Prove that $\lim_{n\rightarrow \infty} \int_{[0,1]^n}\frac{|x|}{\sqrt{n}}=\frac{1}{\sqrt{3}}$

Prove that $$\lim_{n\rightarrow \infty} \int_{[0,1]^n}\frac{|x|}{\sqrt{n}}=\frac{1}{\sqrt{3}},$$
where $[0,1]^n=[0,1]\times \dots \times [0,1]$ is the unit cube in $\mathbb{R}^n$. 

Answer

Answers can be viewed only if
  1. The questioner was satisfied and accepted the answer, or
  2. The answer was disputed, but the judge evaluated it as 100% correct.
View the answer
The answer is accepted.