∫ ex( sin x+2 cos x) sin x d x is equal to

Question

∫ ex( sin x+2 cos x) sin x d x is equal to (A)  ex cos x +C (B)  ex sin x +C (C)  ex sin2 x +C (D)  ex sin 2x +C

Tardigrade Admin · Accepted Answer

Let I =∫ ex( sin x+2 cos x) sin x d x =∫ undersetIIex undersetI sin2 x d x+∫ 2 ex sin x cos x d x Applying integration by parts in first integral, we get I=ex sin 2 x-∫ 2 sin x cos x ex d x +∫ 2 ex sin x cos x d x+C =ex sin 2 x+C

Q. $\int e^{x} (sin x + 2 cos x) sin x d x$ is equal to

$e^{x} cos x + C$

$e^{x} sin x + C$

$e^{x} sin^{2} x + C$

$e^{x} sin 2 x + C$

$e^{x} (cos x + sin x) + C$

Let $I = \int e^{x} (sin x + 2 cos x) sin x d x$
$= \int II e^{x} I sin^{2} x d x + \int 2 e^{x} sin x cos x d x$
Applying integration by parts in first integral,
we get
$I = e^{x} sin^{2} x - \int 2 sin x cos x e^{x} d x + \int 2 e^{x} sin x cos x d x + C$
$= e^{x} sin^{2} x + C$

Q. ∫ex(sinx+2cosx)sinxdx is equal to

excosx+C

exsinx+C

exsin2x+C