Hey, I am trying to solve this problem, but I'm not sure if I'm on the right track.

Evaluate

$\displaystyle

\iint_D \frac{1}{(x^2 + y^2)^{\frac{n}{2}}}dA

$

where $\displaystyle n$ is an integer and $\displaystyle D$ is the region bounded by the circles with center the origin and radii $\displaystyle r_n$ and $\displaystyle R_n$, $\displaystyle 0<r_n<R_n$.

What I have so far is the idea to use polar coordinates. In this case we would have

$\displaystyle

D = \{(r, \theta)\, |\, r_n\leq r \leq R_n\mbox{ , }0\leq\theta\leq2\pi\}

$

(i added the n subscripts to I can differentiate between $\displaystyle r_n$ and $\displaystyle r$)

I transformed the function using $\displaystyle x = r\cos\theta$ and $\displaystyle y = r\sin\theta$. The resulting expression is

$\displaystyle

\int^{2\pi}_0\int^{R_n}_{r_n}\frac{1}{(r^2\cos^2 \theta + r^2\sin^2\theta)^{\frac{n}{2}}}\,r\,dr\,dx

$

Do you think this is a good approach?

Also, now I don't really know how to solve this integral (how to start). Any help? Thank you.