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Trigonometric substitution

From AoPSWiki

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}

Contents

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.


\sqrt{a^2-x^2}

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


\sqrt{x^2-a^2}

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



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