I am trying to do the comparison lemma on 2 integrals, and I need to evaluate the following integral for all $p \gt0$, or show the integral diverges. $$\int_{0}^{\frac{1}{2}} \frac{1}{x(\ln(\frac{1}{x}))^p}dx = \left [\frac{-\ln x}{(p-1)(-\ln x)^p} \right]_{0}^{{\frac{1}{2}}}$$
How do I do this? I know that I will need to do a substitution using $u=-\ln x$, giving me $dx=-x~du$. However, when I change the limits in the substitution, $-\ln 0$ is undefined, is this sufficient to show that the integral diverges?
Update: I currently have
$$\frac{(\ln 2)^{1-p}}{p-1} + \lim_{k \to 0^+} \left(\frac{\ln k}{(p-1)(-\ln k)^p} \right)$$
and I am stuck on the conclusion. Is it because $ \ln 0$ diverges, hence the integral diverges?
Any help is much appreciated.