The value of $\int\limits_0^{2\pi } {\left[ {\sin 2x\left( {1 + \cos 3x} \right)} \right]} dx$, where [t] denotes the greatest integer function is :

Question

The value of $\int\limits_0^{2\pi } {\left[ {\sin 2x\left( {1 + \cos 3x} \right)} \right]} dx$,
where [t] denotes the greatest integer function is :

2$\pi $