I want to solve the integration
$$ \int \int \frac{rdrd\theta}{1+r^2 /s}$$
over the a disk centered at $(r_t,\theta_t)$ with radius $r_0$, where $r_t \neq 0$ and $\theta_t \neq 0$.
The problem here is that the integral limits are complicated in both $r$ and $\theta$. To simplify matters I did the following instead
$$ \int_0^{2\pi} \int_0^{r_0} \frac{(r-r_t)drd\theta}{1+(r-r_t)^2/s} =2\pi\int_0^{r_0} \frac{(r-r_t)dr}{1+(r-r_t)^2/s} $$
which shifts the disk to the origin and hence greatly simplifying the limits.
Is it valid? If not how to proceed?