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

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

EXAMPLE 1

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EXAMPLE 2

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EXAMPLE 3

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EXAMPLE 4

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EXAMPLE 5

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EXAMPLE 6

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EXAMPLE 7

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EXAMPLE 8

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BC ONLY

EXAMPLE 9

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EXAMPLE 10

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BC ONLY

EXAMPLE 11

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BC ONLY

EXAMPLE 12

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SOLUTION: We use the method of partial fractions and set

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Solving for A and B yields Image Thus,

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EXAMPLE 13

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EXAMPLE 14

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EXAMPLE 15

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EXAMPLE 16

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EXAMPLE 17

Given Image find F (x).

SOLUTION:

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EXAMPLE 18

If Image find F (x).

SOLUTION: We let u = cos x. Thus

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EXAMPLE 19

Find Image

SOLUTION:

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Here we have let Image and noted that

Image

where

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The limit on the right in the starred equation is, by definition, the derivative of F(x), that is, f (x).

EXAMPLE 20

Reexpress Image in terms of u if Image

SOLUTION: When Image 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

Image

EXAMPLE 21

If g is continuous, find Image

SOLUTION:

Image

Note that the expanded limit is, by definition, the derivative of g(x) at c.