[Math] Evaluating a double integral: $\iint \exp(\sqrt{x^2+y^2})\:dx\:dy$

Question

How to evaluate the following integral? $$\iint \exp\left(\sqrt{x^2+y^2} \right)\:dx\:dy$$

I'm trying to integrate this using substitution and integration by parts but I keep getting stuck.

Answer 1

If you switch to polar coordinates, you end out integrating $re^r \,dr \,d\theta$, which you should be able to integrate over your domain by doing the $r$ integral first (via integration by parts).