Integrating a function of second order partial derivatives

Question

Let f be a $C^2$ function with a compact support. Prove that:$$ \int_{\mathbb{R}^2}\ln(x^2+y^2)\left(\frac{\partial^2f}{\partial x^2}+\frac{\partial^2f}{\partial y^2}\right)d\ell_2=4\pi f(0) $$ The hint is to use the polar coordinates, but substituting $\Delta f=\frac{\partial^2f}{\partial r^2}+\frac{1}{r}\frac{\partial f}{\partial r}+\frac{1}{r^2}\frac{\partial^2f}{\partial \theta^2}$ doesn't seem like a simplification to me and I have no idea how to integrate $\frac{\partial^2f}{\partial \theta^2}$ with respect to r.