0-sin-px-x-dx-p-R-

Question Number 128314 by 676597498 last updated on 06/Jan/21

\int_{0}^{\infty} \frac{\sin (px)}{\sqrt{x}} dx

\forall p \in R

Answered by Dwaipayan Shikari last updated on 06/Jan/21

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

= \frac{1}{i} \int_{0}^{\infty} e^{{i p t}^{2}} - e^{- {i p t}^{2}} d t {p t}^{2} = i j {p t}^{2} = - i g

= \frac{1}{2 p} \int_{0}^{\infty} {(\frac{i j}{p})}^{- \frac{1}{2}} e^{- j} d j + {(\frac{- i g}{p})}^{- \frac{1}{2}} e^{- g} d g

= \frac{1}{2 \sqrt{p}} Γ (\frac{1}{2}) (e^{\frac{π}{4} i} + e^{- \frac{π}{4} i}) = \sqrt{\frac{π}{2 p}}

Answered by mnjuly1970 last updated on 06/Jan/21

s o l u t i o n : Ψ = \int_{0}^{\infty} \frac{s i n (p x)}{x^{\frac{1}{2}}} d x

Ψ \overset{p x = ξ}{=} \frac{1}{p^{\frac{1}{2}}} \int_{0}^{\infty} \frac{s i n (ξ)}{ξ^{\frac{1}{2}}} d ξ =

= \frac{1}{p^{\frac{1}{2}}} * \frac{π}{2 Γ (\frac{1}{2}) s i n (\frac{π}{4})} = \frac{π}{\sqrt{p} 2 \sqrt{π} \frac{\sqrt{2}}{2}}

= \frac{π}{\sqrt{2 p π}} = \frac{\sqrt{2 p π}}{2 p} = \sqrt{\frac{π}{2 p}}