$\int\int_{D}\frac{1}{x^2+y^2}dx \ dy$

Question

Given the next domain $D=\{(x,y)\in \mathbb R^2 :1\leq x^2+y^2\leq4,0\leq y \}$ I have to compute this integral: $$\int\int_{D}\frac{1}{x^2+y^2}dx \ dy$$

I know that $x=r\cos\phi$ and $y=r\sin\phi$ and $x^2+y^2=r^2$, but this is not enough. Can somebody help me,please?

Answer 1

Hint: use polar coordinates. This is the integral of $\frac{1}{r^2}$ over the upper semi-circle of radius $2$ minus the upper semi-circle of radius $1$.