If f ( x ) satisfies the relation f ( x ) = e x + 0 x ( y + x e x ) f ( y ) d y , then e + f ( 0 ) is equal to _______ . [2026]

Question

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 _______ . [2026]

Smart Achievers · Accepted Answer

(2)

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

$f (x) = e^{x} + A + B x e^{x}$

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

$A = \frac{A}{2} + 0 - (- 1) + B (e - 1)$

$\frac{A}{2} + B (1 - e) = 1$

$B = \int_{0}^{1} f (y) d y$

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

$B = (e - 1) + A + B (0 - (- 1))$

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

$f (x) = e^{x} + A + B x e^{x}$

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

$e + f (0) = 2$

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