Antiderivative of this function?

This is a tricky one. Let's work with the hardest part first, the power of the exponential. Let u = x^3 ==> u^2 = x^6.

If u = x^3, then du= 3x^2 dx.

Substituting back into the original yields. (1/3) ∫(3x^2)x^6 *e^(x^2) dx gives us (1/3) ∫ u^2 e^u du.

Use integration by parts let w = u^2 and dv = e^u du.
w*v - ∫v dw = u^2 *e^u - ∫2u e^u du.

Continue on in the same way and use integration by parts again for the remaining integral.