I'm going to assume that the t was only raised to the third and that it was multiplied by e^t because if t was raised to (3e^t), I'd probably kill myself (okay, I won't but it would be something that I cannot solve due to it either being impossible or me not doing calculus for while).
Anyway, with that assumption in mind, it's easy to solve this. Just use integration by parts three times. I'm not going to show my work because I don't have the proper math symbols and that would drive me nuts. Unless I did my math wrong, it should end up being:
t^3*e^t - 3*t^2*e^t + 6*t*e^t - 6*e^t
