sin-x-3-c-dx-sinh-x-3-c-dx-

Question Number 75089 by MJS last updated on 07/Dec/19

\int \sin (x^{3} + c) d x = ?

\int \sinh (x^{3} + c) d x = ?

Commented by mathmax by abdo last updated on 07/Dec/19

\int s i n (x^{3} + c) d x = I m (\int e^{i (x^{3} + c)} d x =) a n d

\int e^{i (x^{3} + c)} d x = e^{i c} \int e^{i x^{3}} d x = e^{i c} \int 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}} \Rightarrow

\int e^{i (x^{3} + c)} d x = \frac{1}{3} e^{i c} \int e^{- t} {(i t)}^{\frac{1}{3} - 1} d t

= \frac{1}{3} e^{i c} i^{\frac{1}{3} - 1} \int t^{\frac{1}{3} - 1} e^{- t} d t

= \frac{1}{3} e^{i c} {(e^{\frac{i π}{2}})}^{- \frac{2}{3}} \int t^{\frac{1}{3} - 1} e^{- t} d t

= \frac{1}{3} e^{i c - 3 i π} Γ_{i n} (\frac{1}{3}) = \frac{1}{3} e^{i (c - 3 π)} Γ_{i n} (\frac{1}{3})

Γ_{i n} i s t h e i n c o m p l e t Γ \dots .

Commented by mathmax by abdo last updated on 07/Dec/19

f o r g i v e \int e^{i (x^{3} + c)} d x = \frac{1}{3} e^{i c - i \frac{π}{3}} Γ_{i n} (\frac{1}{3}) \Rightarrow

I = \frac{1}{3} Γ_{i n} (\frac{1}{3}) s i n (c - \frac{π}{3})

Commented by MJS last updated on 07/Dec/19

thank you

Commented by mathmax by abdo last updated on 14/Dec/19

y o u a r e w e l c o m e s i r .