Evaluation of Definite Trigonometric Integrals: A Comparative Study of Methods and Applications

Abstract

The first type of calculus problem is a definite integral of a trigonometric function, and it has many applications. However, the types of trigonometric integrals are not all the same, and different solution methods have varying levels of effectiveness depending on the structure of the integrand. This paper studies and compares several typical methods for evaluating trigonometric definite integrals, such as basic integration formulas, trigonometric identity transformations, u-substitution, power-reduction formulas and symmetry properties. First, classify the integrals by their attributes, and then, according to the three criteria of computational steps, applicability and ease of simplification, evaluate each method. Based on the above results, u-substitution can be used for many difficult trigonometric integrals, and symmetry-based methods are generally more convenient when applicable. In addition, this paper shows some applications of trigonometric definite integrals in simple harmonic motion, alternating current circuits and acoustic wave analysis. This paper establishes a systematic framework for selecting integration methods, clarifying which techniques are most suitable for different types of trigonometric integrals.