Originally Posted by

**angela87** I have a couple of integrals that I need to evaluate - they are in the context of Lebesgue integrals. They seem similar to questions regarding contour integration that I've done before, but I'm not sure if that applies here.

I need to show the following:

1) $\displaystyle \int_{0}^{\infty}\frac{\sin(\alpha x)}{e^x-1}dx=\sum_{n=1}^{\infty}\frac{\alpha}{\alpha^2 + n^2}$

2) $\displaystyle \int_{0}^{1}\frac{\log x}{1+x}dx=\sum_{n=1}^{\infty}\frac{(-1)^n}{n^2}$

Thanks for any help!