Since the time derivative of the velocity function is acceleration, d dt v ( t ) = a (t ), we can take the indefinite integral of both sides, finding ∫ d dt v (t )dt = ∫ a( t )dt + C 1, where C 1 is a constant of integration. Since ∫ d dt v (t ) dt = v (t ), the velocity is given by v (t ) = ∫ a( t )dt + C 1. Similarly, the time derivative of the position function is the velocity function, d dt x ( t ) = v (t ). Thus, we can use the same mathematical manipulations we just used and find x (t ) = ∫ v ( t )dt + C 2, where C 2 is a second constant of integration. We can derive the kinematic equations for a constant acceleration using these integrals. With a(t ) = a a constant, and doing the integration in (Figure) , we find v (t ) = ∫ adt + C 1 = at + C 1. If the initial velocity is v (0) = v 0, then v 0 = 0 + C1 . Then, C1 = v 0 and v (t ) = v 0 + at ,