Difference between revisions of "Trigonometric substitution"
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== Examples == | == Examples == | ||
=== <math>\sqrt{a^2+x^2}</math> === | === <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> | + | 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> === | === <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>. | Making use of the identity <math>\sin^2\theta+\cos^2\theta=1</math>, simply let <math>x=a\sin\theta</math>. | ||
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=== <math>\sqrt{x^2-a^2}</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>. | Since <math>\sec^2(\theta)-1=\tan^2(\theta)</math>, let <math>x=a\sec\theta</math>. | ||
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{{stub}} | {{stub}} | ||
+ | [[Category: Trigonometry]] |
Latest revision as of 18:42, 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.