$$\require{cancel}$$
There are not really any new principles; it is just a matter of practice with slightly more difficult integrals. We assume that the resistive force per unit mass is $$k \dot{x}^{2}$$. Here, although $$k$$ plays a somewhat similar role to the $$\gamma$$ of Section 3, it is not exactly the same thing as $$\gamma$$, and indeed it is not dimensionally the same as $$\gamma$$. What are the dimensions, and the SI units, of $$k$$?