Why does $\int_1^\infty\frac{\sin^2(x)}{x}\mathrm d x$ diverge?
For example because, for every $n\geqslant1$, $$ \int_{n\pi+\pi/4}^{n\pi+3\pi/4}\frac{\sin^2x}x\mathrm dx\geqslant\int_{n\pi+\pi/4}^{n\pi+3\pi/4}\frac{\frac12}{(n+1)\pi}\mathrm dx=\frac1{4(n+1)}$$