Let $D:= \{{(x_1, x_2, x_3, ..., x_n) \mid x_i \geq 0, x_1+x_2+...+x_n \leq 1}\}$
If $i_1, i_2, ..., i_n$ are any nonnegative integers,
How can we show that $\int\int...\int x_1^{i_1}...x_n^{i_n} d(x_1, x_2, ..., x_n)$ is
$ i_1!i_2!...i_n!/(n+i_1+i_2+...+i_n)!$