Approximation
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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
$$
\int u dv = uv - \int vdu
$$
$$
d(uv) = vdu + udv
$$
$$ u = \ln x,~ dv = 1,~ du = {1 \over x},~v=x $$
$$ \int_{x=1}^n \ln x~dx = [{x \ln x - \int x {1 \over x} ~dx}]^n_{x=1} $$
$$ = [x \ln x - x ]^n_{x=1} = n \ln n - n + 1 $$