I was reading these slides (very good by the way!), and if you skip all the way to the end the author compares all the different integrators presented.
In one way or another, they all fall short except for Improved Euler Integration and Runge Kutta Integration, which both pass all tests.
I suppose I should mention that I am working on a 2D game that isn't very physics intensive. I was just curious as to where Improved Euler Integration would fall short and Runge Kutta would have to be used instead.
My game mostly just consists of simple gravity (jumping and falling), movement along the X and Y axis, and bounding box collision. Would it be worth it to use Runge Kutta in that case or would Improved Euler integration be sufficient? I see in many discussions where users of Euler Integration are chastised but from what I can see it seems equivalent in simple 2D matters (the improved one, that is). Also, I'd imagine it'd be quite faster than Runge Kutta.
Edit: So apparently Euler Integration and Improved Euler Integration are a part of the family of Runge Kutta integration. That may have been confusing. When I am referring to Runge Kutta in this post I mean RK4.