TITLE: Evaluating the integral, $\int_{0}^{\infty} \ln\left(1 - e^{-x}\right) \,\mathrm dx $
QUESTION [43 upvotes]: I recently got stuck on evaluating the following integral. I do not know an effective substitution to use. Could you please help me evaluate:
$$\int_{0}^{\infty} \ln\left(1 - e^{-x}\right) \,\mathrm dx $$

REPLY [2 votes]: break $\ln(1-e^{-x})$ like $\ln(1-t) = y-y^2/2-y^3/3- \cdots$ where $y = e^{-x}$
then break at the addition/subtraction point in integration.
then evaluate.