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.
SOLUTION: We use the method of partial fractions and set
Solving for A and B yields Thus,
Given find F ′(x).
If find F ′(x).
SOLUTION: We let u = cos x. Thus
Here we have let and noted that
The limit on the right in the starred equation is, by definition, the derivative of F(x), that is, f (x).
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
If g ′ is continuous, find
Note that the expanded limit is, by definition, the derivative of g(x) at c.