Given-f-x-f-x-2-x-R-if-0-2-f-x-dx-k-then-0-1010-f-x-2a-dx-for-a-Z-

Question Number 125841 by bramlexs22 last updated on 14/Dec/20

G i v e n f (x) = f (x + 2) \forall x \in R

i f \underset{0}{\int^{2}} f (x) d x = k t h e n \underset{0}{\int^{1010}} f (x + 2 a) d x ?

f o r a \in Z

Commented by mr W last updated on 14/Dec/20

505

Commented by bramlexs22 last updated on 14/Dec/20

s t e p b y s t e p s i r

Commented by bramlexs22 last updated on 14/Dec/20

i g o t 505 k s i r . n o t 505

Answered by liberty last updated on 14/Dec/20

f (x) = f (x + 2) i t f o l l o w t h a t f (x) i s

a p e r i o d i c f u n c t i o n w i t h p e r i o d e i s 2

t h e n \underset{0}{\int^{1010}} f (x) d x = \underset{0}{\int^{2}} f (x) d x + \underset{2}{\int^{4}} f (x) d x +

\dots + \underset{1008}{\int^{1010}} f (x) d x

w h e r e \underset{0}{\int^{2}} f (x) d x = \underset{2}{\int^{4}} f (x) d x = \dots = \underset{1008}{\int^{1010}} f (x) d x

s o w e g e t \underset{0}{\int^{1010}} f (x) d x = k + k + k + \dots + k, 505 t i m e s

\underset{0}{\int^{1010}} f (x) d x = 505 k .

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