Q1. Let $f(x) = \frac{1}{\log x} $ and $\Omega = (0,\frac{1}{2})$. Show that $f \in L^p (\Omega)$ for any $p \geq 1$ and $f' \in L^p(\Omega)$ only for $p=1$.
Q2. Show that $g(x) = \log( \log ( 1 + {|x|}^{-1} ) $ belongs to $W^{1,n}(D)$ for $n>1$ where $D = B(0,1)$ in $\mathbb{R}^n$ .
The hint I got for both questions was to use polar or spherical coordinates. I tried integrating $f(x)$ for Q1 by u-substituion with $u= \log(x) $ which gave me a sum of terms. How do I go about showing $f(x)$ is integrable for all $p \geq 1$ with that? I differentiated $g(x)$ wrt $x_i$ for Q2 which wasn't very helpful in seeing a workaround to using polar coordinates. Q2 is a generalization of this. Any tips please.