Calculus AB and Calculus BC

CHAPTER 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 Cis called the constant of integration. The indefinite integral of f (x) is written as Image thus

Image

Indefinite integral

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 Image then

F(x) − G(x) = C (C a constant).