# How would I evaluate the following integral in terms of Catalan's constant $\int_{0}^{\infty} \frac{xe^x}{1+e^{2x}}$?

How would I evaluate the following integral in terms of Catalan's constant? $$\int_{0}^{\infty} \frac{xe^x}{1+e^{2x}}$$ I am pretty sure you have to turn the integral into a series somehow, because we can write catalans constant as a series. $$G=\sum_{0}^{\infty} \frac{(-1)^n}{(2n+1)^2}$$

I am just not sure how to go about turning the integral into a series, I think I would need to do a $$u$$ substituion first, but I don't think $$u=e^x$$ works, and I couldn't think of any other plausible $$u$$ subs.

\begin{align} \int_0^\infty \frac{xe^{x}}{1+e^{2x}}\,dx&=\int_0^\infty \frac{xe^{-x}}{1+e^{-2x}}\,dx\\\\ &=\int_0^\infty x\sum_{n=0}^\infty (-1)^n e^{-(2n+1)x}\,dx\\\\ &=\sum_{n=0}^\infty (-1)^n\int_0^\infty x e^{-(2n+1)x}\,dx\\\\ &=\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)^2}\\\\ &=G \end{align}
• I prefer using $C$ for Catalan's constant; is this also common? – Parcly Taxel Jul 3 '19 at 17:49