Compute an indefinite integral through change of variables

Question

I'm trying to compute the following integral:

$\int_{-\infty}^\infty \int_{-\infty}^\infty e^{x_1^2 + x_2^2} dx_1 dx_2$

I have been thinking about a change of variables with cylindric coordinates, which gives me :

$\iint e^{x_1^2 + x_2^2} dx_1 dx_2 = \iint -\frac{1}{2}e^{-r^2} drd\theta$

Which is already much easier to compute. However I have trouble finding the boundaries of the new integral.

EDIT : corrected a dumb mistake how the integration process, thank you for your help!

Answer 1

$$\lim_{R\rightarrow\infty}\int_0^{2\pi} \int_0^R re^{-r^2} drd\theta$$