Time integration for displacement

Hi,
In dem_doc_2012-1.pdf (doe.gov), the slope terms / (change rate) of total force or torque are based on t. However, this term is based on (t+∆t) only for displacement. Why ?

This is first order time integration. First we update the velocity, then we use the updated velocity to update position.

Thank you. I mixed it with the second-order technique, in which expressions for translational velocity, position and angulay velocity are similar.