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$$\int x^2e^{x^3} \, dx =\frac{x^3}{3}e^{x^3} - \int x^5e^{x^3} \, dx $$

How to solve this exercise? Integration by parts only makes a larger mess...

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What makes you think you need to use integration by parts? – Michael Albanese Jan 5 '13 at 4:44
The integration by parts was not necessary. But it was useful. Now you know how to find $\int x^5e^{x^3}\,dx$. – André Nicolas Jan 5 '13 at 4:52
up vote 2 down vote accepted

The integral can be evaluated by using the substitution $t=x^3$

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Make the substitution $x^3 = t$

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