Question

Solve it:
$\int \sin \frac{x}{2}\sin \frac{x}{3}\cos \frac{x}{4}dx$ .

Answer 1

$I=\int \sin \frac{x}{2}\sin \frac{x}{3}\cos \frac{x}{4}dx$

$\sin \frac{x}{2}\sin \frac{x}{3}\cos \frac{x}{4}$

$=\frac{1}{2}\left(2\sin \frac{x}{2}\sin \frac{x}{3}\right)\left(\cos \frac{x}{4}\right)$

$=\frac{1}{2}\left(\right(\cos \frac{x}{6}-\cos \frac{5x}{6}\left)\cos \frac{x}{4}\right)$

$=\frac{1}{2}(\cos \frac{x}{6}\cos \frac{x}{4}-\cos \frac{5x}{6}\cos \frac{x}{4})$

$=\frac{1}{2}\left\{\frac{1}{2}\right(2\cos \frac{x}{6}\cos \frac{x}{4})-\frac{1}{2}(2\cos \frac{5x}{6}\cos \frac{x}{4}\left)\right\}$

$=\frac{1}{4}(\cos \frac{5x}{12}+\cos \frac{x}{12}-\cos \frac{13x}{12}-\cos \frac{7x}{12})$

$I=\frac{1}{4}\int (\cos \frac{5x}{12}+\cos \frac{x}{12}-\cos \frac{13x}{12}-\cos \frac{7x}{12}dx$

$=\frac{1}{4}[\frac{12}{5}\sin \frac{5x}{12}+\frac{12}{1}\sin \frac{x}{12}-\frac{12}{13}\sin \frac{13x}{12}-\frac{12}{7}\sin \frac{7x}{12}]+C$