The function f(x), that satisfies the condition
f(x) = x + ∫₀^(π /2) sin x.cos y f(y) dy, is :

Solution

Maths sir · 4 steps

The integral is just a constant times sin x, so call it k and solve for k.

1. Pull out sin x. The integration is in y, so sin x comes outside. Write k = ∫₀^(π/2) cos y f(y) dy, a fixed number. Then f(x) = x + k sin x.
2. Feed f back in. Put f(y) = y + k sin y inside the definition of k. This gives one equation in k alone.

   k = ∫₀^(π/2) y cos y dy + k∫₀^(π/2) sin y cos y dy

3. Evaluate both integrals. By parts, ∫₀^(π/2) y cos y dy = [y sin y + cos y]₀^(π/2) = π/2 - 1. Also ∫₀^(π/2) sin y cos y dy = 1/2. So k = π/2 - 1 + k/2.
4. So. k/2 = π/2 - 1, so k = π - 2 and f(x) = x + (π - 2)sin x, which is option (d).

The idea to keep. When the unknown function sits inside a definite integral with constant limits, that integral is a constant. Name it k, substitute the form of f back in, and solve for k.

Answer from the NTA final answer key. Solution by PrepWiser.