Chapter 11.11, Problem 8P

Sketch or computer plot a graph of $y=\ln x$ for $x>0.$ Show that $\ln n!$ is between the values of the integrals ${\displaystyle {\int }_2^{n+1}\ln }xdx$ and ${\displaystyle {\int }_1^n\ln }xdx.$ (Hint: $\ln n!=\ln 1+\ln 2+\ln 3+\cdots$ is the sum of the areas of rectangles of width 1 and height up to the $\ln x$ curve at $x=1,2,3,\cdots ,$ By considering the values of the two integrals for very large $n$ as in Problem $3,$ show that $\ln n!=n\ln n-n$ approximately for large $n.$