# Math Help - Integral Inequality Proof

1. ## Integral Inequality Proof

Prove:

$ln({n!}) < \int_{1}^{n+1} \ln(x) dx$

2. Originally Posted by Paul616
Prove:

$ln({n!}) < \int_{1}^{n+1} \ln(x) dx$
Prove
$\ln k = \int_{k}^{k+1} \ln k \ dx < \int_{k}^{k+1} \ln x \ dx$

$\sum \ln k = \sum \int_{k}^{k+1} \ln x \ dx$

3. Originally Posted by Paul616
Prove:

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

This is basically all you need to argue.