Calculating Integrals Pertaining Trigonometric Function and Polynomial Function by Residue Theorem

Within the domain of complex analysis, the residue theorem, occasionally referred to as Cauchy's residue theorem, emerges as a formidable and indispensable technique for the assessment of line integrals along closed curves involving analytic functions. Remarkably, this unique trait makes it a prized tool in the mathematician's arsenal, which frequently finds application in the calculation of both real integrals and infinite series. Cauchy's Residue Theorem simplifies the solution of complex integrals by focusing on singularity points, offering a streamlined and efficient alternative approach to the complexities associated with conventional integral methods. This paper utilizes trigonometric and polynomial functions as illustrative examples to underscore the significance of the residue theorem in the assessment of improper integrals. Furthermore, it offers in-depth examinations of three distinct examples for each category. These comprehensive evaluations serve to emphasize the importance of the residue theorem in the field of complex analysis and highlight its versatility in addressing a wide range of mathematical challenges.