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.