Prove:
$\displaystyle ln({n!}) < \int_{1}^{n+1} \ln(x) dx$
Notice that $\displaystyle \ln n! = \sum_{k=1}^n \ln k$.
Consider the function $\displaystyle f(x) = \ln x$.
This function is positive and decreasing for $\displaystyle x>0$.
Therefore, the sum of rectangular areas $\displaystyle [1,n]$ by using right-end points is smaller than the area.
This is basically all you need to argue.