Let $f$ be a continuous function on R satisfying 

$f(x+y)=f\left(x\right)+f\left(y\right)\text{ for all }x,y\in \mathbf{R}\text{ with }f\left(1\right) =2$ and g be a function satisfying $f\left(x\right)+g\left(x\right)=$ $e^x$ then the value of the integral  ${\int }_0^1 f\left(x\right)g\left(x\right)dx$ is

• A

  $\frac{1}{e}-4$

• B

  $\frac{1}{4}(e-2)$

• C

  2/3 

• D

  $\frac{1}{2}(e-3)$