Evaluate $\sum_{n=1}^\infty 1/n^2$ using $\int_0^1 \int_0^1 \frac{\mathrm{d}x \, \mathrm{d}y}{1-xy}$

I think if you use Change of Variables in Double Integrals by using Jacobian such that $$u=y+x,v=y-x\;\; u\Big|_{0}^{\sqrt{2}},v\Big|_{0}^{\sqrt{2}}$$ your integral would be easy. See http://www.math24.net/change-of-variables-in-double-integrals.html for more.