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Approximation

In Probability,

lnโกn!=lnโกโˆk=1nk=โˆ‘kโˆ’1nlnโกkโ‰ˆโˆซx=1nlnโกxย dx\ln n! = \ln \prod\limits_{k=1}^n k = \sum\limits_{k-1}^n \ln k \approx \int_{x=1}^n \ln x ~dx

Using integration by parts

โˆซuย dv=uvโˆ’โˆซvย du\int u ~dv = uv - \int v~du d(uv)=vย du+uย dvd(uv) = v~du + u~dv u=lnโกx,ย dv=1,ย du=1x,ย v=xu = \ln x,~ dv = 1,~ du = {1 \over x},~v=x โˆซx=1nlnโกxย dx=[xlnโกxโˆ’โˆซx1xย dx]x=1n\int_{x=1}^n \ln x~dx = [{x \ln x - \int x {1 \over x} ~dx}]^n_{x=1} =[xlnโกxโˆ’x]x=1n=nlnโกnโˆ’n+1= [x \ln x - x ]^n_{x=1} = n \ln n - n + 1

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