A calculus problem by Jose Sacramento

Calculus Level 3

Let $$f: [0,1] \to \mathbb R$$ be a continuous and positive function satisfying $$f(x) f(1-x) = 1$$ for $$0 \leq x \leq 1$$.

Compute the closed form of $$\displaystyle \int_0^1 \dfrac1{1+f(x)} \, dx$$.

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