let-x-0-t-x-1-e-t-dt-with-x-gt-0-1-prove-that-x-1-x-pi-sin-pix-2-find-the-value-of-0-e-x-2-dx-

Question Number 33895 by math khazana by abdo last updated on 26/Apr/18

l e t Γ (x) = \int_{0}^{\infty} t^{x - 1} e^{- t} d t w i t h x > 0

1) p r o v e t h a t Γ (x) Γ (1 - x) = \frac{π}{s i n (π x)}

2) f i n d t h e v a l u e o f \int_{0}^{\infty} e^{- x^{2}} d x .

Answered by tanmay.chaudhury50@gmail.com last updated on 27/Apr/18

2 . l e t t = x^{2}

d t / d x = 2 x

= 2 \sqrt{t}

\underset{0}{\int^{\infty}} e^{- t} . 1 / 2 . t^{- 1 / 2} d t

= 1 / 2 \underset{0}{\int^{\infty}} e^{- t} t^{1 / 2 - 1} d t

= 1 / 2 ⌈ (1 / 2)

= 1 / 2 . \sqrt{Π}

Answered by tanmay.chaudhury50@gmail.com last updated on 27/Apr/18

Commented by tanmay.chaudhury50@gmail.com last updated on 28/Apr/18

d o y o u l i k e t h e d i f f e r e n t m e t h o d t o f i n d t h e s o l u t i o n

\dots i t i s n o t a v a i l a b l e i n a p p s . .

Commented by NECx last updated on 28/Apr/18

w h i c h a p p i s t h i s p l e a s e ?

Answered by tanmay.chaudhury50@gmail.com last updated on 28/Apr/18