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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*?2012-11-04
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    This is tagged [tag: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?2013-11-25

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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).

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    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.2013-11-25
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    I assume he didn't mean an indefinite integral. Note the question is from over a year ago.2013-11-25
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    Yeah, I just noticed that it was retagged by Harry Peter. Sorry. I have removed that tag.2013-11-25