Past exam of the options II course of the University of Oxford / 2008 printed variant / D / D4 / b / ii / Solution

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.