double integral with square costh

[maxandri]

Hello,
I need to find
$\displaystyle \int \frac{\sqrt{|x|}}{x^2+y^2}$ in a circle with r=1, origin in 0,0.
If I use polar eq. I get $\displaystyle \sqrt{|cos\vartheta |}$ which I can't solve. Any idea?

And if I have the same integral but defined in a SQUARE (0<x<1 and 0<y<1) I can't use polar eq, so I find arctan y/x?!? Really I have no idea. Thank you.