JEE Main 2026 Jan 24 Shift 2, Mathematics Q21: Some Properties of Definite Integrals

JEE Main2026Jan 24, Shift 2Mathematics

If $f (x)$ satisfies the relation $f (x) = e^{x} + \int_{0}^{1} (y + x e^{x}) f (y) d y$ , then $e + f (0)$ is equal to ________.

Your answer

Solution

Since $x$ is constant inside the integral, write

$f (x) = e^{x} + A + B x e^{x}$ , where $A = \int_{0}^{1} y f (y) d y$ and $B = \int_{0}^{1} f (y) d y$ .

Compute $B$ : $B = \int_{0}^{1} (e^{y} + A + B y e^{y}) d y = (e - 1) + A + B \cdot 1$ , since $\int_{0}^{1} y e^{y} d y = 1$ .

$B - B = e - 1 + A \Rightarrow A = 1 - e$ .

(The value of $B$ itself is also needed for $f$ , but $f (0)$ depends only on $A$ :)

$f (0) = e^{0} + A + B \cdot 0 \cdot e^{0} = 1 + A = 1 + (1 - e) = 2 - e$ .

$∴ e + f (0) = e + (2 - e) = 2$ .

Concept behind this question

Concept-wise practice with instant solutions on Fundamenthol.