## 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 *C*is called the *constant of integration.* The indefinite integral of *f* (*x*) is written as thus

**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 then

*F*(*x*) − *G*(*x*) = *C* (*C* a constant).