TITLE: Calculate the following integral
QUESTION [15 upvotes]: $$\int_{[0,1]^n} \max(x_1,\ldots,x_n) \, dx_1\cdots dx_n$$
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?

REPLY [6 votes]: 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 
$$E[\max X_i] = E[Y] = \int_\mathbb R xf_Y(x) dx = \int_0^1 nx^n dx = \frac n{n+1}$$