Trigonometric substitution: Difference between revisions

Latest revision as of 17:42, 30 May 2021

Trigonometric substitution is the technique of replacing variables in equations with $\sin \theta\,$ or $\cos {\theta}\,$ or other functions from trigonometry.

In calculus, it is used to evaluate integrals of expressions such as $\sqrt{a^2+x^2},\sqrt{a^2-x^2}$ or $\sqrt{x^2-a^2}$

Examples

$\sqrt{a^2+x^2}$

To evaluate an expression such as $\int \sqrt{a^2+x^2}\,dx$ , we make use of the identity $\tan^2x+1=\sec^2x$ . Set $x=a\tan\theta$ and the radical will go away. However, the $dx$ will have to be changed in terms of $d\theta$ : $dx=a\sec^2\theta$ $d\theta$

$\sqrt{a^2-x^2}$

Making use of the identity $\sin^2\theta+\cos^2\theta=1$ , simply let $x=a\sin\theta$ .

$\sqrt{x^2-a^2}$

Since $\sec^2(\theta)-1=\tan^2(\theta)$ , let $x=a\sec\theta$ .

This article is a stub. Help us out by expanding it.

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 == 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  d\theta</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]]

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