Finding moment generating function from pdf

Question

How do you find the mgf of $f(x) = 3x^2$ for $0 < x < 1$?

I tried multiplying the function with $e^{tx}$. My answer turned out to be $(3e^t)/t$

According to the book, that's just part of an answer.

Answer 1

Per Sean Roberson's comment. If your PDF is $$f(x) = \begin{cases} 3x^2 \; &0<x<1\\ 0 \; &\text{otherwise} \end{cases}$$ then the MGF is $$M(t) = E(e^{tx}) = \int_{-\infty}^{\infty}e^{tx}f(x) dx = 3\int_{0}^{1}e^{tx}x^2 dx.$$

The rest is integration by parts (remember, we're integrating with respect to $x$, so treat $t$ as a constant.)

As a little side note (to talk about something other than a raw calculation), I do like how this Wikipedia article explains why MGFs are useful.