calculate-0-sin-x-6-dx-

Question Number 91621 by abdomathmax last updated on 02/May/20

c a l c u l a t e \int_{0}^{\infty} s i n (x^{6}) d x

Commented by mathmax by abdo last updated on 02/May/20

\int_{0}^{\infty} s i n (x^{6}) d x = I m (\int_{0}^{\infty} e^{{i x}^{6}} d x) w e h a v e b y c h a n g e m e n t {i x}^{6} = - t

\Rightarrow x^{6} = i t \Rightarrow x = {(i t)}^{\frac{1}{6}} = e^{\frac{i π}{12}} t^{\frac{1}{6}} \Rightarrow

\int_{0}^{\infty} e^{{i x}^{6}} d x = e^{\frac{i π}{12}} \int_{0}^{\infty} e^{- t} \times \frac{1}{6} t^{\frac{1}{6} - 1} d t

= \frac{1}{6} e^{\frac{i π}{12}} \int_{0}^{\infty} t^{\frac{1}{6} - 1} e^{- t} d t w e h a v e Γ (x) = \int_{0}^{\infty} t^{x - 1} e^{- t} d t

= \frac{1}{6} Γ (\frac{1}{6}) (c o s (\frac{π}{12}) + i s i n (\frac{π}{12})) \Rightarrow \int_{0}^{\infty} s i n (x^{6}) d x = \frac{1}{6} Γ (\frac{1}{6}) s i n (\frac{π}{12})