Let f(x) be a differentiable function defined on [0,2], such that f'(x)=f'(2-x) for all x(…

Question

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

Accepted Answer

f'(x) = f'(2 - x)

On integrating both side f(x) = -f(2 - x) + c

put x = 0

f(0) + f(2) = c $$ \Rightarrow $$ c = 1 + e2

$$ \Rightarrow $$ f(x) + f(2 - x) = 1 + e2 ..... (i)

$$I = \int\limits_0^2 {f(x)dx}  = \int\limits_0^1 {\{ f(x) + f(2 - x)\} dx = (1 + {e^2})} $$

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