Difference between revisions of "Trigonometric substitution"
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Trigonometric substitution is the technique of replacing unknown variables in equations with <math>\sin \theta\,</math> or <math>\cos {\theta}\,</math> | Trigonometric substitution is the technique of replacing unknown variables in equations with <math>\sin \theta\,</math> or <math>\cos {\theta}\,</math> | ||
− | + | In calculus, it is used to evaluate integrals with expressions such as <math>\sqrt{a^2+x^2},\sqrt{a^2-x^2}</math> | |
+ | and <math>\sqrt{x^2-a^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. | ||
− | + | ||
+ | == <math>\sqrt{a^2-x^2}</math> == | ||
+ | Making use of the identity <math>\displaystyle\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>\displaystyle\sec^2(\theta)-1=\tan^2(\theta)</math>, let <math>x=a\sec\theta</math>. | ||
+ | |||
+ | == Examples == | ||
+ | |||
+ | |||
+ | |||
+ | {{stub}} |
Revision as of 00:39, 8 July 2006
Trigonometric substitution is the technique of replacing unknown variables in equations with or
In calculus, it is used to evaluate integrals with expressions such as
and![]()
Contents
To evaluate an expression such as , we make use of the identity
. Set
and the radical will go away.
Making use of the identity , simply let
.
Since , let
.
Examples
This article is a stub. Help us out by expanding it.