My instructor has a fondness for asking questions regarding the convergence of such integrals: $ \int_{0}^{1} \frac{\ln(x)}{x^{1/2}}\,\mathrm dx $
$ \int_{0}^{1} \frac{\ln(x)}{x^{3/2}}\,\mathrm dx $
What is the best way to determine the convergence of such integrals? Comparison tests require that $ f(x), g(x) \ge 0 $ so let's consider $-\ln x$.
How can I proceed from here?