I have two questions about integration with polar coordinates.
1.
If I have a indefinite integral like $$\int \int e^{x^2 + y^2} \mathrm{d}x \mathrm{d}y$$
Do I still have indefinite integrals after polar transformation? $$\int \int re^{r^2} \mathrm{d}r \mathrm{d}\varphi=\frac{e^{x^2}}{2}+C$$
2.
If I have an definite integral like $$\int\limits_{a}^b \int\limits_{m}^n e^{x^2 + y^2} \mathrm{d}x \mathrm{d}y$$
What exactly happens with the integration limits when I use polar transformation? $$\int\limits_?^? \int\limits_?^? re^{r^2} \mathrm{d}r \mathrm{d}\varphi$$ I know that when $[-\infty,\infty]\times [-\infty,\infty] \implies [-0,2\pi]\times [0,\infty]$