Calculate the following integral $$\int_{[0,1]^n} \max(x_1,\ldots,x_n) \, dx_1\cdots dx_n$$
My work:
I know that because all $x_k$ are symmetrical I can assume that $1\geq x_1 \geq \cdots \geq x_n\geq 0$ and multiply the answer by $n!$ so we get that $\max(x_1\ldots,x_n)=x_1$ and the integral that we want to calculate is
$n!\int_0^1 x_1 \, dx_1 \int_0^{x_1}dx_2\cdots\int_0^{x_{n-1}} \, dx_n$ and now it should be easier but I'm stuck..
Can anyone help?
 A: One can see by induction that:
$$\int_0^1 x_1 dx_1 \int_0^{x_2 } dx_2 \cdots = \int_0^1 x_1 \frac{x_{n - (n-1)}^{n-1}}{(n-1)!} dx_1 =\frac{1}{(n+1) \times (n-1)!}$$
Therefore, the original integral is:
$$n! \times \frac1{(n+1) \times (n-1)!} = \frac{n}{n+1}$$
A: Note that $\max(x_1,x_2,\dots,x_n)=\max(x_1,\max(x_2,\dots,x_n))$.  In addition, note that
$$\begin{align}
\int_0^1\int_0^1\max(x,y)\,dx\,dy&=\int_0^1\left(\int_0^y y\,dx+\int_y^1 x\,dx\right)\,dy\\\\
&=\int_0^1 \left(\frac12+\frac12 y^2\right)\,dy
\end{align}$$
Now, we can write
$$\begin{align} \int_0^1  \max(x_1,x_2,\dots,x_n)\,dx_1&=\int_0^1  \max(x_1,\max(x_2,\dots,x_n))\,dx_1\\\\
&=\int_0^{\max(x_2,\dots,x_n)}\max(x_2,\dots,x_n)\,dx_1+\int_{\max(x_2,\dots,x_n)}^1x_1\,dx_1\\\\
&=\frac12 +\frac12\left(\max(x_2,\dots,x_n)\right)^2
\end{align}$$
Then, observe that 
$$\begin{align}
\frac{1}{k}\int_0^1 \left(\max(x_k,\dots ,x_n)\right)^{k}\,dx_k&=\int_0^{\max(x_{k+1},\dots ,x_n)}\left(\max(x_{k+1},\dots ,x_n)\right)^k\,dx_k+\int_{\max(x_{k+1},\dots ,x_n)}^1 x_k^k\,dx_k\\\\
&=\frac{1}{k}\left(\frac{k}{k+1}\left(\max(x_{k+1},\dots,x_n)\right)^{k+1}+\frac{1}{k+1}\right)\\\\
&=\frac{1}{k(k+1)}+\frac{1}{k+1}\left(\max(x_{k+1},\dots,x_n)\right)^{k+1}
\end{align}$$
Proceeding inductively, we find that 
$$\begin{align}
\int_0^1\cdots \int_0^1 \max(x_1,x_2,\dots,x_n)\,dx_1\cdots dx_n&=\frac1{(2)(1)}+\frac{1}{(3)(2)}+\frac{1}{(4)(3)}+\cdots +\frac{1}{(n+1)(n)}\\\\
&=\sum_{k=1}^n \frac{1}{k(k+1)}\\\\
&=\sum_{k=1}^{n}\left(\frac{1}{k}-\frac{1}{k+1}\right)\\\\
&=\frac{n}{n+1}
\end{align}$$
A: This is pretty straightforward using probabilistic methods: What you're looking for is 
$$E[\max_{1 \le i \le n} X_i] = \int_{[0,1]^n} \max x_i \ \ dx_1\dots dx_n$$
where $X_i$ are iid uniformly distribuited on $[0,1]$.
Now call $Y = \max_i X_i$; its distirbution function is given by 
$$F_Y(x) = P(\max X_i \le x) = \prod_i P(X_i \le x) = x^n \ \ \ \text{  $x \in [0,1]$}$$
Hence $Y$ is absolutely continuos and its density is given by 
$$f_Y(x) = nx^{n-1}1_{[0,1]}(x)$$
Hence we find that 
$$E[\max X_i] = E[Y] = \int_\mathbb R xf_Y(x) dx = \int_0^1 nx^n dx = \frac n{n+1}$$
A: This is an alternative approach.
Let $X_i$ ($i=1,\cdots , n$) be independent uniform random variable in $[0,1]$. 
What is the PDF of $M=\max (X_1, \cdots, X_n)$? 
Then what is $\mathbf{E}[M]$?
