$\lim_{n \to \infty }n\int_{0}^{\pi}\left \{ x \right \}^{n}dx$ Calculate
$$\lim_{n \to \infty }n\int_{0}^{\pi}\left \{ x \right \}^{n}dx$$
My try: $x-1<[x]<x$
$-x<-[x]<1-x$
$0<x-[x]<1$ then I applied an integral but I get the result pi which is the wrong answer.

$\int_0^1 x^n + \int_1^2 (x-1)^n + \int_2^3 (x-2)^n +\int_3^\pi (x-3)^n$ then 
$\frac{1}{n+1}+\frac{1}{n+1}+\frac{1}{n+1}+\frac{(\pi-3)^{n+1}}{n+1}$ 
$\lim \frac{3n+n(\pi-3)^{n+1}}{n+1}$
