I'm looking for a generic closed formula (possibly including finite sums) for integral of form
$$ \int_{-1}^{+1} \int_{-1}^{+1} \dfrac{x^n y^m}{\left|x-y\right|} \space dx \space dy $$
my skills in multiple integrals are pretty rusty, unfortunately, so I have no idea except that for each particular m and n the values should be fairly easy to calculate.
assume that m and n are non-negative integers.
I would say it is infinite (at least the positive part where $x,y >0$). Consider just a neighborhood of $\{x = y\}$. Let $(t,u) = (x-y, x+y)$, then the integral is larger than something like
$$ \alpha \int_{t = -\epsilon}^\epsilon \big(\int_{u=1}^{\frac{3}{2}} du\big) \frac{1}{|t|} dt $$ where $\alpha$ is the minimum of $x^n y^m$ in this subregion.