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

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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

$\blacksquare$


Sources