Find all the values of $\alpha$ and $\beta$ for which the following integral converges:
$$\int_0^1 \frac{\cos(\frac{1}{x})}{x^{\alpha}(1-x^2)^{\beta}} \,dx$$
My attempt:
$\left|\frac{\cos(\frac{1}{x})}{x^{\alpha}(1-x^2)^{\beta}}\right| \leqslant \frac{1}{x^{\alpha}(1-x^2)^{\beta}}$, hence the integral converges if $\alpha < 1$ and $\beta < 1.$
And now we have to prove that there are no other values of of $\alpha$ and $\beta$ for which the integral converges absolutely and find values of of $\alpha$ and $\beta$ for which the integral converges conditionally. I have no idea how to do it.