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GATE 2024 CS (CS2) – Question 16

Engineering Mathematics · Calculus · 1 mark · Multiple choice

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 $\int_0^2 f(x)dx$?

  1. 0
  2. 1
  3. 2
  4. $-1$

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Show answer and explanation

Correct answer: (B) 1

Explanation

Let I = ∫₀² f(x)dx. Substituting x→2−x gives I = ∫₀² f(2−x)dx = ∫₀²(1−f(x))dx = 2 − I, so I = 1.