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.

  • 7
    $\begingroup$ integration...*over what*? $\endgroup$ – DonAntonio Nov 4 '12 at 17:30
  • $\begingroup$ 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? $\endgroup$ – 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).

  • $\begingroup$ 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. $\endgroup$ – robjohn Nov 25 '13 at 14:02
  • $\begingroup$ I assume he didn't mean an indefinite integral. Note the question is from over a year ago. $\endgroup$ – Zarrax Nov 25 '13 at 14:05
  • $\begingroup$ Yeah, I just noticed that it was retagged by Harry Peter. Sorry. I have removed that tag. $\endgroup$ – robjohn Nov 25 '13 at 14:07

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