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# Annotation of /trunk/NationalProblemLibrary/Indiana/Indiana_setIntegrals10InvTrig/ur_in_10_3.pg

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 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))**($m/2)"; 42 : 43 : TEXT(EV2(<"antider")); 60 : jj 209 61 :$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 :

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