And it'll probably require multiple integration by parts or integration by parts and a substitution if you can figure it out.
First integration by parts:
dv = e^t u = e^[t^2]
v = e^t du = 2*t*e[t^2]
So:
e^[t^2] * e^t - S(e^t * 2 * t * e^[t^2] dt) =
e^[t^2] * e^t - 2 * S(t * e^[t^2 + t] dt)
Second starts to get really messy, which is why I think you need substitution. I worked the second out, but it won't help too much. Gots ta get back to my own work..
Edit: That's probably wrong, but who knows. I haven't take a math course in a while.