We fix $x > 0$ and for all natural integers $n$, we define $F_{n}(x)$ by:
$$F_{n}(x) = \ln\left(\frac{n! \, n^{x+1}}{(x+1)(x+2) \ldots (x+n+1)}\right)$$
Show that for all natural integers $i$:
$$\int_{x+i}^{x+i+1} \ln t \, dt = \ln(x+i) - \int_{i}^{i+1} \frac{u-i-1}{u+x} du$$