answer-to-26024-let-put-c-0-cos-ax-2-dx-and-c-0-sin-ax-2-dx-ew-have-c-is-0-e-iax-2-dx-2-1-R-e-iax-2-dx-and-i-put-x-1-2-r-x-notation-

Question Number 26107 by abdo imad last updated on 19/Dec/17

a n s w e r t o 26024 l e t p u t c = \int_{0}^{\infty} c o s ({a x}^{2}) d x a n d c = \int_{0}^{\infty} s i n ({a x}^{2}) d x

e w h a v e c - i s = \int_{0}^{\infty} e^{- {i a x}^{2}} d x = 2^{- 1} \int_{R} e^{- {i a x}^{2}} d x a n d i p u t

x^{1 / 2} = r (x) (n o t a t i o n) s o 2 (c - i s) = \int_{R} e^{- {(r (i a) x)}^{2}} d x

a n d b y t h e c h a n g e m e n t t = r (i a) x w e f i n d

2 (c + i s) = {(r (i a))}^{- 1} \int_{R} e^{- t^{2}} d t = r (π) / r (i a) b u t

r (i a) = r (i) r (a) = r (a) e^{} - - > 2 (c + i s) = r (π) r {(a)}^{- 1} e^{- i π / 4}^{)}

- - > c = r (2 π) /_{4 r (a)} a n d s = r (2 π) /_{4 r (a)}