How would I evaluate the following integral by using integration by parts?

Integral of: (t^3)(e^x)?

You mean (x^3)(e^x)?

x^3 exp(x) dx =

x^3 d[exp(x)] =

d[x^3 exp(x)] - exp(x) d[x^3] =

d[x^3 exp(x)] - 3 x^2 exp(x) dx

So, if you integrate this you get

x^3 exp(x) - 3 Integral of x^2 exp(x) dx

You must repeat this two more times the power on front of the exp(x) will get zero and then you have the answer.

It's simpler to calculate the integral by evaluating the integral of Exp(ax) and then to differentiate the answer three times w.r.t. a in order to get the x^3 in front of the exponential.