integration - does $int_0^infty x/(1+x^2 sin^2x) mathrm dx$ converge or diverge?

$$\int_0^\infty x/(1+x^2\sin^2x) \mathrm dx$$

I'd be very happy if someone could help me out and tell me, whether the given integral converges or not (and why?). Thanks a lot.

Answer

Hint: The integral diverges, there is trouble when $x$ is large. For detail, use the fact that if $x \ge 1$, then
$1+x^2\sin^2 x \le 2x^2$ and therefore

$$\frac{x}{1+x^2\sin^2 x} \ge \frac{1}{2x}.$$