TITLE: Show that,$\int_0^\pi \left|\frac{\sin nx}{x}\right|\mathrm{d}x \ge \frac{2}{\pi}\left(1+\frac12+\cdots+\frac{1}{n}\right)$
QUESTION [7 upvotes]: Show that,$$\int_0^\pi \bigg|\dfrac{\sin nx}{x}\bigg|\mathrm{d}x \ge \dfrac{2}{\pi}\bigg(1+\dfrac12+\cdots+\dfrac{1}{n}\bigg)$$  
I could not approach the problem at all. Please help.

REPLY [8 votes]: $$
\begin{align}
I &=\sum_{i=0}^{n-1} \int_{i\pi/n}^{(i+1)\pi/n} \frac{|\sin{nx}|}{|x|}\,dx \\
&=\sum_{i=0}^{n-1} \int_{0}^{\pi/n} \frac{|\sin{nx}|}{i\pi/n+ x}\,dx \\
&\gt \sum_{i=1}^{n} \int_{0}^{\pi/n} \frac{|\sin{nx}|}{i\pi/n}\,dx \\
&= \sum_{i=1}^{n} \frac{2/n}{i\pi/n} \\
&= \frac{2}{\pi}\sum_{i=1}^{n} \frac{1}{i}
\end{align}
$$