Integrable function on positive axis.

Question

Suppose that $f(x)$ is continuous on $(0,\infty)$ and that $$\int_0^{\infty} \frac{f(x)}{x} \,dx < \infty.$$ Does it follow that $$\int_0^{\infty}\frac{f(x)^2}{x} \,dx < \infty ?$$

Answer 1

$$\int_0^{\infty} \frac{\sin x}{x} \,dx =\frac{\pi}{2},$$ but $\int_0^{\infty} \frac{\sin^2 x}{x} \,dx $ is divergent, because $$\frac{\sin^2 x}{x}=\frac{1-cos 2x}{2x}.$$