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Let $\alpha \in \mathbb R$ and let's call $I:=[0,1]$. Evaluate
$ \int_{I^n} \left( \min_{1\le i \le n}x_i \right)^{\alpha}\,\, dx. $

Well, the case $n=1$ is easy and the integral equals $\frac{1}{\alpha+1}$, for every $\mathbb R \ni \alpha \ne - 1$.

I've done also the case $n=2$ and, if I'm not wrong, it's $\displaystyle \frac{2}{(\alpha+1)(\alpha+2)}$.

My big problem is that I cannot understand how to deal with the general case. Any ideas?

Thanks in advance.

1 Answers 1

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Denote $a_n^{\alpha}$ the integral. We have, since $\alpha\neq -1$, that \begin{align} a_n^{\alpha}&=n!\int_{\{0$a_n^{\alpha}=n!a_1^{\alpha+n-1}\prod_{j=1}^{n-1}\frac 1{\alpha+j}=n!\prod_{j=1}^n\frac 1{\alpha+j}.$