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

JEE Main · 2026 · 24 Jan 2026 · Shift 2 · Definite Integrals

f (x)

satisfies the relation

f (x) = e^{x} + \int_{0}^{1} (y + xe^{x}) f (y) d y

, then

e + f (0)

is equal to ______.

Answer: 2

Separating the kernel,

f (x) = e^{x} + \int_{0}^{1} y f (y) d y + x e^{x} \int_{0}^{1} f (y) d y = 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

. Then

B = \int_{0}^{1} (e^{y} + A + B y e^{y}) d y = (e - 1) + A + B

, using

\int_{0}^{1} y e^{y} d y = 1

, which forces

A = 1 - e

. Hence

f (0) = 1 + A = 2 - e

, and

e + f (0) = 2

JEE Main · 2026 · 24 Jan 2026 · Shift 2 · Integer answer

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