What question did this study set out to answer?

This research aims to clarify the use of integration by parts, particularly when integrals reappear due to cyclic patterns.

June 15, 2026Open Access

Cyclic Integration by Parts

Key Points

This research aims to clarify the use of integration by parts, particularly when integrals reappear due to cyclic patterns.
Developed a general theorem applicable to the family of integrals eax sin(bx)dx.
Worked through progressively complex examples to demonstrate the theorem's application.
Created a classification table for better understanding.
Demonstrated that using integration by parts can lead to functions reappearing cyclically.
Established that no error is made when encountering loops, necessitating algebraic solutions.
Provided a historical context highlighting sec3 xdx as a classical example related to the technique.

Abstract

Integration by parts is a tool used by every calculus student.The technique gives goodresults more often than not,when it does not, it is usually because the result yielded a newawkward integral that takes the same form as the original one resulting in a loop.Whenintegration by parts is applied to certain inetgrals, the original integral sometimes reappear,this leads to confusion sometimes, making students question if they chose the right techniqueor there are mistakes.Turns out there is no mistake after all as now we have to solve theintegral algebracally. We present a general theorem for the family eax sin(bx)dx, workthrough examples of increasing complexity, and conclude with a classification table. A briefhistorical note familiarise the reader with the technique and explains why sec3 xdx is aclassical example

Cyclic Integration by Parts

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Abstract

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