I was trying to find the values of $r>0$ for which the integral
$\int^{\pi/2}_{0}\csc^{r}x\,\mathrm{d}x$
exists. Unfortunately, I'm short on ideas on how this could be done. If this may serve as any guidance, to be crude, I tried punching different values for $r$ in Mathematica, and it looks like we are looking for $r\in(0,1)$. In any case, I would be thankful for advice on how we can determine and prove that.