In this graduate thesis we have presented the main conceptual frameworks for one of the most important mathematical concept, integral of functions of a single variable. The mathematical background of (definite and indefinite) integral of a function of a single variable is overviewed from different perspectives at different levels of education. We have concentrated on one of the crucial theorems of the calculus within which integrals of functions of a single variable are a part, The fundamental theorem of calculus (FTC) and the underlying concepts. Together with the aforementioned ideas as our main goal, we have tried to emphasize the conceptual links of the rate of change and the accumulated change of the quantity, that is that a certain integral rate of change of a function (dependence of two quantities) is equal to the total accumulated change in function value on an observed interval. Such resonance allows us to develop conceptual understanding of the underlying concepts of the integral calculus and its application in various disciplines that use mathematics (e.g. physics, economy) and in situations in world around us, implying that the concept of integral is a significant part of the high school education. Besides, the goal of this thesis was to present different didactical realizations of the concept of integral calculus from university and high school textbooks in Croatia, and examples of didactical activities for pupils and students from European and world literature. In this way we highlight the diverse and broad possibilities of development of conceptual understanding of the calculus at all levels of education, especially in high school mathematics education.