Double integral of a rational function Consider the region $D$ given by $1\leq x^2+y^2\leq2\land0\leq y\leq x$. Compute $$\iint_D\frac{xy(x-y)}{x^3+y^3}dxdy$$
Attempt: The region $D$ is part of a ring in the first quadrant below the line $y=x$
Any hints are wellcome.

$$x=r\cos \phi$$
$$y=r\sin \phi$$
$$J=r$$
$$\int_{0}^{\pi /4}\int_1^{\sqrt2}\frac{r^4\cos \phi \sin \phi (\cos \phi - \sin \phi)}{r^3(\cos^3 \phi + \sin^3 \phi)}drd\phi=$$
Can you take it from here? 
