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Let $f$ be a continuous function on R satisfying
$f (x + y) = f (x) + f (y) for all x, y \in R with f (1) = 2$ and g be a function satisfying $f (x) + g (x) =$ $e^{x}$ then the value of the integral $\int_{0}^{1} f (x) g (x) d x$ is

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1e−4

14(e−2)

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12(e−3)

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detailed solution

Correct option is C

First show that f(x)=2x,x∈RNow ∫01 f(x)g(x)dx=∫01 2xex−2xdx=2xex01−2∫01 exdx−43x301=2e−2(e−1)−43=23.

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Let f be a continuous function on R satisfying f(x+y)=f(x)+f(y) for all x,y∈R with f(1) =2 and g be a function satisfying f(x)+g(x)= ex then the value of the integral ∫01 f(x)g(x)dx is

Remember concepts with our Masterclasses.

Ready to Test Your Skills?

free study material

Let $f$ be a continuous function on R satisfying
$f (x + y) = f (x) + f (y) for all x, y \in R with f (1) = 2$ and g be a function satisfying $f (x) + g (x) =$ $e^{x}$ then the value of the integral $\int_{0}^{1} f (x) g (x) d x$ is