Actually I had almost the same question in my 3u exam, the hard part in this question for a lot of my mates was actually to realise the fact that you have to implement natural log in the integral
There is a very
VERY subtle trick placed there that reveals that an exponential might've been needed.
If you look closely at the initial equation, recalling that v is just dx/dt, this is actually motion in the form of exponential decay. The final answer was virtually doomed to have an exponential within it. The consequence is that because we find dt/dx before dx/dt, we find t = f(x) + C first. Because f(t) and g(x) are bound to be mutually inverse functions (or even just by looking at the inverse function theorem dx/dt dt/dx = 1) it's essentially unavoidable that the natural log must appear halfway through.
You can try explaining this to those kids at your school if you wish