limits=>[1,2]));

ENDDOCUMENT();
________________________________________________________________________________

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

## AmberHolden tagged
## Shotwell cleaned

## DBsubject('Calculus')
## DBchapter('Techniques of Integration')
## DBsection('Integration by Partial Fractions')
## Date('6/3/2002')
## Author('')
## Institution('')
## TitleText1('Calculus Early Transcendentals')
## EditionText1('4')
## AuthorText1('Stewart')
## Section1('7.4')
## Problem1('55,56')


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

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

$showPartialCorrectAnswers = 1;

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

TEXT(beginproblem());
BEGIN_TEXT
Evaluate the indefinite integral.
\[ \int \frac{x + $a}{x^2 + $b x +$c} \, dx \] $BR
Answer: \{ans_rule( 40) \} \(+C\)

END_TEXT

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

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