show-that-0-xsin-x-3-dx-1-3-pi-1-3-

Question Number 110112 by mathdave last updated on 27/Aug/20

s h o w t h a t

\int_{0}^{\infty} x \sin (x^{3}) d x = \frac{1}{3} ∙ \frac{π}{Γ (\frac{1}{3})}

Answered by mnjuly1970 last updated on 27/Aug/20

x^{3} = t \Rightarrow {_{d x = \frac{1}{3} t^{\frac{- 2}{3}} d t}^{x = t^{\frac{1}{3}}}

w e k n o w :: \int_{0}^{\infty} \frac{s i n (x)}{x^{p}} d x \overset{0 < p ⩽ 1}{=} \frac{π}{2 Γ (p) s i n (\frac{p π}{2})}

Ω = \int_{0}^{\infty} x s i n (x^{3}) d x = \frac{1}{3} \int_{0}^{\infty} \frac{s i n (t)}{t^{\frac{1}{3}}} d t

Ω = \frac{1}{3} * \frac{π}{2 Γ (\frac{1}{3}) s i n (\frac{π}{6})} = \frac{π}{3 Γ (\frac{1}{3})} ♣

M . N . J u l y 1970

Commented by mathdave last updated on 27/Aug/20

u d i d w e l l m a n

Commented by mnjuly1970 last updated on 27/Aug/20

g r a t e f u l \dots s i r \dots .