for $\int_0^{\pi/2} \frac{\cos^\alpha(x)}{x\sqrt{\sin^\beta(x)}} \,dx$ to converge, $\alpha,\beta=?$

I need to find all the values of alpha and beta for which this integral converges:
$$\int_0^{\pi/2} \frac{\cos^\alpha(x)}{x\sqrt{\sin^\beta(x)}} \,dx$$ I'm at a bit of a loss about how to approach this problem - I know that the function is undefined at x=0, and that $\alpha$ should be greater than 0 or else the function will be undefined at $\frac{\pi}{2}$ as well.
I don't know what are the implications of that are on the integrals convergence.
I know I should use a Taylor approximation to find a similar function and do a limit convergence test but I don't understand why the solution I found does the test with x-->0+ instead of $∞$.
the solution split this function into two areas - 0 to $\frac{\pi}{4}$ and $\frac{\pi}{4}$ to $\frac{\pi}{2}$ I don't really understand why.
I just want to understand the logic behind the solution.

Solution 1:

There is no actual need to use Taylor approximations here. The key idea here to use convergence tests where x approches the “problematic points” from the left, or from the right.

What one should keep in mind here is the fact we have two variables here $\alpha$, $\beta$, we need to formulate two conditions on them where our convergence depends on it.

One approach is to define two functions $g$, $h$ where each of them depends on different variables that we know WHEN they converge (depends on the value of $\alpha,\beta$) and use the convergence tests involving the limits of $\frac{f}{g}$ or $\frac{f}{h}$ to establish the conditions.