I read the derivation of $v=u+at$ using integration. The steps are as follows -

$$\frac{dv}{dt}=a dt$$
$$dv=adt$$
$$\int dv=\int a dt$$
$$\int dv=a\int dt$$
$$v=at+c$$
My questions are as follows -

**1. On integrating $dt$ on the RHS we get a $+c$ (constant of integration) but why is there no $+c$ on the LHS while integrating $dv$?**

**2.Can we take out something of the integration symbol as $a$ is taken out is the second last step?**

**3.Why in the the last step it is $at+c$ and not $at+ac$?**