Let f(x) be a continuous function from R to R such that f(x) = 1 - f(2 - x) Which one of the following options is the CORRECT value of ∈t_0^2 f(x)\,dx?

GATE CSE 2024 Session 2 Q16: Integral of f over [0, 2] when f(x) = 1 − f(2 − x)

Let $f(x)$ be a continuous function from $\mathbb{R}$ to $\mathbb{R}$ such that

$f(x) = 1 - f(2 - x)$

Which one of the following options is the CORRECT value of $\displaystyle\int_0^2 f(x)\,dx$ ?

Try it: multiple choice: exactly one correct option

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Answer: B (official GATE 2024 final answer key)

Concept used: The substitution $x \to 2 - x$ leaves $\int_0^2$ unchanged.

First step: Let $I = \int_0^2 f(x)\,dx = \int_0^2 f(2-x)\,dx$ ; adding gives $2I = \int_0^2 [f(x) + f(2-x)]\,dx = \int_0^2 1\,dx$ .

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Source: official GATE 2024 Session 2 CS question paper and final answer key, organised by IISc Bangalore. Question paper · Answer key