Weierstrass Substitution

Proof Technique
The Weierstrass Substitution is an application of Integration by Substitution.

The substitution is:


 * $u \leftrightarrow \tan \dfrac \theta 2$

for $-\pi < \theta < \pi$, $u \in \R$.

It yields:

This can be stated:


 * $\displaystyle \int F \left({\sin \theta, \cos \theta}\right)\ \mathrm d \theta = 2 \int F \left({\frac {2 u} {1 + u^2}, \frac {1 - u^2} {1 + u^2} }\right) \frac {\mathrm d u} {1 + u^2}$

where $u = \tan \dfrac \theta 2$.

Proof
Let $\displaystyle u = \tan \frac \theta 2$ for $-\pi < \theta < \pi$.

From Shape of Tangent Function, this substitution is valid for all real $u$.

Then:

The result follows from Integration by Substitution.

Also known as
This technique is also known as tangent half-angle subsitution.

Also see

 * Hyperbolic Tangent Half-Angle Substitution