Do you see where it says
m(dv/dt)v = (d/dt)(.5mv^2)
If it's an integration which I don't think it is then I would think it should be
(d/dt)((mv^3)/3) because you're taking the two v's and adding an additional power. I don't think it is an integration because it says right there in the book that you can't integrate, so if it's now then where does the 1/2 come from?