Evaluate the given integral by changing to polar coordiantes.
$\int\int_R \sin(x^2+y^2)\, dA$, where $R$ is the region in the first quadrant between the circles with center the origin and radii $1$ and $3$.
I know how to set up the integral but when I have $\int_1^3 \sin(r^2)r\,dr\,d\theta$ I'm not sure how to integrate with respect to $r$. I tried using $u$ substitution for both $\sin$ and $r^2$, but it didn't seem to eliminate the right variables. I don't think I can treat $\sin(r^2)$ as a constant, or if can just use u substitution on $r^2$ to get $-\cos r^2$. Can someone tell me if either of these are ok or which are the correct ways to integrate this? I couldn't find any examples leading me to believe either of these are ok.
Let $u=r^2$, $\frac{du}{dr}=2r$
$$\int \sin(r^2) r \, dr=\int\sin(u)\frac12\, du$$