= Integral bounds for a logarithmic factorial
{title2=$N\ln N-N+1\leq\ln(N!)\leq(N+1)\ln(N+1)-N+1-2\ln2$}
For integer $N\geq2$, write $\ln(N!)=\sum_{k=2}^N\ln k$. Monotonicity gives $\int_{k-1}^k\ln x\,dx\leq\ln k\leq\int_k^{k+1}\ln x\,dx$. 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.
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