In this equation $v(t)$ represents the velocity of an object falling to the ground, $m$ is the mass of the object, $g$ is the gravitational acceleration and $k > 0$ is a constant related to the air resistance.
The problem is taken from "Nonlinear Dynamics and Chaos" by Steven Strogatz (problem 2.2.13 on page 38). While the focus of the chapter is on solving this kind of nonlinear equation grafically, the first part of this question asks to solve it analitically. Maybe I'm missing some obvious trick, so can someone please give me a hint.
Observe that we can separate the equation as follows. Observe that $m\dot{v} = m \frac{dv}{dt} = mg - k v^2$, and so dividing one side by the other and integrating we obtain
$$ \int \frac{dv}{g - \frac{k}{m} v^2} \;\; =\;\; \int dt. $$
Take the substitution $v= \sqrt{\frac{gm}{k}}\sin u$ and we get that
$$ \int\sqrt{\frac{gm}{k}} \frac{\cos u}{g (1 - \sin^2u)} du \;\; =\;\;\sqrt{\frac{m}{gk}} \int \sec udu \;\; =\;\; t + c. $$
The question here is are you expected to compute this integral by hand (which is admittedly a little ad hoc) or are you allowed to look it up?