GATE CS 2024 Set 2 Q.16: Calculus

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

$-1$

Official answer and solution

Solution

Use the substitution $u = 2 - x$ (or $\int_0^a f(x)\,dx = \int_0^a f(a-x)\,dx$ ) and add the two integrals.

Let $I = \int_0^2 f(x)\,dx$ . Using the property $\int_0^a f(x)\,dx = \int_0^a f(a - x)\,dx$ with $a = 2$ :

I = \int_0^2 f(2 - x)\,dx

Adding the two expressions for $I$ and using $f(x) + f(2 - x) = 1$ :

2I = \int_0^2 \bigl(f(x) + f(2 - x)\bigr)\,dx = \int_0^2 1\,dx = 2 \implies I = 1

Final answer

Option (B)

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

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