Determining substitution method or Integration by parts

Question

Harry claims that the substitution method works for the following problem. Is he correct? If yes, solve the problem, if no, what method could you use. $\int x^2 e^{x^2} dx$.

Sorry for the formatting of the question. I have tried accomplishing this problem through substitution and clearly, it is not working. Would integration by parts work for this problem? Please help me figure this out and a reason as to why it would work or would not work would be appreciated.

Thank you

Answer 1

That function has not an elementary primitive. Although, you can integrate over $\mathbb R$. Are you sure you didn't miss the interval of integration?

Answer 2

Integration by parts works if you happen to know an antiderivative of $e^{x^2}$ (see the erfi function).