CHAPTER XI - Primitives

Abstract

This chapter discusses the technique of finding the primitives of a large class of functions. It explains a theorem rule of integration by parts, which is useful in evaluating the primitives of products. The chapter describes the way by which the primitives of rational functions can be determined in terms of the elementary functions—that is, the rational, trigonometric, and exponential functions and their inverses. A rational function is a quotient of two polynomials. A rational function is called proper if the degree of the numerator is less than that of the denominator; otherwise it is called improper. By division, an improper rational function can be expressed as the sum of a polynomial and a proper rational function. The primitives of polynomials are found readily. The chapter discusses the evaluation of primitives of proper rational functions with quadratic denominators. It also presents a unique factorization theorem, which explains that every polynomial with real coefficients can be factored in one and only one way into linear and irreducible quadratic factors, some of which may be repeated.