Question

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

• A. $ e^{x} \cos \,x +C$
• B. $ e^{x} \sin\, x +C$
• C. $ e^{x} \sin^{2} \,x +C$
• D. $ e^{x} \sin \,2x +C$
• E. $ e^{x} \left(\cos\, x+\sin x\right) +C$

Answer 1

Answer: (C)

Let $I =\int e^{x}(\sin \,x+2 \,\cos \,x) \sin \,x \,d x $
$ =\int \underset{II}{e^{x}} \, \underset{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 $