1i feel the inequality is otherwise
\int_{0}^{n\pi}{|\frac{\sin x}{x}|dx}=\int_{0}^{\pi}{\sum_{r=1}^{n}{|\frac{\sin x}{x+(r-1)\pi}|}dx}\geq \int_{0}^{\pi}\sum_{r=1}^{n}{\frac{|\sin x|dx}{r\pi}}= \frac{2}{\pi}\sum_{r=1}^{n}{\frac{1}{r}}
Prove that
\int_{0}^{n\Pi}{\left|\frac{sinx}{x} \right|}dx < \frac{2}{\Pi}\left(1 + \frac{1}{2} + \frac{1}{3} + ......\frac{1}{n} \right)
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