Find all the values $\alpha \in(0,\infty)$ such that the improper integral $$\int\limits_0^\infty \frac{\Bbb dx}{1+x^{\alpha}\sin^2x}$$ is convergent.
My attempt is to analyze the cases (i) $\alpha =1$, (ii) $\alpha >1$, and $\alpha <1$.
The case (i) gives divergence. I don’t know how to handle the other two cases. Does anyone have any suggestions for these cases?