We’ll use the method of integration by parts to find this integral. Let u=x³ and e^xdx=dv then du=3x²dx and e^x=v. Thus we have

We need again method of integration by parts to evaluate ∫x²e^xdx. Let u=x² and dv=e^xdx which means du=2xdx and v=e^x. Hence 

and similarly we have

Thus, we obtain 

and finally we get