Many years ago I managed to complete my undergraduate calculus syllabus (mostly on integration methods) one lecture ahead of schedule, resulting in the need to give a additional lecture. As a consequence, I improvised a lecture on integration strategy as opposed to integration technique. Regarding integration by parts, for instance, I mentioned that some functions "like" to be differentiated, such as polynomials or (particularly) logarithms, some are "indifferent" to differentiation, such as exponentials, and some functions "prefer" to be integrated, such as total derivatives like ( 2x e^{x^2} ). The strategy of integration by parts is then to "move" the derivatives from functions that like to be integrated (or are at least indifferent) to those that like to be differentiated. Thus, for instance, if trying to use integration by parts to integrate 𝑥²log𝑥, it would make sense to integrate the 𝑥² factor to move a derivative onto the log𝑥 factor. (In analysis (particularly in PDE), there is a similar strategy of integrating oscillatory or rough factors to move derivatives onto smooth factors.) I had several students come up to me afterwards to tell me that this ad hoc lecture was the most useful one of the entire class... (3/3)