Question

Integrate the function: $xsin\ 3x$

Answer 1

$Let\ I=∫xsin\ 3xdx$

Taking x as first function and $sin\ 3x$ as second function and integrating by parts, we obtain

$I= x∫sin\ 3x dx-∫{(\frac {d}{dx}x)∫sin\ 3x dx}$

$I = x(\frac {-cos\ 3x}{3})-∫1.(\frac {-cos\ 3x}{3})dx$

$I = \frac {-xcos\ 3x}{3}+\frac 13∫cos\ 3x dx$

$I = \frac {-xcos \ 3x}{3}+\frac 19sin\ 3x+C$