= Solution
A real measurable function is <Lebesgue integrable> precisely when the integral of its absolute value is finite. For every integer $k\geq1$, on $I_k=[k+1/6,k+5/6]$ one has $|\sin\pi x|\geq1/2$ and $x\leq k+1$. Consequently
$$
\int_{I_k}\frac{|\sin\pi x|}{x}\,dx\geq\frac{1}{3(k+1)}.
$$
The intervals have disjoint interiors. Countable additivity of the nonnegative <Lebesgue integral>, or the <monotone convergence theorem> applied to their finite unions, gives
$$
\int_1^\infty\frac{|\sin\pi x|}{x}\,dx\geq\frac13\sum_{k=1}^\infty\frac1{k+1}=\infty.
$$
The <harmonic series> diverges, so the function is \b[not <Lebesgue integrable>]. Cancellation between adjacent oscillations cannot give absolute integrability.
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