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How do you integrate $e^{x^2}$?

When I try to integrate (indefinite) $e^{x^2}$, supposing $x^2 = t$, and integrating by parts, the solutions seem to be never-ending. Is there any other way to integrate this?

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marked as duplicate by Martin Sleziak, Nameless, sdcvvc, Stefan Hansen, Gerry Myerson Jan 20 '13 at 10:23

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You can use maclaurin series of $e^{x^2}$ to integrate

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No. It is well-known that there is no elementary antiderivative for $e^{x^2}$.

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can i know why? And a glimpse of how can it be done in 'advanced' way? –  cipher Jan 20 '13 at 9:53
By "elementary", we mean "elementary functions", such as taking powers and multiplying and addition. There is no expression for the integral $e^{x^2}$. We can only give it a name (specfically the "error function"). I don't know how one can prove this. –  akkkk Jan 20 '13 at 10:07
@akkkk: The error function is related to the integral of $e^{-x^2}$, not $e^{x^2}$. –  Mårten W Jan 20 '13 at 21:06
@MårtenW: I must admit I missed that, but of course this can be solved by some appropriate complex substitution. –  akkkk Jan 20 '13 at 21:52

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