A Lecture on Integration by Parts

Integration by parts (IBP) has acquired a bad reputation. While it allows us to compute a wide variety of integrals when other methods fall short, its implementation is often seen as plodding and confusing. Readers familiar with tabular IBP understand that, in particular cases, it has the capacity to significantly streamline and shorten computation. In this paper, we establish a tabular approach to IBP that is completely general and, more importantly, a powerful tool that promotes exploration and creativity.