Divergence of $\int_{0}^{1/2} 1/(|\sqrt{x}\ln(x)|)^{p} dx$ The matter of interest is
$$\int_{0}^{1/2} \frac{1}{|\sqrt{x}\ln(x)|^p}\, dx$$
I am aware that this integral converges for $p=2$ (that's not too hard to show). I also believe that this integral diverges for $p>2$...but how can I show that using elementary calculus and related techniques (comparison test etc)? 

A simple change of variables does the trick: $y=1/x$
$$\int_0^\frac12\frac1{(-\sqrt x\log(x))^p}dx = \int_2^\infty\frac1{-y^{-p/2}\log(\frac1y)^p}\frac1{y^2}=\\
=\int_2^\infty \frac{y^{\frac p2-2}}{\log(y)^p}dy$$
The last integral converges iff $\frac p2-2\leq-1$, which is equivalent to $p\leq2$.
