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

$$Givenf\left(x\right)=f(x+2),\mathrm{\forall }x\in \mathbb{R}$$$$If\underset{0}{\stackrel{2}{\int }}f\left(x\right)dx=pthen\underset{0}{\stackrel{2020}{\int }}f(x+2a)dx=?$$$$fora\in {\mathbb{Z}}^{+}$$

Answered by talminator2856791 last updated on 17/Dec/21

$$$$$$f(x+2a)=f(x+2a-2k)k\in \mathbb{Z}$$$$\Rightarrow f(x+2a)=f\left(x\right)$$$$$$$${\int }_0^{2020}f(x+2a)dx={\int }_0^{2020}f\left(x\right)dx$$$$$$$${\int }_0^{2}f\left(x\right)dx={\int }_z^{z+2}f\left(x\right)dxz\in \mathbb{R}$$$$$$$${\int }_0^{2020}f\left(x\right)dx=\underset{k=0}{\sum ^{1009}}{\int }_{2k}^{2k+2}f\left(x\right)dx$$$$$$$$\Rightarrow {\int }_0^{2020}f(x+2a)dx=1010p$$$$$$

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