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Integral is a most powerful and useful tool for solving various problems in natural sciences such as mathematics, physics and so on.We proceed strictly in a logical order, with the focus on the traditional approach of analytics in developing the theory of integration. We have focused on mathematical rigour, summarized the theory of integration with this focus, and more clearly described the integrability with singular integrals.First, we introduce the concept of antiderivative, and described the definition and properties of the indefinite integral, the calculus and, in particular, the integration of some elementary functions.Next, we discuss the definition of the definite integral the integrability criterion, and its properties. In the improper integral we describe improper of bounded function in infinite interval and unbounded function in bounded interval , and convergence criterion of improper integral .We also discuss the application of integration to geometric and physical quantities.