Proving $n! = \Theta\bigr(\sqrt n \; \bigr( \frac{n}{e}\bigr)^n \bigl)$ by integration

Question

I'm trying to prove: $$n! = \Theta\bigr(\sqrt n \; \bigr( \frac{n}{e}\bigr)^n \bigl)$$

using integration. I'v found a proof but it is very long and not elegant. Any ideas of a short and elegant proof of the above?

Answer 1

Hint

Try to express $$\ln n!=\sum _{k=1}^{n}\ln k$$in terms of a Reimann's integral.