Lecture 8. The Method of Change of Variable (Substitution) in the Indefinite Integral.

This lecture is devoted to one of the main methods of integration change of variable (substitution). The substitution formula is rigorously justified, and its special case, the method of bringing under the differential sign, is introduced. Linear substitutions for integrating functions of a linear argument are covered first. Nonlinear substitutions are then discussed, particularly for eliminating irrationalities involving roots of different degrees. Special attention is given to integrals containing a quadratic trinomial: completing the square followed by a linear substitution reduces them to tabular forms involving arctangents, arcsines, or logarithms. Numerous examples illustrate each technique, including splitting the numerator to separate the derivative of the denominator. The lecture concludes with a summary table of typical substitutions and practical recommendations. This material prepares students for more advanced integration methods.