I'm doing a perturbation methods course where we have shown that often (but not always - e.g. if dominant contribution is not at an end point of the integral, or if boundary term diverges etc), doing integration by parts succesively will generate an asymptotic approximation to the integral.
Whereas for Laplace's method and the method of stationary phase (etc) I have a 'mental picture' of why I would expect these methods to generate an asymptotic approximation, I have nothing for this integration by parts method.
My question is, is integration by parts purely an 'almost random method that often works' to generate an asymptotic approximation, or is there some mental picture you can give me for why we expect it to sometimes do this?