Improper Integral in an old manuscript

Question

I was reading through an old manuscript and came across the following "elementary exercise:"

$\int^\infty_0 \dfrac{1}{1+x^2 \sin^2 x} dx$.

Anyone have a clever way of seeing this? I haven't done decent integration in so long, I have a feeling I'm just overlooking something simple.

Thanks for any help.

Answer 1

The value is $\infty$. Note that for positive integers $n$, $$ \int_{(n-1/2) \pi}^{n \pi} \dfrac{dx}{1 + x^2 \sin^2 x} \ge \int_{(n-1/2) \pi}^{n\pi}\dfrac{1}{1 + (n \pi)^2 \sin^2 x} = \dfrac{\pi}{2\sqrt{n^2 \pi^2 + 1}} $$