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So far, I determined that the integral converges for every $q>p+1$.

I noticed that for example for the values $p=5, q=3$ the integral still converges. There are some values for which the integral diverges, too.

I have also tried to apply the Dirichlet test and limit tests with various functions.

Any insight on how to determine the exact values of p and q for which the integral diverges/converges will be much appreciated!

Yonatan Izutskiver
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By setting $x=e^{-t}$ we have: $$\int_{0}^{1} x^p (-\log x)^q\,dx = \int_{0}^{+\infty} t^q e^{-(p+1)t}\,dt = \frac{\Gamma(q+1)}{(p+1)^{q+1}} $$ as soon as both $p$ and $q$ are greater than $-1$.

Jack D'Aurizio
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  • I wasn't familiar with the gamma function, up until now. Are there more ways to approach this problem? – Yonatan Izutskiver May 12 '16 at 11:13
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    @YonatanIzutskiver: you may study the convergence of the middle integral also without knowing that its value (when convergence holds) is related with $\Gamma(q+1)=q!$. It is enough to ensure integrability in a right neighbourhood of zero ($q>-1$) and in a left neighbourhood of $+\infty$ ($p>-1$). – Jack D'Aurizio May 12 '16 at 11:25