﻿ ﻿PROPERTIES OF DEFINITE INTEGRALS - Definite Integrals - Calculus AB and Calculus BC

## CHAPTER 6 Definite Integrals

### B. PROPERTIES OF DEFINITE INTEGRALS

The following theorems about definite integrals are important.

Fundamental Theorem of calculus The evaluation of a definite integral is illustrated in the following examples. A calculator will be helpful for some numerical calculations.

EXAMPLE 1 EXAMPLE 2 EXAMPLE 3 EXAMPLE 4 EXAMPLE 5 EXAMPLE 6 EXAMPLE 7 EXAMPLE 8 BC ONLY

EXAMPLE 9 EXAMPLE 10 BC ONLY

EXAMPLE 11 BC ONLY

EXAMPLE 12 SOLUTION: We use the method of partial fractions and set Solving for A and B yields Thus, EXAMPLE 13 EXAMPLE 14 EXAMPLE 15 EXAMPLE 16 EXAMPLE 17

Given find F (x).

SOLUTION: EXAMPLE 18

If find F (x).

SOLUTION: We let u = cos x. Thus EXAMPLE 19

Find SOLUTION: Here we have let and noted that where The limit on the right in the starred equation is, by definition, the derivative of F(x), that is, f (x).

EXAMPLE 20

Reexpress in terms of u if SOLUTION: When u2 = x − 2, and 2u du = dx. The limits of the given integral are values of x. When we write the new integral in terms of the variable u, then the limits, if written, must be the values of u that correspond to the given limits. Thus, when x = 3, u = 1, and whenx = 6, u = 2. Then EXAMPLE 21

If g is continuous, find SOLUTION: Note that the expanded limit is, by definition, the derivative of g(x) at c.

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