How do you integrate f(x,y$\displaystyle )=\frac{ln(x^{2}+y^{2})}{x^{2}+y^{2}}\$ over the region $\displaystyle 1{\le}x^{2}+y^{2}{\le}e^{2}$?
following your lead, and with the aid of Mush's post we convert to polar coordinates. we obtain the integral $\displaystyle \int_0^{2 \pi} \int_1^{e^2} \frac {\ln r^2}{r^2} \cdot r ~dr~d \theta = 2 \int_0^{2 \pi} \int_1^{e^2} \frac {\ln r}r ~dr~d \theta$
you should be able to handle that