Trigonometric Substitution

Integrals of the form

$$\int f(\cos\theta, \sin\theta)\,d\theta$$ (1)

can be solved by making the substitution $z = e^{i\theta}$ so that $dz = ie^{i\theta}\,d\theta$ and expressing

$$\cos\theta = {e^{i\theta}+e^{-i\theta}\over 2} = {z+z^{-1}\over 2}$$ (2)

$$\sin\theta = {e^{i\theta}-e^{-i\theta}\over 2i} = {z-z^{-1}\over 2i}.$$ (3)

The integral can then be solved by Contour Integration.

Alternatively, making the substitution $t\equiv \tan(\theta/2)$ transforms (1) into

$$\int f\left({{2t\over 1+t^2}, {1-t^2\over 1+t^2}}\right){2\,dt\over 1+t^2}.$$ (4)

The following table gives trigonometric substitutions which can be used to transform integrals involving square roots.

Form	Substitution
$\sqrt{a^2-x^2}$	$x = a \sin\theta$
$\sqrt{a^2+x^2}$	$x = a \tan\theta$
$\sqrt{x^2-a^2}$	$x = a \sec\theta$

See also Hyperbolic Substitution