How can the integrating factor method be derived to give a solution to a differential equation?

Consider the general equationdy/dx + Py = Qwhere P and Q are functions of x.R (which will be introduced later) is also a function of x.
So, all of a sudden, we are going to just state the product rule.We can say that d/dx (u X v) = u dv/dx + v du/dx.We can write this as d/dx (y X R) = y dR/dx + R dy/dx (*)If we compare this to our initial differential equation,
If we multiply our equation by our function R, we will getR dy/dx +RPy=RQNow, if we can find the function R such that if we differentiate this, we get RP then we can write the equation asd/dx (Ry)=RQ. We can then solve this to obtain y in terms of x.
But first, we need to find the said function R under the conditions.As dR/dx = RP, we can re-arrange this to give1/R dR/dx=PIf we rearrange this, we get 1/R dR = P dx.We can then integrate both sides to give us ln(R) = Int(P)dx.In this case, i will assume that constants of integration are negligible.This is also the seperation of variables methods for solving the equation dR/dx = RP.
So we know that ln(R) = Int(P)dx.raising each side to the power of e will give usR = e^(Int(P)dx).This is the integrating factor which i am sure you are all familiar with. hopefully this has given some conceptual ideas of where it comes from.