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Q.

$\int {xe}^{x} \sin xdx$ is equal to

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a

$\frac{e^{x}}{2} {x (\sin x - \cos x) + \cos x} + C$

b

$\frac{e^{x}}{2} {x (\cos x - \sin x) + \sin x} + C$

c

$\frac{e^{x}}{2} {x (\sin x - \cos x) - \cos x} + C$

d

$\frac{e^{x}}{2} {x (\cos x - \sin x) - \sin x} + C$

answer is A.

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Detailed Solution

$\begin{array}{l} I = \int x (e^{x} \sin x) dx = x \frac{e^{x}}{2} (\sin x - \cos x) - \int \frac{e^{x}}{2} (\sin x - \cos x) dx \\ = x \frac{e^{x}}{2} (\sin x - \cos x) - \frac{e^{x}}{2} (- \cos x) + c = \frac{e^{x}}{2} [x (\sin x - \cos x) + \cos x] + c \end{array}$

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