# Definite Integral to Infinity of x over Exponential of x plus One

## Theorem

$\displaystyle \int_0^\infty \frac x {e^x + 1} \rd x = \frac {\pi^2} {12}$

## Proof

 $\displaystyle \int_0^\infty \frac x {e^x + 1} \rd x$ $=$ $\displaystyle \int_0^\infty \frac {x^{2 - 1} } {e^x + 1} \rd x$ $\displaystyle$ $=$ $\displaystyle \map \eta 2 \map \Gamma 2$ Integral Representation of Dirichlet Eta Function in terms of Gamma Function $\displaystyle$ $=$ $\displaystyle \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} } {n^2} } 1!$ Definition of Dirichlet Eta Function, Gamma Function Extends Factorial $\displaystyle$ $=$ $\displaystyle \frac {\pi^2} {12}$ Sum of Reciprocals of Squares Alternating in Sign

$\blacksquare$