calculate ∫_0 ^∞ e^(−2x) sin{π[x]}dx .


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Question Number 37812 by prof Abdo imad last updated on 17/Jun/18
$calculate ∫_0 ^∞ e^(−2x) sin{π[x]}dx .$
$c a l c u l a t e \int_{0}^{\infty} e^{- 2 x} s i n {π [x]} d x .$
Commented by abdo mathsup 649 cc last updated on 19/Jun/18
$∫_0 ^∞ e^(−2x) sin{π[x]}dx=Σ_(n=0) ^∞ ∫_n ^(n+1) e^(−2x) sin(nπ)dx =Σ_(n=0) ^∞ sin(nπ)∫_n ^(n+1) e^(−2x) dx =0$
$\int_{0}^{\infty} e^{- 2 x} s i n {π [x]} d x = \sum_{n = 0}^{\infty} \int_{n}^{n + 1} e^{- 2 x} s i n (n π) d x$ $= \sum_{n = 0}^{\infty} s i n (n π) \int_{n}^{n + 1} e^{- 2 x} d x = 0$
	Answered by tanmay.chaudhury50@gmail.com last updated on 18/Jun/18

	$[x] = 0 1 > x ⩾ 0$ $[x] = 1 2 > x ⩾ 1$ $[x] = 2 3 > x ⩾ 2$ $t h u s p u t t i n g t h e v a l u e o f [x] w e g e t i n t r e g a l$ $m u l t i p l e o f Π . . . t h u s v a l u e o f s i n {Π [x]} = 0$ $s o t h e i n t r d g a t i o n v a l u e i s z e r o$ $R e f e r f l o o r f u n c t i o n / g r e a t e s t i n t e g e r f u n c t i o n$
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