Past exam of the options II course of the University of Oxford / 2008 / D / D4 / b

For fixed $n>0$, Tonelli and the allowed exponential integral giveThe point $x=0$ is irrelevant to the integral. Fubini therefore permits reversing the signed integrals. Evaluating the elementary inner integral yieldsThe first term has integral $\pi /2$. The absolute value of the remainder is at mostConsequently the improper integral tends to $\pi /2$. The interchange was justified on each finite strip, without asserting absolute integrability on the infinite quadrant.

It is not Lebesgue integrable, because its absolute integral diverges. On $I_k=[k\pi +\pi /6,k\pi +5\pi /6]$, one has $|\sin x|\ge 1/2$ and $x\le (k+1)\pi$. ThusThese disjoint-interval lower bounds have a divergent harmonic sum. The finite improper value from part (i) is therefore conditional convergence, not Lebesgue integrability.