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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*?2012-04-13
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    it could not be solved using elementary functions2012-04-13
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    You could relate it to an [incomplete Gamma-function](http://en.wikipedia.org/wiki/Incomplete_gamma_function). And this function has been well studied.2012-04-13
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    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](http://en.wikipedia.org/wiki/Elementary_function). But for this one, it cannot. See [this post](http://math.stackexchange.com/questions/155/how-can-you-prove-that-a-function-has-no-closed-form-integral) 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?2012-04-13

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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.2012-04-13
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    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".2012-04-13
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    @aldo: You might want to edit your question to account for this new development...2012-04-15
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Take t = x^3 
dt = 3x^2 dx and x = t^(1/3)

and substitute x and find integration by using LIATE

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    We get $(1/3)\int e^{-t} t^{-1/3}\,dt$, unfortunately still not an "elementary" integral.2012-04-13
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    Its look like ∫f(x)g(x)dx.2012-04-13
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    Its look like ∫f(t)g(t)dt. Take f(t)= e^(-t) and g(t) = t^(-1/3)2012-04-13
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    Doesn't help...2012-04-13