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I want to prove

$$\displaystyle\int_0^{\infty} \frac{\sin x}x \,\mathrm{d}x = \frac \pi 2$$

and

$$\displaystyle\int_0^{\infty} \frac{|\sin x|}x \,\mathrm{d}x \to \infty$$

And I found in wikipedia, but I don't know, can't understand. I didn't learn differential equations, Laplace transforms, and even inverse trigonometric functions.

So tell me easy, please.

Rodrigo de Azevedo
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Fourier
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    Why do you think there is an "easy" way to compute this? See http://math.stackexchange.com/questions/5248 for many possible solutions. (The second question follows from estimating the integral over each half-period of $\sin$ and comparing with a harmonic series. This may also require more than your background.) – mrf Jul 11 '13 at 14:25
  • https://math.stackexchange.com/questions/13344/proof-of-int-0-infty-left-frac-sin-xx-right2-mathrm-dx-frac-pi2?noredirect=1&lq=1 – Guy Fsone Jan 02 '18 at 10:30

1 Answers1

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About the second integral: Set $x_n = 2\pi n + \pi / 2$. Since $\sin(x_n) = 1$ and $\sin$ is continuous in the vicinity of $x_n$, there exists $\epsilon, \delta > 0$ so that $\sin(x) \ge 1 - \epsilon$ for $|x-x_n| \le \delta$. Thus we have: $$\int_0^{+\infty} \frac{|\sin x|}{x} dx \ge 2\delta\sum_{n = 0}^{+\infty} \frac{1 - \epsilon}{x_n} = \frac{2\delta(1-\epsilon)}{2\pi}\sum_{n=0}^{+\infty} \frac{1}{n + 1/4} \rightarrow \infty $$

Yuandong
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