$ \int e^{x \sin x}\left(x^{2} \cos x+x \sin x+1\right) d x $, equals (where $ C $ is constant of integration)

Question

$ \int e^{x \sin x}\left(x^{2} \cos x+x \sin x+1\right) d x $, equals (where $ C $ is constant of integration), $ x e^{x^{2} \sin x}+C $, $ x^{2} e^{x \sin x}+C $, $ x e^{x \sin x}+C $, $ 2 x^{2} e^{x \sin x}+C $

MARKS App · Accepted Answer

Let I = ∫ e x sin x x 2 cos x + x sin x + 1 d x ⇒ I = ∫ x . x cos x + sin x e x sin x + 1 . e x sin x d x Let e x sin x = t ⇒ e x sin x sin x + x cos x d x = d t Using integration by parts, I = ∫ x d t d x + t d x = x t + C = x e x sin x + C , where C is the constant of integration.

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