## 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 *u*^{2} = *x* − 2, and 2*u 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 when*x* = 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*.