I don't know how to solve:
$$\int \!\!\! \int\_D x^3y^2 \ln (x^2+y^2) \, dxdy$$
over the area:
$$D=\{ (x,y): 4 \le x^2+y^2 \le 25, \,\, x,y \ge 0 \}$$

Obviously we are doing a double integration over the first quadrant of two circles, one with radius $2$ and one with radius 4. 

I changed 
$$\left[
\begin{align}
dx\,dy=r\,dr\,d\theta\\
x=r \cos\theta\\
y=r \sin\theta\\
\end{align}
\right]
$$

Now we have:
$$ 2\leq r \leq 5$$
$$ 0\leq \theta \leq \frac{\pi}{4} $$

$$\int\int\_D (r\cos\theta)^3 (r\sin\theta)^2\ln (r^2) \ r \, dr\ d\theta$$

I replaced $\sin^2$ by $1-\cos^2$

$$\int\int\_D r^6 \cos^3\theta (1-\cos^2\theta) \ln (r^2) \, dr\, d\theta$$

$$\int\int\_D r^6 \ \left(\cos^3\theta-\cos^5\theta)\right)\ln (r^2)\,dr\,d\theta$$

And after that I tried a double integration with the hope that I'll be back to $\cos^3-\cos^5$ after two cycles but $\ln(r^2)$ is crumping, so I am NOT getting anywhere with that technique. What shall I do?