Advanced-Calculus-sin-1-x-3-x-dx-solution-1-x-t-2-0-sin-t-3-1-t-dt-t

Question Number 136803 by mnjuly1970 last updated on 26/Mar/21

\dots . . A d v a n c e d ◂ \dots \dots \dots \dots . ▸ C a l c u l u s \dots . .

ϕ = \int_{- \infty}^{\infty} \frac{s i n (\frac{1}{x^{3}})}{x} d x = \dots \dots ? ? ?

s o l u t i o n ::

ϕ \overset{\frac{1}{x} = t}{=} 2 \int_{0}^{\infty} \frac{s i n (t^{3})}{\frac{1}{t}} . \frac{d t}{t^{2}} \dots \dots \dots \dots .

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

\overset{t^{3} = y}{=} \frac{2}{3} \int_{0}^{\infty} \frac{s i n (y)}{y^{\frac{1}{3}}} . \frac{d y}{y^{\frac{2}{3}}} = \frac{2}{3} \int_{0}^{\infty} \frac{s i n (y)}{y}

\overset{⟨ \int_{0}^{\infty} \frac{s i n (r)}{r} d r = \frac{π}{2} ⟩}{=} \frac{π}{3} \dots \dots \dots . .

\dots \dots . . ϕ = \int_{- \infty}^{\infty} \frac{s i n {(\frac{1}{x})}^{3}}{x} d x = \frac{π}{3} \dots \dots . . ✓ ✓ ✓

Answered by Ar Brandon last updated on 26/Mar/21

ϕ = \int_{- \infty}^{\infty} \frac{\sin (\frac{1}{x^{3}})}{x} dx = 2 \int_{0}^{\infty} x^{- 1} \sin (x^{- 3}) dx

= 2 \cdot \frac{1}{∣ - 3 ∣} \cdot \frac{π}{2 Γ (1 - \frac{1 - 1}{- 3}) \sin [(1 - \frac{1 - 1}{- 3}) \frac{π}{2}]} = \frac{π}{3}