The function $f(x)$, that satisfies the condition $f(x)=x+\int_{0}^{\pi/2} \sin x \cdot \cos y f(y) dy$ is:

(A) $x+\frac{2}{3}(\pi-2) \sin x$

JEE Main · 2021 · 01 Sep 2021 · Shift 2 · Definite Integrals

The function

f (x)

, that satisfies the condition

f (x) = x + \int_{0}^{π /2} sin x \cdot cos y f (y) d y

is:

Answer: (D)

Let

C = \int_{0}^{π /2} cos y f (y) d y

(a constant). Then

f (x) = x + C sin x

. Substituting back:

C = \int_{0}^{π /2} cos y (y + C sin y) d y = \int_{0}^{π /2} y cos y d y + C \int_{0}^{π /2} cos y sin y d y

. Using integration by parts:

\int_{0}^{π /2} y cos y d y = [y sin y]_{0}^{π /2} - \int_{0}^{π /2} sin y d y = \frac{π}{2} - 1

. And

\int_{0}^{π /2} cos y sin y d y = \frac{1}{2}

. So

C = \frac{π}{2} - 1 + \frac{C}{2}

, giving

C = π - 2

. Therefore

f (x) = x + (π - 2) sin x

JEE Main · 2021 · 01 Sep 2021 · Shift 2 · Single correct option