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 $d\theta$ : $dx=a\sec^2\theta$ $d\theta$ .