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Q)

$\begin{align*}\int \frac{\sin 4x}{ \sin x} dx \end{align*}$


$(A)\;2 [\frac{\sin 3x}{3} - \sin x] + c$
$(B)\; [\frac{\sin 3x}{3} - \sin x] + c$
$(C)\; [\frac{\sin 3x}{3} + \sin x] + c$
$(D)\;2 [\frac{\sin 3x}{3} + \sin x] + c$

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A)
$\int \frac{sin 4x}{\sin x} dx = \int \frac{2 \sin 2x \cos 2x}{ \sin x} dx$
$ = \int \frac{4 \sin x \cos x \cos 2x }{\ sin x} dx$
$=2 \int 2 \cos x cos 2x dx $
$= 2 \int (\cos 3x + \cos x) dx $
$= 2 [\frac{\sin 3x}{3} + \sin x] + c$
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