TITLE: does $\int_0^\infty x/(1+x^2 \sin^2x) \mathrm dx$ converge or diverge?
QUESTION [7 upvotes]: $$\int_0^\infty x/(1+x^2\sin^2x) \mathrm dx$$
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.

REPLY [2 votes]: The answer for the convergence/divergence may be easily obtained by using the following trivial inequality:
$$\int_{0}^{\infty} \frac{x}{1+x^{2}\sin^{2}{x}} \geq \int_{0}^{\infty} \frac{x}{1+x^{2}}= \bigl(\frac{1}{2}\log(x^2+1)\bigr)_{0}^{\infty}  \to \infty\ $$
The proof is complete.