Let f : R → R be a continuous function

Question:

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that $f(\mathrm{x})+f(\mathrm{x}+1)=2$, for all $\mathrm{x} \in \mathbb{R}$. If

$\mathrm{I}_{1}=\int_{0}^{8} f(\mathrm{x}) \mathrm{dx}$ and $\mathrm{I}_{2}=\int_{-1}^{3} f(\mathrm{x}) \mathrm{d} \mathrm{x}$, then the value

of $\mathrm{I}_{1}+2 \mathrm{I}_{2}$ is equal to________.

Solution:

$f(x)+f(x+1)=2$

$\Rightarrow f(x)$ is periodic with period $=2$

$\mathrm{I}_{1}=\int_{0}^{8} f(\mathrm{x}) \mathrm{dx}=4 \int_{0}^{2} f(\mathrm{x}) \mathrm{d} \mathrm{x}$

$=4 \int_{0}^{1}(f(x)+f(1+x)) d x=8$

Similarly $\mathrm{I}_{2}=2 \times 2=4$

$\mathrm{I}_{1}+2 \mathrm{I}_{2}=16$

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