Lecture 7. The Indefinite Integral. Antiderivative.

This lecture introduces the concept of the indefinite integral as the set of all antiderivatives of a function. The main theorem on antiderivatives is presented, showing that any two antiderivatives differ by a constant. The geometric interpretation (a family of integral curves) and the basic properties of linearity are discussed. A comprehensive table of basic integrals is provided, which students must memorize. Special attention is given to the method of direct integration transforming the integrand into tabular form using algebraic manipulations and trigonometric identities. Each step is illustrated with worked examples, including verification by differentiation. The lecture lays the foundation for further study of more advanced integration techniques.