■
■
■■
■5 Antidifferentiation
CONCEPTS AND SKILLS
In this chapter, we review
indefinite integrals,
formulas for antiderivatives of basic functions,
and techniques for finding antiderivatives (including substitution).
For BC Calculus students, we review two important techniques of
integration:
integration by parts,
and integration by partial fractions.A. ANTIDERIVATIVES
The antiderivative or indefinite integral of a function f (x) is a function F(x)
whose derivative is f (x). Since the derivative of a constant equals zero, the
antiderivative of f (x) is not unique; that is, if F(x) is an integral of f (x), then so
is F(x) + C, where C is any constant. The arbitrary constant C is called the
constant of integration. The indefinite integral of f (x) is written as ; thus
.The function f (x) is called the integrand. The Mean Value Theorem can be used
to show that, if two functions have the same derivative on an interval, then they
differ at most by a constant; that is, if , then
F(x) − G(x) = C (C a constant).B. BASIC FORMULAS
Familiarity with the following fundamental integration formulas is essential.