A calculus problem by Hussein Eid

Using dominated convergence theorem : $\int_0^1\int_0^1 \frac{\mathrm{d}x\mathrm{d}y}{1-xy} = \sum_{k=0}^{\infty}\int_0^1\int_0^1 x^k y^k \ \mathrm{d}x = \sum_{k=0}^{\infty} \frac{1}{(k+1)^2}$ The last sum is well known to equal $\frac{\pi^2}{6}$ (google for the Basel problem).