mathematical-analysis-prove-that-0-sin-3-x-x-3-dx-3pi-8-

Question Number 132987 by mnjuly1970 last updated on 17/Feb/21

\dots . m a t h e m a t i c a l a n a l y s i s \dots

p r o v e t h a t :: ϕ = \int_{0}^{\infty} \frac{{s i n}^{3} (x)}{x^{3}} d x = \frac{3 π}{8}

* * * * \dots \dots \dots .

Answered by Dwaipayan Shikari last updated on 17/Feb/21

\int_{0}^{\infty} \frac{{s i n}^{3} x}{x^{3}} d x

= \frac{3}{4} \int_{0}^{\infty} \frac{s i n x}{x^{3}} - \frac{1}{4} \int_{0}^{\infty} \frac{s i n 3 x}{x^{3}} d x

= \frac{3}{4} . \frac{π}{4 s i n (\frac{3 π}{2})} - \frac{9}{4} \int_{0}^{\infty} \frac{s i n u}{u^{3}} d u \int_{0}^{\infty} \frac{s i n x}{x^{μ}} d u = \frac{π}{2 Γ (μ) s i n (\frac{π}{2} μ)}

= - \frac{9}{4.4 s i n (\frac{3 π}{2})} - \frac{3 π}{16} = \frac{9 π}{16} - \frac{3 π}{16} = \frac{3 π}{8}

Commented by mnjuly1970 last updated on 18/Feb/21

t h a n k s a l o t m r p o w e r \dots

Commented by Dwaipayan Shikari last updated on 18/Feb/21

M r p o w e r ? ?

h a h a a a \dots . s i r

Commented by mnjuly1970 last updated on 18/Feb/21

s o r r y s o r r y \dots

m a s t e r D w a i p a y a n . .

Answered by mnjuly1970 last updated on 18/Feb/21

a n o t h e r s o l u t i o n \dots

φ (a) = \int_{0}^{\infty} \frac{{s i n}^{3} (a x)}{x^{3}} d x

ϕ = φ (1) \dots

φ^{'} (a) = \frac{\partial}{\partial a} (\int_{0}^{\infty} \frac{{s i n}^{3} (a x)}{x^{3}} d x)

= \int_{0}^{\infty} \frac{3 x c o s (a x) {s i n}^{2} (a x)}{x^{3}} d x

= 3 \int_{0}^{\infty} \frac{{s i n}^{2} (a x) c o s (a x)}{x^{2}} d x

= \frac{3}{2} \int_{0}^{\infty} \frac{s i n (2 a x) s i n (a x)}{x^{2}} d x

= \frac{3}{4} \int_{0}^{\infty} \frac{c o s (a x) - c o s (3 a x)}{x^{2}} d x

φ^{″} (a) = \frac{3}{4} \int_{0}^{\infty} \frac{- s i n (a x)}{x} d x + \frac{9}{4} \int_{0}^{\infty} \frac{s i n (3 a x)}{x} d x

= \frac{3 π}{4} \Rightarrow φ^{'} (a) = \frac{3}{4} π a + C

φ^{'} (0) = 0 = C

φ^{'} (a) = \frac{3}{4} π a \Rightarrow φ (a) = \frac{3}{8} π a^{2} + K

φ (0) = 0 = K

φ (a) = \frac{3}{8} π a^{2} \Rightarrow ϕ = φ (1) = \frac{3}{8} π ✓ ✓