"Let f : R $\to$ R be a continuous function such that f(x) + f(x + 1) = 2, for all x$\in$R. If ${I_1} = \int\limits_0^8 {f(x)dx}$ and ${I_2} = \int\limits_{ - 1}^3 {f(x)dx}$, then the value of I1 + 2I2 is equal to ____________."

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Accepted Answer

"$f(x) + f(x + 1) = 2$ .... (i)

$x \to (x + 1)$

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

by (i) & (ii)

$f(x) - f(x + 2) = 0$

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

$\therefore$ f(x) is periodic with T = 2

${I_1} = \int_0^{2 \times 4} {f(x)dx} = 4\int_0^2 {f(x)dx}$

$${I_2} = \int_{ - 1}^3 {f(x)dx} = \int_0^4 {f(x + 1)dx} = \int_0^4 {(2 - f(x))dx} $$

$\Rightarrow$ ${I_2} = 8 - 2\int_0^2 {f(x)dx}$

$\Rightarrow$ ${I_2} = 8 -$${{{I_1}} \over 2}$

$\Rightarrow$ ${I_1} + 2{I_2} = 16$"

Let f : R $\to$ R be a continuous function such that f(x) + f(x + 1) = 2, for all x$\in$R.

If ${I_1} = \int\limits_0^8 {f(x)dx}$ and ${I_2} = \int\limits_{ - 1}^3 {f(x)dx}$, then the value of I₁ + 2I₂ is equal to ____________.

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Let f : R $\to$ R be a continuous function such that f(x) + f(x + 1) = 2, for all x$\in$R. If ${I_1} = \int\limits_0^8 {f(x)dx}$ and ${I_2} = \int\limits_{ - 1}^3 {f(x)dx}$, then the value of I1 + 2I2 is equal to ____________.

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More Definite Integration questions

Drill 25 more like these. Every day. Free.

Let f : R $\to$ R be a continuous function such that f(x) + f(x + 1) = 2, for all x$\in$R.

If ${I_1} = \int\limits_0^8 {f(x)dx}$ and ${I_2} = \int\limits_{ - 1}^3 {f(x)dx}$, then the value of I₁ + 2I₂ is equal to ____________.