Test for divergence of $\int_{0}^{\infty} \frac{\sin^2(x)}{x}dx$ without evaluating the integral I would like to prove that $$\int_{0}^{\infty} \frac{\sin^2(x)}{x}dx$$ diverges without actually evaluating the integral. Is there a convergence test from calculus or real analysis that can show that this integral diverges?
Thanks.
Edit: Someone pointed out that this is a possible duplicate. However, the question put forth as a possible duplicate asks about $\sin(x^2)$, not about $\sin^2(x)$.

$$
\begin{align}
\int_0^\infty\frac{\sin^2(x)}{x}\,\mathrm{d}x
&=\sum_{k=1}^\infty\int_{(k-1)\pi}^{k\pi}\frac{\sin^2(x)}{x}\,\mathrm{d}x\\
&\ge\sum_{k=1}^\infty\frac1{k\pi}\int_{(k-1)\pi}^{k\pi}\sin^2(x)\,\mathrm{d}x\\
&=\sum_{k=1}^\infty\frac1{2k}
\end{align}
$$
