TITLE: Solving an $n$-dimensional integral
QUESTION [5 upvotes]: Let $p> - 1$. Find the integral
$$ \int\limits_{(0,1)^n} \left( \frac{\min(x_1,...,x_n)}{\max(x_1,...,x_n)} \right)^p dx_1...dx_n. $$

REPLY [3 votes]: There are $n(n-1)$ pairs $j,k$ so that $x_j=\min\limits_ix_i$ and $x_k=\max\limits_ix_i$. On each of these identical regions, the integral is
$$
\int_0^1\int_0^b\left(\frac ab\right)^p(b-a)^{n-2}\,\mathrm{d}a\,\mathrm{d}b
$$
So the sum over the $n(n-1)$ identical regions would be
$$
\begin{align}
&n(n-1)\int_0^1\int_0^b\left(\frac ab\right)^p(b-a)^{n-2}\,\mathrm{d}a\,\mathrm{d}b\\
&=n(n-1)\int_0^1b^{n-1}\int_0^1a^p(1-a)^{n-2}\,\mathrm{d}a\,\mathrm{d}b\\
&=n(n-1)\int_0^1b^{n-1}\frac{\Gamma(p+1)\Gamma(n-1)}{\Gamma(p+n)}\,\mathrm{d}b\\
&=(n-1)!\frac{\Gamma(p+1)}{\Gamma(p+n)}\\
&=\frac{(n-1)!}{(p+1)(p+2)\cdots(p+n-1)}\\
&=\frac1{\binom{p+n-1}{n-1}}
\end{align}
$$