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>. | ||
| + | |||
| + | {{stub}} | ||
| + | [[Category: Trigonometry]] | ||
Latest revision as of 17:42, 30 May 2021
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 
.
This article is a stub. Help us out by expanding it.
