Solution
= Solution
\b[It is not Lebesgue integrable], because its absolute integral diverges. On $I_k=[k\pi+\pi/6,k\pi+5\pi/6]$, one has $|\sin x|\geq1/2$ and $x\leq(k+1)\pi$. Thus
$$
\int_{I_k}\frac{|\sin x|}x\,dx\geq\frac1{3(k+1)}.
$$
These disjoint-interval lower bounds have a divergent harmonic sum. The finite improper value from part (i) is therefore conditional convergence, not Lebesgue integrability.