Post History

$$c\neq x_0 + \frac{a_0}{2}t^2$$

$$\dot{x} (t)=\int \ddot{x} t dt=\ddot{x}t+c$$

$$\dot{x}(t)=\ddot{x}t+\dot{x}(0)$$

$$x(t)=\int \dot{x} (t) dt$$

$$=\int \ddot{x}t+\dot{x_0} dt$$

$$=\dot{x_0}t+\frac{\ddot{x}}{2}t^2+c$$
$$=\dot{x_0}t+\frac{\ddot{x}}{2}t^2+x_0$$

Helped by [PF](https://www.physicsforums.com/threads/how-to-find-constant-equal-to-what-in-integration.1006206/post-6529345)