$$\int_0^\infty x/(1+x^2\sin^2x) \mathrm dx$$
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.
[Math] does $\int_0^\infty x/(1+x^2 \sin^2x) \mathrm dx$ converge or diverge
convergence-divergenceimproper-integralsintegration
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Best 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 $$\frac{x}{1+x^2\sin^2 x} \ge \frac{1}{2x}.$$