Calculate the following integral
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 $\mathbf{E}[M]$?
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:
$$n! \times \frac1{(n+1) \times (n-1)!} = \frac{n}{n+1}$$