Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2009 / Paper 1 / 16 / b

Set $t=\log (1/x)$, so $x=e^{-t}$ and $dx=-e^{-t}dt$. The improper integral becomesThe Gamma function recurrence givesThe exponential-transformed integrand also proves convergence at the logarithmically singular endpoint.