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

Theorem

 * $\displaystyle \int_0^\infty \map \ln {\frac {e^x + 1} {e^x - 1} } \rd x = \frac {\pi^2} 4$

Proof
We can write:

Let:


 * $u = \coth \dfrac x 2$

We have, by Derivative of Hyperbolic Cotangent Function:


 * $\dfrac {\d u} {\d x} = -\dfrac 1 2 \csch^2 \dfrac x 2$

From Difference of Squares of Hyperbolic Cotangent and Cosecant, this can be written:


 * $\dfrac {\d u} {\d x} = \dfrac 1 2 \paren {1 - \coth^2 \dfrac x 2} = \dfrac 1 2 \paren {1 - u^2}$

From Limit to Infinity of Hyperbolic Cotangent Function, we have:


 * as $x \to \infty$, $u \to 1$.

We also have:


 * as $x \to 0^+$, $u \to \infty$.

With this, we have: