Weierstrass Substitution

Proof Technique
The Weierstrass Substitution, or the Tangent Half-Angle Subsitution, is an application of Integration by Substitution.

The substitution is:


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

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

It yields:

This can be stated:

Proof
Let $\displaystyle x = \tan \frac \theta 2$, $-\pi < \theta < \pi$.

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

Let $2u = \theta$.

Then $x = \tan u$.

Next:

Also,

Lastly,

Summarizing: