Show that for all integer $n > 1$, $$\int_{1}^{n} \frac{q(u)}{u} \mathrm{~d}u = \ln(n!) + (n-1) - n\ln(n) - \frac{1}{2}\ln(n) = \ln\left(\frac{n! e^n}{n^n \sqrt{n}}\right) - 1.$$