Wow. That’s really cool. It strikes me like integration by parts from another perspective: you are essentially replacing one integral with the sum of a function and another integral, hoping that the new integral is easier to compute, and you find it by playing around with the product rule and chain rule.
I just tried to use this technique to integrate f'(g(x)) in general, and noticed that the “second” approximation will work when g”(x) / (g'(x))^3 is a constant. This is precisely the situation we have here with g(x) = sqrt(x).

]]>