I still don’t understand. How is taking the integral useful? You’ve moving from a function whose second derivative’s changes of sign are the points of interest to a function whose third derivative’s changes of sign are the points of interest. It’s all the bother of taking the derivative, except in the other direction, further away from what you’re looking for!
For example, consider the function 4x[sup]3[/sup] + 9x[sup]2[/sup] + 1. Suppose I want to know where its inflection point is and I start by taking the integral, x[sup]4[/sup] + 3x[sup]3[/sup] + x + C. Now what?