I'm having trouble in the following problem
Find the values of $p$ for which the integral $\int_0^1 \frac{dx}{x^p~\ln(x)} $ converges.
So far I separated the integral into two separate integrals, $\int_0^\frac{1}{2} \frac{dx}{x^p~\ln(x)} $ and $\int_\frac{1}{2}^1 \frac{dx}{x^p~\ln(x)} $, for the first one I found that it converges for values of $p$ where $p\le0$, however I don't seem to find the way of proving the convergence of the second integral, tried using the power series for $\ln(x)$ but I'm not sure how to proceed, any insight would be appreciate it, Thanks.