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| 1 : | jj | 209 | ##DESCRIPTION |
| 2 : | ##KEYWORDS('integrals', 'trig', 'substitution') | ||
| 3 : | ##ENDDESCRIPTION | ||
| 4 : | |||
| 5 : | DOCUMENT(); # This should be the first executable line in the problem. | ||
| 6 : | |||
| 7 : | loadMacros( | ||
| 8 : | "PG.pl", | ||
| 9 : | "PGbasicmacros.pl", | ||
| 10 : | "PGchoicemacros.pl", | ||
| 11 : | "PGanswermacros.pl", | ||
| 12 : | "PGauxiliaryFunctions.pl" | ||
| 13 : | ); | ||
| 14 : | |||
| 15 : | TEXT(&beginproblem); | ||
| 16 : | $showPartialCorrectAnswers = 1; | ||
| 17 : | |||
| 18 : | $a = random(2,9,1); | ||
| 19 : | $k = random(4,9,1); | ||
| 20 : | |||
| 21 : | $aa = $a*$a; | ||
| 22 : | $n = 2*$k + 3; | ||
| 23 : | $m = 2*$k + 1; | ||
| 24 : | $p = 2*$k; | ||
| 25 : | |||
| 26 : | $ans = "(1/($aa*$m)) * (x^2/($aa - x^2))**($m/2)"; | ||
| 27 : | |||
| 28 : | TEXT(EV2(<<EOT)); | ||
| 29 : | |||
| 30 : | Evaluate the indefinite integral | ||
| 31 : | $BR \[\int \frac{ x^{$p} \, dx}{($aa - x^2)^{$n/2}} \] | ||
| 32 : | $BR $BR \{ans_rule( 50) \} | ||
| 33 : | $BR | ||
| 34 : | EOT | ||
| 35 : | |||
| 36 : | &HINT(EV3(<<'EOT')); | ||
| 37 : | $HINT $BR | ||
| 38 : | |||
| 39 : | Make a trigonometric substitution. To attack the resulting | ||
| 40 : | integral, remember that \( (\sin x)/(\cos x) = \tan x \) | ||
| 41 : | and that \( 1/(\cos x) = \sec x \). | ||
| 42 : | EOT | ||
| 43 : | |||
| 44 : | &ANS(function_cmp_up_to_constant($ans)); | ||
| 45 : | |||
| 46 : | $soln_den = $aa*$m; | ||
| 47 : | |||
| 48 : | &SOLUTION(EV3(<<'EOT')); | ||
| 49 : | |||
| 50 : | $SOL $BR $BR | ||
| 51 : | |||
| 52 : | We recall from the table on page 484 that this circumstance calls for a | ||
| 53 : | substitution of \(x=$a \sin\theta\), and of course its inverse | ||
| 54 : | \(\theta=\sin^{-1}\left(\frac{x}{$a}\right)\). | ||
| 55 : | |||
| 56 : | \[ | ||
| 57 : | \begin{align*} | ||
| 58 : | \int \frac{x^{$p}dx}{\left($aa-x^2\right)^{$n/2}} | ||
| 59 : | &= \int \frac{x^{$p}dx}{\left(\sqrt{$aa-x^2}\right)^{$n}} \\\\ | ||
| 60 : | &= \int \frac{\left($a \sin\theta\right)^{$p} \cdot $a \cos\theta d\theta} | ||
| 61 : | {\left(\sqrt{$a^2-$a^2\sin^2\theta}\right)^{$n}} \\\\ | ||
| 62 : | &= \int \frac{$a^{$m}\sin^{$p}\theta\cos\theta d\theta} | ||
| 63 : | {\left(\sqrt{$a^2\cos^2\theta}\right)}^{$n}} \\\\ | ||
| 64 : | &= \int \frac{$a^{$m}\sin^{$p}\theta\cos\theta d\theta} | ||
| 65 : | {\left($a \cos\theta\right)^{$n}} \\\\ | ||
| 66 : | &= \frac{1}{$aa} \int \tan^{$p}\theta \sec^2\theta d\theta \\\\ | ||
| 67 : | &= \frac{\tan^{$m}\theta}{$aa \cdot $m} \\\\ | ||
| 68 : | &= \frac{\tan^{$m}\left(\sin^{-1}\left(\frac{x}{$a}\right)\right)} | ||
| 69 : | {$soln_den} | ||
| 70 : | \end{align*} | ||
| 71 : | \] | ||
| 72 : | |||
| 73 : | Now this is not in the same form as the answer given by WeBWorK, so to see | ||
| 74 : | that they're the same, we can perform the following manipulations: | ||
| 75 : | |||
| 76 : | \[ | ||
| 77 : | \begin{align*} | ||
| 78 : | \frac{\tan^{$m}\left(\sin^{-1}\left(\frac{x}{$a}\right)\right)}{$soln_den} | ||
| 79 : | &= \frac{ \left( | ||
| 80 : | \frac{\sin\left(\sin^{-1}\left(\frac{x}{$a}\right)\right)} | ||
| 81 : | {\cos\left(\sin^{-1}\left(\frac{x}{$a}\right)\right)} | ||
| 82 : | \right)^{$m} } { $soln_den } \\\\ | ||
| 83 : | &= \frac{ \left( \frac{ \left(\frac{x}{$a}\right)^2 } | ||
| 84 : | { \cos^2\left(\sin^{-1}\left(\frac{x}{$a}\right)\right) } | ||
| 85 : | \right)^{\frac{$m}{2}} } { $soln_den } \\\\ | ||
| 86 : | &= \frac{ \left( \frac{ \left(\frac{x}{$a}\right)^2 } | ||
| 87 : | { 1 - \sin^2\left(\sin^{-1}\left(\frac{x}{$a}\right)\right) } | ||
| 88 : | \right)^{\frac{$m}{2}} } { $soln_den } \\\\ | ||
| 89 : | &= \frac{ \left( \frac{ \left(\frac{x}{$a}\right)^2 } | ||
| 90 : | { 1 - \left(\frac{x}{$a}\right)^2 } | ||
| 91 : | \right)^{\frac{$m}{2}} } { $soln_den } \\\\ | ||
| 92 : | &= \frac{ \left( \frac{x^2}{$a^2-x^2} \right)^{$m/2} } { $soln_den } | ||
| 93 : | \end{align*} | ||
| 94 : | \] | ||
| 95 : | |||
| 96 : | EOT | ||
| 97 : | |||
| 98 : | ENDDOCUMENT(); # This should be the last executable line in the problem. | ||
| 99 : |
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