Integrate the function x sin 3x - Vidyadip

Question

Integrate the function x sin 3x

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Answer

$\int {x\sin 3xdx} $

$= 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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