Cauchy integral theorem belongs to an extremely important part of complex functions, which is a fundamental bridge, and people can derive Cauchy integral theorem from the residue theorem. Cauchy's integral theorem is generally applied in many higher mathematics, is an important theorem concerning path integrals of fully pure functions. It claims that if there are two different paths from a point to another point and the function is fully pure between these two different paths, then it can be derived that the two path integrals of the function are equal. It is widely believed that the a generalization of Cauchy integral theorem and Cauchy integral formula is just like the residue theorem, and the flexible use of the residue theorem can easily solve some difficult problems in complex functions. Therefore, this paper will use the definition and derivation process of Cauchy residue theorem and integral theorem to demonstrate the specific connection between them and their practical application.