CHAPTER 4 - Indefinite Integrals of Algebraic Functions

Abstract

This chapter describes the indefinite integrals of algebraic functions. A function f (x) is said to be algebraic if a polynomial P(x, y) in the two variables x, y can be found with the property that P(x, f(x)) = 0, for all x for which f(x) is defined. Functions that are not algebraic are called transcendental functions. Fundamental difference between algebraic and transcendental functions is that whereas algebraic functions can only have a finite number of zeros, a transcendental function can have an infinite number. Thus, for example, sin x has zeros at x = ±nπ, n = 0, 1, 2, Transcendental functions arise in many different ways, one of which is as a result of integrating algebraic functions. The simple algebraic functions can be integrated by first expressing their integrands in terms of partial fractions, and then integrating the result term by term. Indefinite integrals are classified according to their integrands as rational functions, nonrational algebraic functions, or transcendental functions.