# Closed form of $\int_{0}^{1}\int_{0}^{1}\frac{1}{1-\frac{xy}{2}}dxdy$

I've tried using the Jacobian matrix, but this method yielded no significant results. Another way is to turn this integral into the $$\sum_{n=0}^{\infty}\frac{1}{(n+1)^22^n}$$

However, I do not know how to find the exact sum of this series.

• – user90369 Jun 13 at 10:01
• Your comment has helped me a lot. I'm going to study Spence's duties. – Mathsource Jun 13 at 10:18
• Try the change of variable $u=(x+y)/2,v=(y-x)/2$ – FDP Jun 13 at 10:33
• The answer should be $\frac{\pi^2}{6}-\ln^2 2$ – Yuriy S Jun 13 at 11:07
As has been mentioned in the comments, the function to use here is $$\mathrm{Li}_2(x)=\sum_{k\ge1}\frac{x^k}{k^2}$$ so that your sum is given by $$S=\frac12\mathrm{Li}_2(\frac12)$$. To evaluate this exactly, we use the formula $$\mathrm{Li}_2(z)+\mathrm{Li}_2(1-z)=\frac{\pi^2}{6}-\ln(z)\ln(1-z)$$ and plug in $$z=1/2$$ to get $$S=\frac{\pi^2}{24}-\frac{\ln^2(2)}{4}.$$