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A notion of the integral is one of the main ideas of mathematical analysis (or simply “analysis”). In some literature integrals are introduced in two steps: initially, the so-called indefinite integral is defined as an inverse operation to differentiation; then a definite integral is introduced via some natural applications such as an area under a curve. In this chapter we shall use a much clearer and more rigorous route based on defining the definite integral first as a limit of an integral sum. Then the indefinite integral is introduced in a natural way simply as a convenience tool.