Definite Integral from 0 to Half Pi of Reciprocal of One plus Power of Tan x

Theorem

 * $\ds \int_0^{\pi/2} \frac {\d x} {1 + \tan^m x} = \frac \pi 4$

where $m$ is a real number.

Proof
So:

giving:


 * $\ds \int_0^{\pi/2} \frac {\d x} {1 + \tan^m x} = \frac \pi 4$