What could be an easy method (Calc 1) to check if a given integral is not integrably in terms of elementary functions?
Take for example:
$$ \int e^{-t^{2}}dt$$
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This definite integral has an elementary closed form: https://en.m.wikipedia.org/wiki/Gaussian_integral. – lhf Jul 27 '15 at 22:22
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Whoops, I meant the indefinite integral. Fixed. – YoTengoUnLCD Jul 27 '15 at 22:34
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See http://math.stackexchange.com/a/404427/589 and also http://math.stackexchange.com/questions/1028569/are-there-ways-of-telling-whether-or-not-an-integral-will-have-a-closed-expressi and http://math.stackexchange.com/questions/155/how-can-you-prove-that-a-function-has-no-closed-form-integral. – lhf Jul 27 '15 at 22:48
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See Liouville's theorem and the Risch algorithm. – Lucian Jul 28 '15 at 00:38
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There isn't one. Calc 1 doesn't come anywhere close to proving that some functions, like $e^{-x^{2}}$, don't have elementary integrals.
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