Prove that $\int \ddot{x}(t)\mathrm dt=v_0 + \frac{F_0}{m}t$
Solution 1:
its not stated anywhere but I am going to assume (like the answer seems to) that $a$ is constant for all $t$ i.e. $a'=0$ now this means that: $$v(t)-v(0)=\int_0^t a\,d\tau\\v(t)=at+v(0)\\v(t)=v_0+at$$ which is what they said. now just use the fact that: $$\sum F=ma$$ which leaves you with: $$v=v_0+\frac{F}{m}t$$