This paper proposes a backstepping nonlinear control to perform time-varying trajectory tracking of a quadrotor UAV, assuming its whole six degree of freedom (6DOF) dynamics. The control law is designed with integral action of the controlled velocities' errors, in order to guarantee convergence to the reference signals. By using the whole-body dynamic model in the backstepping approach and assuming the desired behavior of the rotation matrix as a part of a virtual control, it is obtained reference trajectory tracking and stabilization of rotational movements, avoiding singularities in the translational control. The stability analysis is performed for the complete system through the Lyapunov Theory. Simulations are carried out to corroborate the efficiency of the proposed controller when tracking a time-varying trajectory of the regulated variables (x, y, z and ).