Application of Cauchy's Residue Theorem in Evaluating Real and Trigonometric Integrals

Abstract

Evaluating definite real and trigonometric integrals remains a fundamental problem in mathematical analysis, particularly when elementary calculus techniques are insufficient or inefficient. This study aims to examine the application of Cauchy’s Residue Theorem as an analytical method for evaluating selected definite real and trigonometric integrals. The analysis transforms the original integrals into appropriate contour integrals in the complex plane, identifies the relevant isolated singularities, computes their residues, and applies the Residue Theorem to derive exact integral values. The results show that this approach provides systematic and exact solutions to integrals that are otherwise difficult to evaluate using conventional real-variable methods. The study concludes that residue theory offers an effective and mathematically elegant framework for definite integral evaluation. These findings reinforce the practical significance of complex analysis and provide a methodological reference for selecting suitable contours and calculating residues in advanced mathematical problem-solving.