Integrating an equation of motion

The form of the E.o.M almost implies straight away there's a conserved quantity: i.e. it looks as if you can write it of the form $frac{d}{dt}(...) = 0$. In the case you presented, $2 dot{r} dot{phi} + r ddot{phi} = 0$, it should be easy to see that this can be written as $$ frac{1}{r} frac{d}{dt} big(r^2 dot{phi} big) =0 ,$$ which is exactly the form we're looking for. It implies that $r^2 dot{phi}$ is a constant and conserved. We introduce the constant $m$ just as we would when doing indefinite integrals, as this doesn't affect the equations of motion. This is how we conclude that $m r^2 dot{phi}=const$. There's also less 'guess work' if you know to look for something of the form $frac{d}{dt}(...) = 0$ in the E.o.M, rather than starting from the solution and just checking it.