Calculation of the Dirichlet Integrals: From Basic to Generalized

This paper conducts research centered on the integral problems involving trigonometric functions, with a focus on exploring the solution methods in the case of improper integrals. Firstly, it reviews the basic processing methods of definite integrals and improper integrals and introduces the commonly used techniques in solving such integrals, including the application of methods such as variable substitution, integration by parts, and "taking partial derivatives and then integrating". Subsequently, it introduces the definition and typical properties of the Dirichlet integral, providing theoretical support for the subsequent solution of specific problems. Through an in-depth analysis of three specific integral problems, it demonstrates how to transform complex integrals into known forms for solution. The research shows that the rational application of the ideas of function transformation and limits not only helps to simplify the calculation process but also effectively improves the accuracy and efficiency of problem-solving. The discussion in this paper has certain reference value for understanding the structural characteristics and solution paths of improper integrals involving trigonometric functions.