Hey phys251.

The MGF(t) = E[e^(tX)]Integral (over all X) e^(tx)*f(x)dx for any continuous random variable.

This means that MGF(t) = Integral [0,1] e^(tx)*1*dx = (e^t)/t - 1/t = (e^t - 1)/t. Wikipedia confirms this:

http://en.wikipedia.org/wiki/Uniform...ating_function