Let f(x) be a differentiable function defined on [0, 2] such that f...

${f}^{'}\left(x\right) = {f}^{'}\left(2 - x\right)$

$f\left(x\right) = - f\left(2 - x\right) + c$

put $x = 0$

$f\left(0\right) = - f\left(2\right) + c$

$c = f\left(0\right) + f\left(2\right) = 1 + {e}^{2}$

so, $f\left(x\right) + f\left(2 - x\right) = 1 + {e}^{2}$

$I = {∫}_{0}^{2}f\left(x\right)dx$

$I = {∫}_{0}^{2}f\left(2 - x\right)dx$

$2I = {∫}_{0}^{2}\left(f\left(x\right) + f\left(2 - x\right)\right)dx$

$2I = \left(1 + {e}^{2}\right){∫}_{0}^{2}dx$

$I = 1 + {e}^{2}$