Let f(x) be a differentiable function satisfying f(x) = ∫[0 to x] (f(t) cos(t)) dt. Then f(x) is equal to:
Correct: A
Differentiating both sides with respect to x, we get f'(x) = f(x) cos(x). So f'(x)/f(x) = cos(x). Integrating both sides with respect to x, we have ∫(f'(x)/f(x)) dx = ∫cos(x) dx => ln|f(x)| = sin(x) + C. Therefore, |f(x)| = e^(sin(x) + C) = e^C * e^(sin(x)) = k * e^(sin(x)) where k = e^C > 0. We are given that f(x) = ∫[0 to x] (f(t) cos(t)) dt. When x = 0, f(0) = ∫[0 to 0] (f(t) cos(t)) dt = 0. Thus, f(0) = k * e^(sin(0)) = k * e^0 = k = 0. Thus f(x) = 0. The constant is 0 so f(x) = 0. However, C is also defined as ∫f'(x)/f(x), so this is wrong. It must be that f(x) = 0