Method to compute $\int_{\sqrt{2}}^{2} \int_{1}^{\sqrt{2}} \frac{(\log(\frac{xy}{2}))^2 (x^2+y^2) }{(x^2-y^2)^{2}}\,dx \,dy$

Question

Can anybody give a hint, how to compute the integral analytically $$ \int_{\sqrt{\,{2}\,}}^{2}\int_{1}^{\sqrt{\,{2}\,}} \log^{2}\left(xy \over 2\right)\, {x^{2} + y^{2} \over \left(\,{x^{2} - y^{2}}\,\right)^{2}} \,\mathrm{d}x\,\mathrm{d}y. $$ Please, I am not looking for computer assisted proofs.

Answer 1

First you need to check if the integrals are in fact interchangeable using Fubini/Tonelli's theorem. Note that the integrand is positive everywhere, so that to verify that $f$ is integrable: $$ \int_{\mathcal{X} \times \mathcal{Y}}f^{+} = \int_{\mathcal{X} \times \mathcal{Y}}\max(f, 0) \leq\int_{\mathcal{X} \times \mathcal{Y}}|f|<\infty $$ The only case when integrand doesn't exist is $x=y=\sqrt{2}$. Taking the upper bound on $X$ and $Y$ we get $$ \int_{\mathcal{X} \times \mathcal{Y}}f^{+}<\int_{\mathcal{X} \times \mathcal{Y}}\frac{(\log 2 )^2}{9}<\infty $$ The last function is integrable because it is defined on a compact set. So the requirement of Fubini's theorem is fulfilled and integrals are exchangeable. Now you can use the substitution trick recommended by @AlexeyBurdin.