This article discusses the application of Cauchy's Residue Theorem to improper integrals of real functions. The theorem states that the integral along a simple closed contour is equal to 2πi times the sum of the residues of the function at single points located inside the contour. Furthermore, the article describes various methods for determining the residues, including the use of Laurent series and Taylor series. In addition, Jordan's lemma is also referenced in this article. Cauchy's Residue Theorem on improper integrals can be employed to resolve integrals that are challenging to compute using traditional real analysis methods. By identifying the residue of the integral at a singularity within a closed contour, the integral along the contour can be evaluated. The application of the residue theorem to improper integrals can be expressed in a specific form to facilitate calculation. This method offers several advantages over conventional methods. Some of the sources consulted in the preparation of this article include the following publications: Complex Analysis, Residue Theorem and Its Applications, and Calculus Applications in Physics Lectures.