Question
Integrate the function x sin 3x
$= x\int {\sin 3xdx - \int {\left( {\frac{d}{{dx}}x\int {\sin 3x} dx} \right)dx} } $
[Applying product rule]
$= x\left( {\frac{{ - \cos 3x}}{3}} \right) - \int {1\left( {\frac{{ - \cos 3x}}{3}} \right)dx + c}$
$ = \frac{{ - 1}}{3}x\cos 3x + \frac{1}{3}\int {\cos 3xdx + c} $
$= \frac{{ - 1}}{3}x\cos 3x + \frac{1}{3}\frac{{\sin 3x}}{3} + c$
$= \frac{{ - 1}}{3}x\cos 3x + \frac{1}{9}\sin 3x + c$
$= - \frac{x}{3}\cos 3x + \frac{1}{9}\sin 3x + c$
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$ \begin{vmatrix} \text{cos 15}^{0}& \text{sin 15}^{0} \\ \text{sin 75}^{0} & \text{cos 75}^{0} \\ \end{vmatrix}$.