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Tic-Tac-Toe and repeated integration by parts: alternative pedagogical perspectives to Lima's integral challenge

ABSTRACT Recently, Lima lamented on the ‘time-consuming and tiresome’ pedagogical nature of repeated integration by parts, throwing down the challenge of providing a pencil-and-paper solution to a related problem in a few seconds. Lima put forth a simple formula as an alternative. In this work, I offer a response to Lima's challenge and his formula. Drawing on the concepts of deconstruction and problematization, I offer alternative pedagogical perspectives regarding repeated integration by parts that involves tabular integration (aka Tic-Tac-Toe). I also explore the case of Lima's formula when an exponent in the integral is purely imaginary and compare this approach with repeated integration by parts. This paper forms alternative pedagogical constructs for students and educators to work with in the space of repeated integration by parts.