find-f-x-0-e-x-1-t-2-1-t-2-dt-interms-ofx-with-x-0-and-calculate-0-e-t-2-dt-

Question Number 27342 by abdo imad last updated on 05/Jan/18

f i n d f (x) = \int_{0}^{\infty} \frac{e^{- x (1 + t^{2})}}{1 + t^{2}} d t i n t e r m s o f x

w i t h x ⩾ 0 a n d c a l c u l a t e \int_{0}^{\infty} e^{- t^{2}} d t .

Commented by abdo imad last updated on 08/Jan/18

a f t e r v e r i f y i n g t h a t f i s d e r i v a b l e i n [0, \propto [

f^{,} (x) = \int_{0}^{\infty} \frac{\partial}{\partial x} (\frac{e^{- x (1 + t^{2})}}{1 + t^{2}}) d t = \int_{0}^{\infty} - e^{- x (1 + t^{2})} d t

= - e^{- x} \int_{0}^{\infty} e^{- {x t}^{2}} d t t h e c h . \sqrt{x} t = u g i v e

f^{,} (x) = - e^{- x} \int_{0}^{\infty} e^{- u^{2}} \frac{d u}{\sqrt{x}} = - \frac{e^{- x}}{\sqrt{x}} \int_{0}^{\infty} e^{- u^{2}} d u

= - \frac{\sqrt{π}}{2} \frac{e^{- x}}{\sqrt{x}} (x > 0)

\Rightarrow f (x) = - \frac{\sqrt{π}}{2} \int_{0}^{x} \frac{e^{- t}}{\sqrt{t}} d t + λ a f t e r t h a t w e u s e t h e c h . \sqrt{t} = u

f (x) = - \frac{\sqrt{π}}{2} \int_{0}^{\sqrt{x}} \frac{e^{- u^{2}}}{u} 2 u d u + λ

= λ - \sqrt{π} \int_{0}^{\sqrt{x}} e^{- u^{2}} d u

{l i m}_{x - >\propto} f (x) = 0 = λ - \sqrt{π} \int_{0}^{\propto} e^{- u^{2}} d u = λ - \frac{π}{2} \Rightarrow λ = \frac{π}{2}

f i n a l l y f (x) = \frac{π}{2} - \sqrt{π} \int_{0}^{\sqrt{x}} e^{- u^{2}} d u

w e h a v e a l o t s o f m e t h o d t o p r o v e t h a t \int_{0}^{\infty} e^{- x^{2}} d x = \frac{\sqrt{π}}{2} .