Techniques and Theories in Evaluating Definite Integrals Involving Trigonometric Functions

Zeming Tao

2025

Abstract

Trigonometric integrals play a significant role in mathematical analysis, particularly in calculus, Fourier analysis, and physics. Their evaluation requires systematic techniques due to the periodicity and oscillatory nature of sine and cosine functions. This paper considers fundamental strategies for evaluating definite trigonometric integrals, including symmetry properties, orthogonality relations, power reduction formulas, and product-to-sum formulas. One of the primary challenges here is simplifying complex integrals of products or powers of trigonometric functions. With reduction formulas and transformation identities, one can break integrals down into manageable terms. In this article, these methods are demonstrated step-by-step with examples, revealing how effectively they work on challenging integrals. The results show that the use of orthogonality properties can often eliminate entire terms, greatly simplifying calculations. Power reduction and product-to-sum identities also allow integration without messy algebraic manipulations. The results in this work have broad applications in physics, engineering, and signal processing, where trigonometric integrals are frequently used. This work provides a systematic way of integrating these integrals, reducing computational time in a wide variety of scientific and mathematical applications.

Paper Citation

in Harvard Style

Tao Z. (2025). Techniques and Theories in Evaluating Definite Integrals Involving Trigonometric Functions. In Proceedings of the 2nd International Conference on Innovations in Applied Mathematics, Physics, and Astronomy - Volume 1: IAMPA; ISBN 978-989-758-774-0, SciTePress, pages 90-93. DOI: 10.5220/0013814500004708

in Bibtex Style

@conference{iampa25,
author={Zeming Tao},
title={Techniques and Theories in Evaluating Definite Integrals Involving Trigonometric Functions},
booktitle={Proceedings of the 2nd International Conference on Innovations in Applied Mathematics, Physics, and Astronomy - Volume 1: IAMPA},
year={2025},
pages={90-93},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0013814500004708},
isbn={978-989-758-774-0},
}

in EndNote Style

TY - CONF

JO - Proceedings of the 2nd International Conference on Innovations in Applied Mathematics, Physics, and Astronomy - Volume 1: IAMPA
TI - Techniques and Theories in Evaluating Definite Integrals Involving Trigonometric Functions
SN - 978-989-758-774-0
AU - Tao Z.
PY - 2025
SP - 90
EP - 93
DO - 10.5220/0013814500004708
PB - SciTePress