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

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.

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integration...*over what*? – DonAntonio Nov 4 '12 at 17:30
This is tagged indefinite-integral. What do you mean by an indefinite double integral? Do you really want a definite integral? If so, what would be its domain? – robjohn Nov 25 '13 at 13:57

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).
Given the proper domain which can be expressed as $r\le f(\theta)$, then we can integrate $\int(f(\theta)-1)e^{f(\theta)}\,\mathrm{d}\theta$. However, the question was asked about an indefinite integral, which doesn't really make sense for a double integral. Never mind, I just saw that Harry Peter retagged it to include that tag. Sorry. – robjohn Nov 25 '13 at 14:02