Difference between revisions of "Trigonometric substitution"
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− | Trigonometric substitution is the technique of replacing | + | '''Trigonometric substitution''' is the technique of replacing [[variable]]s in equations with <math>\sin \theta\,</math> or <math>\cos {\theta}\,</math> or other [[function]]s from [[trigonometry]]. |
− | + | In [[calculus]], it is used to evaluate [[integral]]s of [[expression]]s such as <math>\sqrt{a^2+x^2},\sqrt{a^2-x^2}</math> or <math>\sqrt{x^2-a^2}</math> | |
− | + | == Examples == | |
+ | === <math>\sqrt{a^2+x^2}</math> === | ||
+ | To evaluate an expression such as <math>\int \sqrt{a^2+x^2}\,dx</math>, we make use of the identity <math>\tan^2x+1=\sec^2x</math>. Set <math>x=a\tan\theta</math> and the radical will go away. However, the <math>dx</math> will have to be changed in terms of <math>d\theta</math>: <math>dx=a\sec^2\theta</math> <math>d\theta</math> | ||
+ | === <math>\sqrt{a^2-x^2}</math> === | ||
+ | Making use of the identity <math>\sin^2\theta+\cos^2\theta=1</math>, simply let <math>x=a\sin\theta</math>. | ||
− | + | ||
+ | === <math>\sqrt{x^2-a^2}</math> === | ||
+ | Since <math>\sec^2(\theta)-1=\tan^2(\theta)</math>, let <math>x=a\sec\theta</math>. |
Revision as of 07:24, 20 August 2019
Trigonometric substitution is the technique of replacing variables in equations with or or other functions from trigonometry.
In calculus, it is used to evaluate integrals of expressions such as or
Contents
Examples
To evaluate an expression such as , we make use of the identity . Set and the radical will go away. However, the will have to be changed in terms of :
Making use of the identity , simply let .
Since , let .