Correct me if I'm wrong, there is no reason to numerically integrate this. This is not a differential equation. The integral solution is a simple function that can just be evaluated.
Leaving it as a differential equation gives you some room to experiment more easily. For example, suppose you want to add a swimming mechanic to your game: you might end up with a completely different closed form solution, but if you left it as a differential equation it could just be a couple extra additive terms.
I think the article is handling it this way for pedagogical reasons. It's more clearly in the realm of "Software Engineering" to say "This is what our solution's trying to approximate, and the easiest approximation causes practical problems, so this is a better way to approximate the same thing (in an only slightly more complicated way) by adding an extra term." The author's approach is easier to understand because it's merely a small patch on an existing design.
If you say "By the way, there's an exact solution to this, it involves something called Integrals that's usually the focus of at least three semesters of Calculus in college, but you can't really understand it without a few courses in Real Analysis, Differential Equations, and Numerical Methods..." then it seems like you're talking too much about Math instead of Software Engineering. As a result, you lose the audience whose main interest is making games for fun and profit, and don't care about math (or so they think). It's much harder to understand because it's a complete redesign of the integrator that relies on a non-trivial body of theory.
