This video introduces a way to solve simple ODEs in the form $y' = f(x)$ simultaneously with an initial value problem, using definite integration. My differential equations book presents what appears to be a similar, definite integration in the section on separable differential equations, but in my opinion doesn't explain the step clearly enough to be helpful.
In the simplest terms possible, why is this algorithm equivalent to solving an initial value problem the traditional way, and how, if at all, can it be applied to ODEs that require more than simple integration (and perhaps more than separation of variables) and/or multiple "initial" values (i.e., problems with both a proportionality constant and an integration constant) to solve?