# Solving integral $\int x e^{-x^3}dx$

I need help with evaluating the integral:

$$\int x e^{-x^3}dx$$

Thanks!

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What do you call to solve an integral? –  Did Apr 13 '12 at 10:18
it could not be solved using elementary functions –  dato datuashvili Apr 13 '12 at 10:29
You could relate it to an incomplete Gamma-function. And this function has been well studied. –  Raskolnikov Apr 13 '12 at 10:37
If it were a definite integral, it could be accurately approximated. If it were $\int x^2e^{-x^3}dx$, the antiderivative can be expressed in terms of elementary functions. But for this one, it cannot. See this post covering a relevant theorem of Liouville and the Risch algorithm for more info. If this is a homework problem, it is an error. What is the precise problem or your real need? –  bgins Apr 13 '12 at 10:58

Typo perhaps? If it is meant to be either $\int x^2 e^{x^{3}}dx$ or $\int x e^{x^{2}} dx$ then these can be evaluated very simply.

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Yes, it was typo in the homework. It was supposed to be $x^2 e^{x^3}dx$. Thanks anyway. –  aldo Apr 13 '12 at 11:28
No problem, glad to be of help. I take it you're good from now on? It should be obvious since I assume the very integral in question is being evaluated as you had most probably been studying "integration by inspection". –  Autolatry Apr 13 '12 at 11:36
@aldo: You might want to edit your question to account for this new development... –  Ｊ. Ｍ. Apr 15 '12 at 16:39
Take t = x^3

We get $(1/3)\int e^{-t} t^{-1/3}\,dt$, unfortunately still not an "elementary" integral. –  GEdgar Apr 13 '12 at 13:05