TITLE: Integrating the multinomial over a hypercube
QUESTION [6 upvotes]: I have come across an integral of the form
$$\int_{b}^{a}\cdots\int_{b}^{a} \left( \sum_{i=1}^{n}x_i\right)^mdx_1d x_2\dots dx_n.$$
I have a solution that makes use of the partition function, but I feel there should be a much nicer solution and I'm sure this has been looked at before.  Does anybody know a reference?
Motivation:
This integral has appeared whilst trying to compute moments of the Voronoi cell of the lattice $A_n$ (see page 462 of Sphere Packings, Lattices and Groups)

REPLY [11 votes]: We want the coefficient of $t^m/m!$ in 
  $$ \int_b^a\cdots \int_b^a e^{(x_1+\cdots+x_n)t}dx_1\dots dx_n =
           \left(\frac{e^{at}-e^{bt}}{t}\right)^n. $$
Expanding by the binomial theorem and then taking the coefficient of $t^m/m!$ from each term will give a formula.