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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.

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  • $\begingroup$ Well, there are certainly other values for which it converges. For example, $\alpha = 2, \beta = \frac12$. $\endgroup$
    – Mark Fischler
    May 7, 2018 at 19:37
  • $\begingroup$ Oops on the above comment; $\alpha = 2-\epsilon$ works, though. $\endgroup$
    – Mark Fischler
    May 7, 2018 at 19:53

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