Integrals \int^{\infty}_0 \ln(x)/\cosh^2(x) dx

• June 4th 2008, 08:28 PM
jibuzhu
Integrals \int^{\infty}_0 \ln(x)/\cosh^2(x) dx
Question 1:
I find the following integral formula from "Table of integrals" but I do not
know how to derive it.
$
\int^{\infty}_{0} \frac{\ln (x)}{\cosh^2(x)} dx = \psi(\frac{1}{2}) + \ln(\pi),
$

where $\psi$ is the Digamma function.

Question 2:
How to get the answer for the slightly modifed integral:
$
\int^{\infty}_{0} \frac{\ln (x)}{\cosh^2(x+a)} dx = ?
$