Approximation

In Probability,

$$\ln n! = \ln \prod\limits_{k=1}^n k = \sum\limits_{k-1}^n \ln k \approx \int_{x=1}^n \ln x ~dx$$

Using integration by parts

$$\int u ~dv = uv - \int v~du$$

$$d(uv) = v~du + u~dv$$

$$u = \ln x,~ dv = 1,~ du = {1 \over x},~v=x$$

$$\int_{x=1}^n \ln x~dx = [{x \ln x - \int x {1 \over x} ~dx}]^n_{x=1}$$

$$= [x \ln x - x ]^n_{x=1} = n \ln n - n + 1$$