Past exam of the Probability, statistics and options II course of the University of Oxford / 2009 / D / D3 / iii / Solution

The integrand is nonnegative and measurable. It is bounded by $x$ near zero and by $x{e}^{-x}$ for $x>0$, so its integral is finite. For every $x>0$ the geometric expansion gives $x/(e^x+1)={\sum }_{n\ge 1}(-1{)}^{n-1}x{e}^{-nx}$. To justify termwise integration, note thatTonelli gives an integrable sum of absolute values; dominated convergence then applies to the signed partial sums. ConsequentlyIf evaluated further, this is $(1-2^{-1})\zeta (2)={\pi }^2/12$.