Differential kinematic is one of the most important solution methods in robot kinematics. The main advantage of the differential kinematic method is that it can be easily implemented any kind of mechanisms. Also, an accurate and efficient kinematic based trajectory tracking application can be easily implemented by using this method. In differential kinematic method, we use Jacobian as a mapping operator in the velocity space. Inversion of Jacobian matrix transforms the desired trajectory velocities, which are the linear and angular velocities of the end effector, into the joint velocities. The joint velocities are required to be integrated to obtain the pose of the robot manipulator. This integration can be evaluated by using numerical integration methods, since the inverse kinematic equations are highly complex and nonlinear. Therefore, the performance of the trajectory tracking application of the robot manipulator is directly affected by the chosen numerical integration method. This paper compares the performances of numerical integration methods in the trajectory tracking application of redundant robot manipulators. Several widely used numerical integration methods are implemented into the trajectory tracking application of the 7-DOF redundant robot manipulator named PA-10 and simulation results are given.