Definite Integral to Infinity of Reciprocal of 1 plus Power of x

Theorem

 * $\displaystyle \int_0^\infty \frac 1 {1+x^n} \, \mathrm dx = \frac \pi n \csc \left({\frac \pi n}\right)$

where:
 * $n$ is a real number greater than 1
 * $\csc$ is the cosecant function.

Proof
From Euler's Reflection Formula:


 * $\displaystyle \Gamma \left({\frac 1 n}\right) \Gamma \left({1 - \frac 1 n}\right) = \pi \csc \left({\frac \pi n}\right)$

Then:

Note we have:

As $\theta \nearrow \frac \pi 2$, $\tan\theta \to \infty$ and $\tan 0 = 0$, so making a substitution of $x = \left(\tan \theta\right)^{\frac 2 n}$ to our original integral:

So we have:


 * $\displaystyle \pi \csc \left({\frac \pi n}\right) = n\int_0^\infty \frac 1 {1+x^n} \, \mathrm d x$

Hence:


 * $\displaystyle \int_0^\infty \frac 1 {1+x^n} \, \mathrm d x = \frac \pi n \csc \left({\frac \pi n}\right)$