Letfx  be a differentiable function defined on0,2  such that f'x=f'2−x for all  x∈0,2,f0=1and f2=e2. Then the value of ∫02fx   dx is

Given f(x) is differentiable function  Given f1(x)=f1(2−x) and f(0)=1,f(2)=e2 Consider I=∫02 f(x)dx=[x⋅f(x)]02−∫02 x⋅f1(x)dx=2f(2)−∫02 (2−x)f1(2−x)dx=2e2−∫02 f1(x)dx+∫02 xf1(x)dx=2e2−2[f(x)]02+2e2−I=2e2−2e2−1+2e2−I2I=2e2−2e2+2+2e2=21+e2I=1+e2