In Evan's PDE book, as part of the proof that convolution with the fundamental solution solves Poisson's equation, he makes the following argument: $$ C_1 \int_{B(0,\varepsilon)}|\Phi(y)|dy\leq C\varepsilon^2|\log\varepsilon|, $$
where $C_1$ is some constant and $$ \Phi(x) = -\frac{1}{2\pi}\log|x|,\qquad x\in \mathbb{R}^2. $$
My issue is that I don't understand at all how he's bounding the integral. I don't even know where to begin. Can someone help me sort this out? He also handles the case $n\geq 3$ (the function isn't a logarithm), which I also don't understand. But I figure I'll be able to figure that out if I can get some help for this case.
Hint:
Use polar coordinates and
$$\int_0^\epsilon r\log r \, dr = \frac{1}{2}\epsilon^2 \log \epsilon - \frac{1}{4} \epsilon^2$$