Show that $\int^{\infty}_0 \frac{\sin^4 x}{x^4}=\frac{\pi} 3$.

Question

Show that $$\int^{\infty}_0 \left( \frac{\sin x}{x}\right)^4=\frac{\pi} 3$$

Although I know the integral with the index is $1$ and $2$, I have no idea on this one. Please help.

Here's an approach that uses Fourier transforms: http://math.stackexchange.com/questions/318037/prove-int-0-infty-frac-sin4xx4dx-frac-pi3/318039#318039 — Ron Gordon, Aug 01 '14 at 15:34

score 3 · Accepted Answer · edited Apr 13 '17 at 12:20

3

A hint:

I think it 's better to go under joriki's light to see what we need here. Moreover, this is a basic fact that $$\sin^4(x)=3/8+(1/8)\cos(4x)-(1/2)\cos(2x)$$

Of course, I am considering a bit complicated way than other post. In fact, we can show that $$\int_0^{\infty}\frac{1-\cos(x)}{x^2}=\pi/2$$

edited Apr 13 '17 at 12:20

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answered Aug 01 '14 at 15:08

Mikasa

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You are always so refreshingly humble and respectful of others' work when you make connections and answer! – amWhy Aug 01 '14 at 15:20
See Rob's general case. – Mikasa Aug 01 '14 at 15:20
@amWhy: The OP wanted to cross an street so Jonas take his hand and they passed the street. While I did it for him after circulating him around the city. :D Thanks – Mikasa Aug 01 '14 at 15:23
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I like circulate with you around this wonderful city! – Aug 02 '14 at 09:01

Show that $\int^{\infty}_0 \frac{\sin^4 x}{x^4}=\frac{\pi} 3$.

1 Answers1