Complex analysis is a branch of mathematics that studies the properties and behaviors of functions of complex variables, where a complex variable is a quantity that has both a real part and an imaginary part. Complex analysis is important in many areas of science, including physics, engineering, and computer science. The importance of complex analysis lies in its ability to solve problems that are difficult or impossible to solve using only real variables alone. For example, because of the complicated integrals involved, many problems in fluid mechanics, electromagnetism, and quantum mechanics can be solved using complex analysis. This paper introduces an important theorem in complex analysis, which is the residue theorem. By applying the residual theorem, the trigonometric function integral and the polynomial integral exponential function integral are calculated, which simplifies the complexity and difficulty. With the help of examples, the application of the residue theorem is demonstrated. This paper contributes to extending the idea of integral calculation and facilitates the efficient solution of integral calculations in practical problems.