Problems on Techniques of Integration

Use the Integration by Parts. Set

u &=&x\\
dv &=& \sin(3x)dx\;.


du &=&dx\\
v &=&\displaystyle - \frac{1}{3}\cos(x)\;.


\begin{displaymath}\int x \sin(3 x) dx = - x \frac{1}{3}\cos(x) - \int - \frac{1}{3}\cos(x) dx \end{displaymath}


\begin{displaymath}\int x \sin(3 x) dx =- \frac{x}{3}\cos(x) + \frac{1}{3}\sin(x) + C \;.\end{displaymath}

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