Let f(x) be a differentiable function defined on [0, 2] such that f^′(x)=f^′(2-x) for all x ∈(0,2), f(0)=1 and f(2)= e^2. Then the value of ∫_0^2…

Let $\displaystyle f(x)$ be a differentiable function defined on [$\displaystyle 0$, $\displaystyle 2$] such that $\displaystyle f^{\prime}(x)=f^{\prime}(2-x)$ for all $\displaystyle x \in(0,2), f(0)=1$ and $\displaystyle f(2)=\mathrm{e}^2$. Then the value of $\displaystyle \int_0^2 f(x) \mathrm{d} x$ is :