I am unable to think on how to derive this inequality. Can someone please help!!
Inequality is -> $\int_1^n logx dx$ $\leq$ log(n!) $\int_2^{n+1} logx dx $ .
I tried using Integration by parts of logx , applying limits and then comparing with log(n!) but that doesn't seems to help.
Evaluate the first integral to get
$$\int_1^n \log x \:dx = x\log x - x \Bigr|_1^n = n\log n - n + 1$$
then by Stirling's approximation we have that
$$n\log n - n + 1 \leq \log(n!)$$
and the integral on the right is greater than $1$, giving the desired inequality.