I'm having a look at a "proof" or motivating illustration of the Integrating Factor method for solving a first-order linear ODE, but can't quite understand the precise way in which one goes from step (1) below to step (2) here. I understand that one is to integrate both sides, but I think I'm a bit unclear as to how the integration is being done from to when the integrand itself is to involve t, and how the term [TEX]I(t_0)w(t_0)[TEX] arrives in the LHS. And presumably, on the RHS we are free to just change to a dummy variable in the integrand while we integrate over t?
Integrating both sides gives
Could anyone please elaborate?