For all integers $k \geqslant 2$, we set: $$u_{k} = \ln k - \int_{k-1}^{k} \ln t \, dt$$ Using two integrations by parts, show that: $$u_{k} = \frac{1}{2}(\ln k - \ln(k-1)) - \frac{1}{2} \int_{k-1}^{k} \frac{(t-k+1)(k-t)}{t^{2}} dt$$
For all integers $k \geqslant 2$, we set:
$$u_{k} = \ln k - \int_{k-1}^{k} \ln t \, dt$$
Using two integrations by parts, show that:
$$u_{k} = \frac{1}{2}(\ln k - \ln(k-1)) - \frac{1}{2} \int_{k-1}^{k} \frac{(t-k+1)(k-t)}{t^{2}} dt$$