\(\int 6 \sin 3x \cos 2x \:dx\)
\begin{equation*}
\begin{split}
& \int 6 \sin 3x \cos 2x \:dx\\\\
& 3 \int 2 \sin 3x \cos 2x \:dx\\\\
& {\color {blue} 2\sin A \cos B = \sin (A+B) + \sin (A - B) }\\\\
& 3 \int \sin (3x+2x) + \sin (3x - 2x) \:dx\\\\
& 3 \int \sin 5x + \sin x \:dx\\\\
& 3 \int \sin 5x \: \frac {d(5x)}{5} + 3 \int \sin x \:dx\\\\
& - \frac{3}{5} \cos 5x - 3 \cos x + C
\end{split}
\end{equation*}