Since it is a form of linear differential equation.

Integrating Factor (I.F) = e^{∫ p dx}

Solution of differential equation is given by

y.(I.F) = ∫ Q.(I.F) dx + C

⇒ y. x = ∫ (sin x).x dx + C

⇒ y. x = ∫ (sin x).x dx + C

Consider integral ∫ (sin x).x dx

Treating x as first function and sin x as second function. So, integrating by Parts we get,

⇒ x. (-cos x) + ∫ 1.cos x dx + C

⇒ – x. cos x + sin x + C

∴ y. x = – x. cos x + sin x + C

⇒ x (y + cos x) = sin x + C = (A) is the required solution.