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Your calculation is not correct. What you calculated is
$$\\int a\\,\mathrm d\textcolor{red}{a}$$
which is something very different from
$$\\int a(t)\\,\mathrm d\textcolor{red}{t}$$
Indeed, in the specific case here, $a = F_0/m$ is time independent, that is, it is a *constant* in respect to time. Now for any constant $k$ we have
$$\int k\\,\mathrm dt = kt+C$$
and in this specific case, $k=F_0/m$, therefore
$$\int a(t)\\,\mathrm dt = \int \frac{F_0}{m}\mathrm dt
= \frac{F_0}{m}t + C$$
Finally we notice that from the context, $C$ here has the meaning of a velocity; indeed, it is the velocity at $t=0$, therefore we name it accordingly, $C=v_0$.

Now obviously whether we write the constant term before or after the linear term is a matter of style only, as addition is commutative.