The value of the integral $|\int\limits_0^{2\pi}[2\sin x] dx|$ is ([.] denotes the greatest integer function)

$π$

$I=\int\limits_0^{π/6}0.dx+\int\limits_{π/6}^{π/2}1.dx+\int\limits_{π/2}^{5π/6}1.dx+\int\limits_{5π/6}^{π}0.dx+\int\limits_{π}^{7π/6}-1dx+\int\limits_{7π/6}^{11π/6}-2dx+\int\limits_{11π/6}^{2π}-1dx=\frac{2π}{3}-\frac{2π}{6}-2\frac{4π}{6}=-π$