$\lim_{n \to \infty }n\int_{0}^{\pi}\left \{ x \right \}^{n}dx$ 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.

Break the integral into four parts; from $0$ to $1$, from $1$ to $2$, from $2$ to $3$ and from $3$ to $\pi$. Then $\{x\}=x,\ \{x\}=x-1,\ \{x\}=x-2,\ \{x\}=x-3$ in those intervals respectively. Then, you can easily calculate the three integrals to find that 
$$n\int_{0}^{\pi}\{x\}^ndx=\frac{3n}{n+1}+\frac{n(\pi-1)^{n+1}}{n+1}.$$ 
Now you can take it from here. Note that $0<\pi-3<1,$ so that $(\pi-1)^{n+1}\rightarrow 0.$
With time delay. I apologize for that, because all the answers here are the same.
