4 - Indefinite Integrals of Algebraic Functions

Abstract

This chapter discusses the indefinite integrals of algebraic functions. The indefinite integrals are classified according to their integrands as rational functions, nonrational (irrational) algebraic functions, or transcendental functions. The chapter reviews the definitions of algebraic and transcendental 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. A fundamental difference between algebraic and transcendental functions is that while the algebraic functions can only have a finite number of zeros, a transcendental function can have an infinite number. The chapter discusses the indefinite integrals of rational functions with integrands involving xn, integrands involving a+bx, integrands involving linear factors, integrands involving a2 ± b2x2, integrands involving a + bx + cx2, integrands involving a + bx3, and integrands involving a + bx4. The chapter also discusses several integrands of nonrational algebraic functions.