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](http://en.wikipedia.org/wiki/Partition_%28number_theory%29), 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](http://www2.research.att.com/~njas/doc/splag.html))



Integrating the multinomial over a hypercube
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.


