The given integral = $\int {\frac{{2\sin \frac{{3x}}{2}\sin \frac{x}{2}}}{{1 + 2 - 4{{\sin }^2}\frac{x}{2}}}dx} $=$\int {\frac{{2\sin \frac{{3x}}{2}\sin \frac{x}{2}}}{{3 - 4{{\sin }^2}\frac{x}{2}}}dx} $
=$ \int {\frac{{2\sin \frac{{3x}}{2}\sin \frac{x}{2}.\sin \frac{x}{2}}}{{3\sin \frac{x}{2} - 4{{\sin }^3}\frac{x}{2}}}dx} $
=$\int {\frac{{2\sin \frac{{3x}}{2}{{\sin }^2}\frac{x}{2}}}{{\sin \frac{{3x}}{2}}}dx} $
=$\int {2{{\sin }^2}\frac{x}{2}dx} $
=$ \int {(1 - \cos x)dx} $
=$ x - \sin x + C$