Yes, I used the notation $v(t)$ to indicate that $v$ is dependent on $t$. And you're right, the integration constant is zero. :)

@knzhou Thanks, I wasn't realizing that. From the explanation in my physics book it seemed to me like they defined $\epsilon_0$ to make that equation simpler (Gauss's law).

@GennaroTedesco For example because it is more efficient for computers to do multiplication than to do division (stackoverflow.com/questions/15745819/…). Thanks for the explanation that it is because of historical reasons. What do you mean exactly with the Coulomb force only representing one $k$?

With easier I mean more logical. It wouldn't make sense to define a constant as the inverse of something if there is no reasons for doing it.