Question-35544

Question Number 35544 by tanmay.chaudhury50@gmail.com last updated on 20/May/18

Commented by prof Abdo imad last updated on 20/May/18

l e t A = \int_{0}^{\infty} \frac{c o s x}{\sqrt{x}} d x a n d B = \int_{0}^{\infty} \frac{s i n x}{\sqrt{x}} d x w e h a v e

A - i B = \int_{0}^{\infty} \frac{e^{- i x}}{\sqrt{x}} d x c h a n g e m e n t \sqrt{x} = t g i v e

\int_{0}^{\infty} \frac{e^{- i x}}{\sqrt{x}} d x = \int_{0}^{\infty} \frac{e^{- {i t}^{2}}}{t} 2 t d t

= 2 \int_{0}^{\infty} e^{- {i t}^{2}} d t =_{t \sqrt{i} = u} 2 \int_{0}^{\infty} e^{- u^{2}} \frac{d t}{\sqrt{i}}

= \frac{2}{e^{i \frac{π}{4}}} \int_{0}^{\infty} e^{- u^{2}} d u = 2 e^{- i \frac{π}{4}} \frac{\sqrt{π}}{2}

= \sqrt{π} {c o s (\frac{π}{4}) - i s i n (\frac{π}{4})} \Rightarrow

\int_{0}^{\infty} \frac{c o s x}{\sqrt{x}} d x = \sqrt{π} . \frac{1}{\sqrt{2}} = \sqrt{\frac{π}{2}}

\int_{0}^{\infty} \frac{s i n x}{\sqrt{x}} d x = \sqrt{π} \frac{1}{\sqrt{2}} = \sqrt{\frac{π}{2}}