calculate-0-sin-x-3-dx-

Question Number 66170 by mathmax by abdo last updated on 10/Aug/19

c a l c u l a t e \int_{0}^{\infty} s i n (x^{3}) d x

Commented by mathmax by abdo last updated on 10/Aug/19

l e t I = \int_{0}^{\infty} s i n (x^{3}) d x \Rightarrow I = - I m (\int_{0}^{\infty} e^{- {i x}^{3}} d x)

c h a n g e m e n t {i x}^{3} = t g i v e x^{3} = - i t \Rightarrow x = {(- i t)}^{\frac{1}{3}} = {(- i)}^{\frac{1}{3}} t^{\frac{1}{3}}

\Rightarrow d x = \frac{1}{3} {(- i)}^{\frac{1}{3}} t^{\frac{1}{3} - 1} \Rightarrow \int_{0}^{\infty} e^{- {i x}^{3}} d x = \frac{1}{3} {(- i)}^{\frac{1}{3}} \int_{0}^{\infty} e^{- t} t^{\frac{1}{3} - 1} d t

= \frac{1}{3} {(e^{- \frac{i π}{2}})}^{\frac{1}{3}} Γ (\frac{1}{3}) = \frac{1}{3} e^{- \frac{i π}{6}} . Γ (\frac{1}{3}) = \frac{1}{3} Γ (\frac{1}{3}) (\frac{\sqrt{3}}{2} - \frac{i}{2}) \Rightarrow

I m (\int_{0}^{\infty} e^{- {i x}^{3}} d x) = - \frac{1}{6} Γ (\frac{1}{3}) \Rightarrow \int_{0}^{\infty} s i n (x^{3}) d x = \frac{1}{6} Γ (\frac{1}{3})