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How to prove that $\displaystyle\int_0^{+\infty}\left|\dfrac{\sin x}{x}\right| \, dx = +\infty$ ? Could any one give some hint ? Thanks.

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This has been answered before. – robjohn Jan 7 '13 at 7:51
up vote 5 down vote accepted

$$\int_{k\pi}^{(k+1)\pi}\left|\frac{\sin x}{x}\right|dx\geq\frac{1}{(k+1)\pi}\int_{k\pi}^{(k+1)\pi}|\sin x|dx.$$

You can bound the integral below by a constant multiple of the harmonic series.

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A nice solution, thx! – ludmofans Jan 7 '13 at 7:34

Hint: $\sum \frac {1}{n} =$ is unbounded. Consider the intervals where $|\sin x| > \frac {1}{2}$.

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