let Γ(x)=∫_0 ^∞ t^(x−1) e^(−t) dt with x>0 1) prove that Γ(x)Γ(1−x)= (π/(sin(πx))) 2) find the value of ∫


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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

	$d o y o u l i k e t h e d i f f e r e n t m e t h o d t o f i n d t h e s o l u t i o n$ $. . . i t i s n o t a v a i l a b l e i n a p p s . .$
	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

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