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.