Use integration by parts to solve:
I = f (x^3)(e^(x^2)) dx
(Let "f" represent the integral)
Let u = (e^(x^2)), dx = (1) / (2x(e^(x^2))) du
Let dv = x^3, v = (x^4) / (4)
I = (1/4)(x^4)(e^(x^2)) - (1/4) f (x^3) (2(e^(x^2))) du
... and now I don't know what to do. I don't know how to take the antiderivative of (e^(x^2)).
You chose the wrong parts to break it up into. The integral of e^x^2 cannot be written in closed form. Use the fact that 2x e^x^2 is the derivative of e^x^2, and choose du = x e^x^2 dx,
u = (1/2)e^x^2
v = x^2
dv = 2x
The answer can be found at
http://www.karakas-online.de/forum/viewtopic.php?t=9515