Weierstrass Substitution

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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:


$\sin \theta = \dfrac {2 u} {1 + u^2}$


$\cos \theta = \dfrac {1 - u^2} {1 + u^2}$


$\dfrac {\d \theta} {\d u} = \dfrac 2 {1 + u^2}$

The above results can be stated:

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

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

Also known as

The technique of Weierstrass Substitution is also known as tangent half-angle substitution.

Some sources call these results the tangent-of-half-angle formulae.

Other sources refer to them merely as the half-angle formulas or half-angle formulae.

Also see

Source of Name

This entry was named for Karl Theodor Wilhelm Weierstrass.