Integration by parts expression help

[AStaunton]

the expression to integrate is:

\int x^{3}e^{x^{2}}dx

and in the spirit of "LIATE" I set my u and dv as the following:

dv=e^{x^{2}}dx

u=x^{3}

however, doing this that I integrate dv=e^{x^{2}}dx in order to get v...and unless I'm missing something, this does not seem like an easy integral! a u substition won't work as i'd need an x^1 term multiplying by the e^x^2...

and going the other way and setting dv=x^3dx and u=e^x^2 and plugging into int,by parts formula gets:

\frac{x^{4}}{4}e^{x^{2}}-\int\frac{x^{4}}{4}2xe^{x^{2}}dx

and I don't think further integration by parts will help with this new integral..

any advice appreciated.

 

[AStaunton]

correction of typo:



however, doing this REQUIRES that I integrate dv=e^{x^{2}}dx in order to get v...

 

[lurflurf]

change variable x^2->u

 

[eumyang]

Perform a u-substitution with u = x³, and then integrate by parts.