Analysis I (part 7) : Indefinite and definite integrals, integration (selected chapters)

Our final theme is integration. We begin by defining the indefinite integral and the definite integral. We introduce the definite integral using Riemann sums and upper and lower sums. We define three important properties of definite integrals: the linearity of the integral, the subdivision of the domain and the monotonicity of the integral. Function integration allows us to define the mean theorem. It states that the mean of a continuous function on a segment is realized as the value of the function at a certain point. We demonstrate this theorem. Finally, we come to the heart of the chapter with the fundamental theorem of integral calculus, which introduces the primitive of a function. We give a few examples of primitive calculations, as well as integration techniques (integration by parts, integration by change of variables, integration by recurrence). We conclude our discussion of integration by presenting the integration of particular functions: the integration of the limited expansion of a function, the integration of integral series and the integration of piecewise continuous functions. These three examples enable us to calculate a function more quickly if it has a special form. Finally, we extend the usual integration to that of generalized integrals. These are defined by passing to the limit in integrals. Three types of generalized integrals are presented, along with examples. The chapter concludes with the integration of rational functions using simple element decomposition.