What do you call *to solve an integral*? – 2012-04-13
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it could not be solved using elementary functions – 2012-04-13
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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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[155] How can you prove that a function has no closed form integral?