Trigonometric substitution

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 $x=a\sec\theta$.

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