Trigonometric substitution

Trigonometric substitution is the technique of replacing unknown variables in equations with $\sin \theta\,$ or $\cos {\theta}\,$

In calculus, it is used to evaluate integrals with expressions such as $\sqrt{a^2+x^2},\sqrt{a^2-x^2}$

and

$\sqrt{a^2+x^2}$

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 $x=a\sec\theta$ .

Examples

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Trigonometric substitution

Contents

$\sqrt{a^2+x^2}$

$\sqrt{a^2-x^2}$

$\sqrt{x^2-a^2}$

Examples