Parent Directory
| 
Revision Log
Revision 269 - (view) (download)
| 1 : | jjholt | 255 | ## DESCRIPTION |
| 2 : | ## Calculus | ||
| 3 : | ## ENDDESCRIPTION | ||
| 4 : | jj | 209 | |
| 5 : | jjholt | 255 | ## KEYWORDS('Substitution' 'Trig Substitution' 'Trigonometric Substitution') |
| 6 : | ## Tagged by tda2d | ||
| 7 : | |||
| 8 : | ## DBsubject('Calculus') | ||
| 9 : | ## DBchapter('Techniques of Integration') | ||
| 10 : | ## DBsection('Trig Substitution') | ||
| 11 : | ## Date('') | ||
| 12 : | ## Author('') | ||
| 13 : | ## Institution('Indiana') | ||
| 14 : | ## TitleText1('') | ||
| 15 : | ## EditionText1('') | ||
| 16 : | ## AuthorText1('') | ||
| 17 : | ## Section1('') | ||
| 18 : | ## Problem1('') | ||
| 19 : | |||
| 20 : | jj | 209 | DOCUMENT(); # This should be the first executable line in the problem. |
| 21 : | |||
| 22 : | loadMacros( | ||
| 23 : | "PG.pl", | ||
| 24 : | "PGbasicmacros.pl", | ||
| 25 : | "PGchoicemacros.pl", | ||
| 26 : | "PGanswermacros.pl", | ||
| 27 : | "PGauxiliaryFunctions.pl" | ||
| 28 : | ); | ||
| 29 : | |||
| 30 : | gage | 269 | TEXT(beginproblem()); |
| 31 : | jj | 209 | $showPartialCorrectAnswers = 1; |
| 32 : | |||
| 33 : | $a = random(2,9,1); | ||
| 34 : | $k = random(4,9,1); | ||
| 35 : | |||
| 36 : | $aa = $a*$a; | ||
| 37 : | $n = 2*$k + 3; | ||
| 38 : | $m = 2*$k + 1; | ||
| 39 : | $p = 2*$k; | ||
| 40 : | |||
| 41 : | $ans = "(1/($aa*$m)) * (x^2/($aa - x^2))**($m/2)"; | ||
| 42 : | |||
| 43 : | TEXT(EV2(<<EOT)); | ||
| 44 : | |||
| 45 : | Evaluate the indefinite integral | ||
| 46 : | $BR \[\int \frac{ x^{$p} \, dx}{($aa - x^2)^{$n/2}} \] | ||
| 47 : | $BR $BR \{ans_rule( 50) \} | ||
| 48 : | $BR | ||
| 49 : | EOT | ||
| 50 : | |||
| 51 : | &HINT(EV3(<<'EOT')); | ||
| 52 : | $HINT $BR | ||
| 53 : | |||
| 54 : | Make a trigonometric substitution. To attack the resulting | ||
| 55 : | integral, remember that \( (\sin x)/(\cos x) = \tan x \) | ||
| 56 : | and that \( 1/(\cos x) = \sec x \). | ||
| 57 : | EOT | ||
| 58 : | |||
| 59 : | gage | 269 | ANS(fun_cmp($ans, mode=>"antider")); |
| 60 : | jj | 209 | |
| 61 : | $soln_den = $aa*$m; | ||
| 62 : | |||
| 63 : | &SOLUTION(EV3(<<'EOT')); | ||
| 64 : | |||
| 65 : | $SOL $BR $BR | ||
| 66 : | |||
| 67 : | We recall from the table on page 484 that this circumstance calls for a | ||
| 68 : | substitution of \(x=$a \sin\theta\), and of course its inverse | ||
| 69 : | \(\theta=\sin^{-1}\left(\frac{x}{$a}\right)\). | ||
| 70 : | |||
| 71 : | \[ | ||
| 72 : | \begin{align*} | ||
| 73 : | \int \frac{x^{$p}dx}{\left($aa-x^2\right)^{$n/2}} | ||
| 74 : | &= \int \frac{x^{$p}dx}{\left(\sqrt{$aa-x^2}\right)^{$n}} \\\\ | ||
| 75 : | &= \int \frac{\left($a \sin\theta\right)^{$p} \cdot $a \cos\theta d\theta} | ||
| 76 : | {\left(\sqrt{$a^2-$a^2\sin^2\theta}\right)^{$n}} \\\\ | ||
| 77 : | &= \int \frac{$a^{$m}\sin^{$p}\theta\cos\theta d\theta} | ||
| 78 : | {\left(\sqrt{$a^2\cos^2\theta}\right)}^{$n}} \\\\ | ||
| 79 : | &= \int \frac{$a^{$m}\sin^{$p}\theta\cos\theta d\theta} | ||
| 80 : | {\left($a \cos\theta\right)^{$n}} \\\\ | ||
| 81 : | &= \frac{1}{$aa} \int \tan^{$p}\theta \sec^2\theta d\theta \\\\ | ||
| 82 : | &= \frac{\tan^{$m}\theta}{$aa \cdot $m} \\\\ | ||
| 83 : | &= \frac{\tan^{$m}\left(\sin^{-1}\left(\frac{x}{$a}\right)\right)} | ||
| 84 : | {$soln_den} | ||
| 85 : | \end{align*} | ||
| 86 : | \] | ||
| 87 : | |||
| 88 : | Now this is not in the same form as the answer given by WeBWorK, so to see | ||
| 89 : | that they're the same, we can perform the following manipulations: | ||
| 90 : | |||
| 91 : | \[ | ||
| 92 : | \begin{align*} | ||
| 93 : | \frac{\tan^{$m}\left(\sin^{-1}\left(\frac{x}{$a}\right)\right)}{$soln_den} | ||
| 94 : | &= \frac{ \left( | ||
| 95 : | \frac{\sin\left(\sin^{-1}\left(\frac{x}{$a}\right)\right)} | ||
| 96 : | {\cos\left(\sin^{-1}\left(\frac{x}{$a}\right)\right)} | ||
| 97 : | \right)^{$m} } { $soln_den } \\\\ | ||
| 98 : | &= \frac{ \left( \frac{ \left(\frac{x}{$a}\right)^2 } | ||
| 99 : | { \cos^2\left(\sin^{-1}\left(\frac{x}{$a}\right)\right) } | ||
| 100 : | \right)^{\frac{$m}{2}} } { $soln_den } \\\\ | ||
| 101 : | &= \frac{ \left( \frac{ \left(\frac{x}{$a}\right)^2 } | ||
| 102 : | { 1 - \sin^2\left(\sin^{-1}\left(\frac{x}{$a}\right)\right) } | ||
| 103 : | \right)^{\frac{$m}{2}} } { $soln_den } \\\\ | ||
| 104 : | &= \frac{ \left( \frac{ \left(\frac{x}{$a}\right)^2 } | ||
| 105 : | { 1 - \left(\frac{x}{$a}\right)^2 } | ||
| 106 : | \right)^{\frac{$m}{2}} } { $soln_den } \\\\ | ||
| 107 : | &= \frac{ \left( \frac{x^2}{$a^2-x^2} \right)^{$m/2} } { $soln_den } | ||
| 108 : | \end{align*} | ||
| 109 : | \] | ||
| 110 : | |||
| 111 : | EOT | ||
| 112 : | |||
| 113 : | ENDDOCUMENT(); # This should be the last executable line in the problem. | ||
| 114 : |
| aubreyja at gmail dot com | ViewVC Help |
| Powered by ViewVC 1.0.9 |