Far East Journal of Mathematical Education

The Far East Journal of Mathematical Education is a peer-reviewed journal focused on mathematical education. It publishes research papers that enhance understanding of mathematical concepts and encourages the use of technology, statistics, algorithms, and simulations in mathematics learning.

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PADÉ APPROXIMANTS FOR EXACT AND APPROXIMATE INDEFINITE INTEGRATION

Authors

  • Tat Leung Yee

Keywords:

indefinite integrals, reduction formulas, elliptic integrals, Padé approximants

DOI:

https://doi.org/10.17654/0973563125014

Abstract

This paper presents explicit closed-form solutions and computational methods for the indefinite integral $\int\left(x^{2 n}+1\right)^{-m} d x$ with positive integers $n, m$. By extending partial fraction decomposition to higherorder poles through complex factorization, we derive a reduction formula that recursively resolves cases $m>1$ to the fundamental $m=1$ solution expressed in logarithmic-arctangent form. Singularity analysis reveals connections to elliptic functions for fractional exponents, validated through Padé approximants and nonlinear ODEs. The methodology provides efficient computation of symmetric rational integrals with applications in nonlinear systems, demonstrated through comprehensive examples.

Received: June 10, 2025
Accepted: July 10, 2025

References

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[9] P. Poláčik, P. Quittner and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems, I: Elliptic equations and systems, Duke Math. J. 139 (2007), 555-579.

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Published

02-08-2025

Issue

Vol. 27 No. 2 (2025)

Section

Articles

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How to Cite

PADÉ APPROXIMANTS FOR EXACT AND APPROXIMATE INDEFINITE INTEGRATION. (2025). Far East Journal of Mathematical Education, 27(2), 109-130. https://doi.org/10.17654/0973563125014