1. ## Unit square integral

Compute the following double integrals taken on an unit square:

• $\int_0^1\int_0^1\sqrt{x^2+y^2}\,dx\,dy$
• $\int_0^1\int_0^1\frac{dx\,dy}{\sqrt{1+x^2+y^2}}$
• $\int_0^1\int_0^1\frac{dx\,dy}{1-xy}$
• $\int_0^1\int_0^1\frac{dx\,dy}{(2-xy)\ln(xy)}$
• $\int_0^1\int_0^1\frac{1-x}{\{-\ln(xy)\}^{5/2}}\,dx\,dy$

2. Originally Posted by Jem
[*] $\int_0^1\int_0^1\frac{dx\,dy}{1-xy}$
$\int_0^1 \int_0^1 \frac{dx~dy}{1-xy} = \int_0^1 \int_0^1 \sum_{n=0}^{\infty} x^ny^n ~ dx~dy = \sum_{n=0}^{\infty} \frac{1}{(n+1)^2} = \frac{\pi^2}{6}$.

See This.

3. Originally Posted by Jem
• $\int_0^1\int_0^1\frac{dx\,dy}{1-xy}$
This one can also be killed by setting $(x,y)=(u-v,u+v).$ From there you compute the Jacobian and make a sketch of the new region to split the new square into two triangles. Finally you get the famous result $\frac{\pi^2}6$ which is the Basel problem.