If int_0^x f(t) dt = x + int_x^1 t f(t) dt, then the value of f(1) is

If int_0^x f(t) dt = x + int_x^1 t f(t) dt, then the value… — JEE Main 1998 Mathematics PYQ

1998

\int_0^x f(t)\, dt = x + \int_x^1 t\, f(t)\, dt

, then the value of

f(1)

A.
$\frac{1}{2}$
B.
$0$
C.
$1$
D.
$-\frac{1}{2}$

Show answer and solution

Correct option: A

Solution

Differentiate both sides with respect to x using Leibniz rule:

f(x) = 1 + (-x·f(x))

So f(x) + x·f(x) = 1, giving f(x) = 1/(1+x).

At x = 1: f(1) = 1/2.

[1998]

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JEE Main 1998 Mathematics question on Definite Integrals and Areas