kinematics, integrate a to v

• Integrate acceleration $a$ to obtain velocity $v$ $$\tag{E1} v(t) - v(t_0) = \int_{t_0}^t a(\tau_1) d\tau_1. $$
• Initial condition $v(t_0) = v_0$, will give $$\tag{E2} v(t) = v_0 + \int_{t_0}^t a(\tau_1) d\tau_1. $$