# 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:

 $\displaystyle \sin \theta$ $=$ $\displaystyle \frac {2 u} {1 + u^2}$ $\displaystyle \cos \theta$ $=$ $\displaystyle \frac {1 - u^2} {1 + u^2}$ $\displaystyle \frac {\d \theta} {\d u}$ $=$ $\displaystyle \frac 2 {1 + u^2}$

This can be stated:

$\displaystyle \int \map F {\sin \theta, \cos \theta} \rd \theta = 2 \int \map F {\frac {2 u} {1 + u^2}, \frac {1 - u^2} {1 + u^2} } \frac {d u} {1 + u^2}$

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

## Proof

Let $u = \tan \dfrac \theta 2$ for $-\pi < \theta < \pi$.

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

Then:

 $\displaystyle u$ $=$ $\displaystyle \tan \dfrac \theta 2$ $\displaystyle \leadsto \ \$ $\displaystyle \theta$ $=$ $\displaystyle 2 \tan^{-1} u$ Definition of Inverse Tangent $\displaystyle \leadsto \ \$ $\displaystyle \dfrac {\d \theta} {\d u}$ $=$ $\displaystyle \dfrac 2 {1 + u^2}$ Derivative of Arctangent Function and Derivative of Constant Multiple $\displaystyle \sin \theta$ $=$ $\displaystyle \dfrac {2 u} {1 + u^2}$ Tangent Half-Angle Substitution for Sine $\displaystyle \cos \theta$ $=$ $\displaystyle \dfrac {1 - u^2} {1 + u^2}$ Tangent Half-Angle Substitution for Cosine

The result follows from Integration by Substitution.

$\blacksquare$

## Also known as

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

## Source of Name

This entry was named for Karl Theodor Wilhelm Weierstrass.