## DESCRIPTION
## Calculus
## ENDDESCRIPTION

## KEYWORDS('improper' 'integral')
## Tagged by tda2d

## DBsubject('Calculus')
## DBchapter('Integrals')
## DBsection('Improper Integrals')
## Date('')
## Author('')
## Institution('Indiana')
## TitleText1('')
## EditionText1('')
## AuthorText1('')
## Section1('')
## Problem1('')

## 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=non_zero_random(-8, 8);
$b=non_zero_random(-8, 8);
$c=non_zero_random(-8, 8);

@functions=("$a x^2 + $b x + $c", "$a x^4", "(x-$a)^2+$b");
$choice=random(0, 2);

$soln = "divergent";

TEXT(EV2(<<EOT));
Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If not, state your answer as "divergent."

$BR \[ \int_{-\infty}^{\infty} ($functions[$choice]) dx \]
$BR $BR \{ans_rule( 30) \}
$BR
EOT
ANS(str_cmp($soln));

if ($choice == 0) {
&SOLUTION(EV3(<<'EOT'));
$SOL $BR $BR
\[
\begin{align*}
\int_{-\infty}^{\infty} ( $a x^2 + $b x + $c ) \; dx
&=	\left \left( \frac{$a x^3}{3} + \frac{$b x^2}{2} + $c x \right)
		\right|_{-\infty}^{\infty}	\\\\
&=	\lim_{x\to\infty} \left( \frac{$a x^3}{3}
			+ \frac{$b x^2}{2} + $c x \right)
		- \lim_{x\to -\infty} \left( \frac{$a x^3}{3}
			+ \frac{$b x^2}{2} + $c x \right)
\end{align*}
\]
Both of these clearly diverge, because they're polynomials,
so the integral in question diverges.
EOT
}
if ($choice == 1) {
&SOLUTION(EV3(<<'EOT'));
$SOL $BR $BR
\[
\begin{align*}
\int_{-\infty}^{\infty} $a x^4 \; dx
&=	\left \frac{$a x^5}{5} \right|_{-\infty}^{\infty}	\\\\
&=	\lim_{x\to\infty} \frac{$a x^5}{5}
		- \lim_{x\to -\infty} \frac{$a x^5}{5}
\end{align*}
\]
Both of these clearly diverge, because they're polynomials,
so the integral in question diverges.
EOT
}
if ($choice == 2) {
$soln_2a = 2 * $a;
$soln_bpa2 = $b + $a**2;
&SOLUTION(EV3(<<'EOT'));
$SOL $BR $BR
\[
\begin{align*}
\int_{-\infty}^{\infty} \left( ( x - $a )^2 + $b \right) \; dx
&=	\int_{-\infty}^{\infty} ( x^2 - $soln_2a x + $soln_bpa2 ) \; dx	\\\\
&=	\left \left( \frac{x^3}{3} - $a x^2 + $soln_bpa2 x \right)
		\right|_{-\infty}^{\infty}	\\\\
&=	\lim_{x\to\infty} \left( \frac{x^3}{3} - $a x^2 + $soln_bpa2 x \right)
		- \lim_{x\to\infty} \left( \frac{x^3}{3}
			- $a x^2 + $soln_bpa2 x \right)
\end{align*}
\]
Both of these clearly diverge, because they're polynomials,
so the integral in question diverges.
EOT
}

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