566
Chapter 5 | Integration
5.4 | Integration Formulas and the Net Change Theorem Learning Objectives 5.4.1 Apply the basic integration formulas.
5.4.2 Explain the significance of the net change theorem. 5.4.3 Use the net change theorem to solve applied problems. 5.4.4 Apply the integrals of odd and even functions.
In this section, we use some basic integration formulas studied previously to solve some key applied problems. It is important to note that these formulas are presented in terms of indefinite integrals. Although definite and indefinite integrals are closely related, there are some key differences to keep in mind. A definite integral is either a number (when the limits of integration are constants) or a single function (when one or both of the limits of integration are variables). An indefinite integral represents a family of functions, all of which differ by a constant. As you become more familiar with integration, you will get a feel for when to use definite integrals and when to use indefinite integrals. You will naturally select the correct approach for a given problem without thinking too much about it. However, until these concepts are cemented in your mind, think carefully about whether you need a definite integral or an indefinite integral and make sure you are using the proper notation based on your choice. Basic Integration Formulas Recall the integration formulas given in the table in Antiderivatives and the rule on properties of definite integrals. Let’s look at a few examples of how to apply these rules. Example 5.23 Integrating a Function Using the Power Rule
Use the power rule to integrate the function ∫ 1 4
t (1+ t ) dt .
Solution The first step is to rewrite the function and simplify it so we can apply the power rule: ∫ 1 4 t (1+ t ) dt = ∫ 1 4 t 1/2 (1+ t ) dt = ∫ 1 4 ⎛ ⎝ t 1/2 + t 3/2 ⎞ ⎠ dt . Now apply the power rule: ∫ 1 4 ⎛ ⎝ t 1/2 + t 3/2 ⎞ ⎠ dt = ⎛ ⎝ 2 3 t 3/2 + 2 5 t 5/2 ⎞ ⎠ | 1 4 = ⎡ ⎣ 2 3 (4) 3/2 + 2 5 (4) 5/2 ⎤ ⎦ − ⎡ ⎣ 2 3 (1) 3/2 + 2 5 (1) 5/2
⎤ ⎦
= 256 15
.
Find the definite integral of f ( x ) = x 2 −3 x over the interval [1, 3].
5.21
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