Given-f-x-f-x-2-x-R-If-0-2-f-x-dx-p-then-0-2020-f-x-2a-dx-for-a-Z-

Question Number 161256 by cortano last updated on 15/Dec/21

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

I f \underset{0}{\int^{2}} f (x) d x = p t h e n \underset{0}{\int^{2020}} f (x + 2 a) d x = ?

f o r a \in Z^{+}

Answered by talminator2856791 last updated on 17/Dec/21

f (x + 2 a) = f (x + 2 a - 2 k) k \in Z

\Rightarrow f (x + 2 a) = f (x)

\int_{0}^{2020} f (x + 2 a) d x = \int_{0}^{2020} f (x) d x

\int_{0}^{2} f (x) d x = \int_{z}^{z + 2} f (x) d x z \in R

\int_{0}^{2020} f (x) d x = \underset{k = 0}{\sum^{1009}} \int_{2 k}^{2 k + 2} f (x) d x

\Rightarrow \int_{0}^{2020} f (x + 2 a) d x = 1010 p