If f'(x) = x |sinx| ∀ x  ∈ (0, π) and f(0) = 1/√3 , then f(x) will be

Q: If f'(x) = x |sinx| ∀ x  ∈ (0, π) and f(0) = 1/√3 , then f(x) will be

(A) – x cosx + sinx + 1/√3

(B) 1/√3 – x cosx + sinx

(C) sinx + x cosx – 2/√3

(D) sinx + x cosx + 2/√3

Sol. f'(x) = x |sinx| ∀ x  ∈ (0, π)

$\large \int f'(x) dx = \int x sinx dx $

$\large f(x) = – x cosx + \int cosx dx + c $

$\large = – x cosx + sinx + c $

f(0) = c = 1/√3

$\large f(x) = – x cosx + sinx + \frac{1}{\sqrt{3}} $

Hence (A) is the correct answer.