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.