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How to find constant equal to what in integration?

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

That's what I did. But, book says

$$x(t)=\int \dot{x}(t)\mathrm dt=x_0+v_0 t+ \frac{F_0}{2m}t^2$$

Seems like, $x_0 + \dfrac{a_0}{2}t^2$ is constant. How to find constant is equal to what?

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1 answer

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

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

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

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

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

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

Helped by PF

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