Integral bounds for a logarithmic factorial ($N\ln N-N+1\le \ln (N!)\le (N+1)\ln (N+1)-N+1-2\ln 2$)

For integer $N\ge 2$, write $\ln (N!)={\sum }_{k=2}^N\ln k$. Monotonicity gives ${\int }_{k-1}^k\ln xdx\le \ln k\le {\int }_k^{k+1}\ln xdx$. Summing and evaluating the endpoint integrals proves the displayed bounds. They also hold with equality when $N=1$. The lower integral starts at one and the upper at two; confusing those endpoints loses the constant in the upper bound. The bounds imply $\ln (N!)=N\ln N-N+O(\ln N)$, while Stirling's formula determines the logarithmic and constant corrections.