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

Proof
From Euler's Reflection Formula:


 * $\map \Gamma {\dfrac 1 n} \, \map \Gamma {1 - \dfrac 1 n} = \pi \, \map \csc {\dfrac \pi n}$

Then:

Note we have:

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

So we have:


 * $\displaystyle \pi \, \map \csc {\frac \pi n} = n \int_0^\infty \frac 1 {1 + x^n} \rd x$

Hence:


 * $\displaystyle \int_0^\infty \frac 1 {1 + x^n} \rd x = \frac \pi n \, \map \csc {\frac \pi n}$