calculate-0-e-2x-sin-pi-x-dx-

Question Number 37812 by prof Abdo imad last updated on 17/Jun/18

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

\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 \dots 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