Lebesgue integral in unit circle

Question

I have this doubt that I cannot solve. $\int \limits_{D}\dfrac{|x−1|^a}{|x^2−y^2|^b} \, dx \, dy$ where $D=\{(x,y)\in \mathbb{R}^2: x^2+y^2<1\}$ If I use polar coordinates I cannot solve anything. Could you help me? Thank you

Answer 1

The $r$ integral in polar coordinates can be solved analytically. It is of the form

$$C\int_{r=0}^1\frac{(\alpha r^2+\beta r+\gamma)^a}{r^{2b}}rdr$$

After that it depends on specific properties of $a$ and $b.$