\displaystyle\frac{1}{2}\sum_{k=1}^{n-1}\frac{1}{n}\ln\frac{k}{n}+\frac{n-1}{2n}\leq n\cdot\sum_{k=2}^{n}\int_{(k-1)/n}^{k/n}\Big(f\Big(\frac{k}{n}\Big)-f(x)\Big)\,dx\leq\\ \leq\frac{1}{2}\sum_{k=2}^{n}\frac{1}{n}\ln\frac{k}{n}+\frac{n-1}{2n}