Definite Integral to Infinity of Reciprocal of Exponential of x minus One minus Exponential of -x over x

Theorem

 * $\displaystyle \int_0^\infty \paren {\frac 1 {e^x - 1} - \frac {e^{-x} } x} \rd x = \gamma$

where $\gamma$ denotes the Euler-Mascheroni constant.

Proof
Let:


 * $x = e^{-t}$

By Derivative of Exponential Function: Corollary 1:


 * $\dfrac {\d x} {\d t} = -e^{-t}$

By Exponential of Zero, we have:


 * as $x \to 1$, $t \to 0$.

By Exponential Tends to Zero and Infinity, we have:


 * as $x \to 0$, $t \to \infty$.

We therefore have: