How to calculate the following integral of an exponential function:
$$I = \int_{0}^{\infty}{ \frac{e^{-ax}}{1-e^{-bx}} }dx,$$
with Residue Theorem? Is the Residue Theorem needed here? In textbooks, one often example of a residue integral with exponentials is this one:
$$I = \int_{0}^{\infty}{ \frac{e^{x}}{1+e^{ax}} }dx = \frac{\pi}{\alpha \sin \frac{\pi}{\alpha}}.$$
This one has a pole in -1, and the upper one in +1, so can it be useful to solve them both via residuals? Or some other kind of method would be better here. Perhaps partial integration or something else. Thank you.