∫_0 ^∞ ((sin^3 (x))/x) dx =?


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Question Number 116500 by bobhans last updated on 04/Oct/20

$\underset{0}{\int^{\infty}} \frac{\sin^{3} (x)}{x} dx = ?$
	Answered by Olaf last updated on 04/Oct/20

	$I = \int_{0}^{\infty} \frac{\sin^{3} x}{x} d x$ $I = \int_{0}^{\infty} - \frac{1}{4} . \frac{\sin 3 x}{x} d x + \int_{0}^{\infty} \frac{3}{4} . \frac{\sin x}{x} d x$ $u = 3 x$ $I = - \frac{1}{4} \int_{0}^{\infty} \frac{\sin u}{u} d u + \frac{3}{4} \int_{0}^{\infty} \frac{\sin x}{x} d x$ $I = \frac{1}{2} \int_{0}^{\infty} \frac{\sin u}{u} d x = \frac{1}{2} \times \frac{π}{2} = \frac{π}{4}$ $(Dirichlet integral)$
	Answered by Bird last updated on 04/Oct/20
	$we hsve sin^3 x =sin^2 x sinx =((1−cos(2x))/2)sinx =(1/2)sinx−(1/2)cos(2x)sinx but cos(2x)sinx =cos(2x)cos((π/2)−x) =(1/2){cos(x+(π/2))+cos(3x−(π/2))} =−(1/2)sinx+(1/2)sin(3x) ⇒ sin^3 x =(1/2)sinx+(1/4)sinx−(1/4)sin(3x) =(3/4)sinx+(1/4)sin(3x) ⇒ ∫_0 ^∞ ((sin^3 x)/x)dx =(3/4)∫_0 ^∞ ((sinx)/x)dx +(1/4)∫_0 ^∞ ((sin(3x))/x)dx(→3x=t) =(3/4).(π/2) +(1/4).3∫_0 ^∞ ((sint)/t)(dt/3) =((3π)/8)+(1/4).(π/2) =((4π)/8) =(π/2)$
	$w e h s v e {s i n}^{3} x = {s i n}^{2} x s i n x$ $= \frac{1 - c o s (2 x)}{2} s i n x$ $= \frac{1}{2} s i n x - \frac{1}{2} c o s (2 x) s i n x$ $b u t c o s (2 x) s i n x = c o s (2 x) c o s (\frac{π}{2} - x)$ $= \frac{1}{2} {c o s (x + \frac{π}{2}) + c o s (3 x - \frac{π}{2})}$ $= - \frac{1}{2} s i n x + \frac{1}{2} s i n (3 x) \Rightarrow$ ${s i n}^{3} x = \frac{1}{2} s i n x + \frac{1}{4} s i n x - \frac{1}{4} s i n (3 x)$ $= \frac{3}{4} s i n x + \frac{1}{4} s i n (3 x) \Rightarrow$ $\int_{0}^{\infty} \frac{{s i n}^{3} x}{x} d x = \frac{3}{4} \int_{0}^{\infty} \frac{s i n x}{x} d x$ $+ \frac{1}{4} \int_{0}^{\infty} \frac{s i n (3 x)}{x} d x (\to 3 x = t)$ $= \frac{3}{4} . \frac{π}{2} + \frac{1}{4} . 3 \int_{0}^{\infty} \frac{s i n t}{t} \frac{d t}{3}$ $= \frac{3 π}{8} + \frac{1}{4} . \frac{π}{2} = \frac{4 π}{8} = \frac{π}{2}$
	Commented by Bird last updated on 04/Oct/20

	$t y p o e r r o r {s i n}^{3} x = \frac{3}{4} s i n x - \frac{1}{4} s i n (3 x)$ $\Rightarrow \int_{0}^{\infty} \frac{{s i n}^{3} x}{x} d x = \frac{3}{4} \int_{0}^{\infty} \frac{s i n x}{x} d x$ $- \frac{1}{4} \int_{0}^{\infty} \frac{s i n (3 x)}{x} d x (\to 3 x = t)$ $= \frac{3}{4} \frac{π}{2} - \frac{1}{4} . 3 \int_{0}^{\infty} \frac{s i n (t)}{t} \frac{d t}{3}$ $= \frac{3 π}{8} - \frac{1}{4} \frac{π}{2} = \frac{3 π}{8} - \frac{π}{8} = \frac{π}{4}$
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