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

$\displaystyle \int_0^\infty \frac x {e^x - 1} \rd x = \frac {\pi^2} 6$
 $\displaystyle \int_0^\infty \frac x {e^x - 1} \rd x$ $=$ $\displaystyle \int_0^\infty \frac {x^{2 - 1} } {e^x - 1} \rd x$ $\displaystyle$ $=$ $\displaystyle \map \zeta 2 \map \Gamma 2$ Integral Representation of Riemann Zeta Function in terms of Gamma Function $\displaystyle$ $=$ $\displaystyle \frac {\pi^2} 6 \times 1!$ Basel Problem, Gamma Function Extends Factorial $\displaystyle$ $=$ $\displaystyle \frac {\pi^2} 6$
$\blacksquare$