Let f:RR be a continuous function such that f(x)+f(x+1)=2, for all xR.

Question

Let $f:\mathbb{R}	o\mathbb{R}$ be a continuous function such that $f(x)+f(x+1)=2$, for all $x\in\mathbb{R}$. If $I_1=\displaystyle\int_0^8 f(x)\,dx$ and $I_2=\displaystyle\int_{-1}^3 f(x)\,dx$, then the value of $I_1+2I_2$ is equal to..........

Answer: 16

Accepted Answer

Given $f(x)+f(x+1)=2\ \ldots(i)$. Putting $x=0$: $f(0)+f(1)=2$; putting $x=1$: $f(1)+f(2)=2\Rightarrow f(2)=f(0)$, and so on. So $f$ is periodic with period 2.

$I_1=\displaystyle\int_0^8f(x)dx=4\int_0^2f(x)dx$

For $I_2$, substitute $x	o x+1$: $I_2=\displaystyle\int_0^4f(x+1)dx=\int_0^4[2-f(x)]dx=8-2\int_0^2f(x)dx\cdot 2$

More precisely, $I_2=\displaystyle\int_0^4[2-f(x)]dx=8-2\int_0^2f(x)dx\Rightarrow 2I_2=16-4\int_0^2f(x)dx=16-I_1$

$	herefore I_1+2I_2=16$

Solutions · 1