244 Chapter 4 Integration 4.1 Antiderivatives and Indefinite Integration Write the general solution of a differential equation and use indefinite integral notation for antiderivatives. Use basic integration rules to find antiderivatives. Find a particular solution of a differential equation. Antiderivatives To find a function F whose derivative is f ͑x͒ ϭ 3x 2, you might use your knowledge of Exploration derivatives to conclude that Finding Antiderivatives d For each derivative, describe F͑x͒ ϭ x3 because ͓x3͔ ϭ 3x2. dx the original function F. The function F is an antiderivative of f. a. FЈ͑x͒ ϭ 2x b. FЈ͑x͒ ϭ x c. FЈ͑x͒ ϭ x2 Definition of Antiderivative Ј͑ ͒ ϭ ͑ ͒ 1 A function F is an antiderivative of f on an interval I when F x f x for all d. FЈ͑x͒ ϭ x in I. x2 Ј͑ ͒ ϭ 1 e. F x 3 x Note that F is called an antiderivative of f, rather than the antiderivative of f. To see f. FЈ͑x͒ ϭ cos x why, observe that What strategy did you use to ͑ ͒ ϭ 3 ͑ ͒ ϭ 3 Ϫ ͑ ͒ ϭ 3 ϩ F1 x x , F2 x x 5, and F3 x x 97 find F? are all antiderivatives of f ͑x͒ ϭ 3x 2. In fact, for any constant C, the function F͑x͒ ϭ x 3 ϩ C is an antiderivative of f. THEOREM 4.1 Representation of Antiderivatives If F is an antiderivative of f on an interval I, then G is an antiderivative of f on the interval I if and only if G is of the form G͑x͒ ϭ F͑x͒ ϩ C, for all x in I where C is a constant. Proof The proof of Theorem 4.1 in one direction is straightforward. That is, if G͑x͒ ϭ F͑x͒ ϩ C, FЈ͑x͒ ϭ f ͑x͒, and C is a constant, then d GЈ͑x͒ ϭ ͓F͑x͒ ϩ C͔ ϭ FЈ͑x͒ ϩ 0 ϭ f ͑x͒. dx To prove this theorem in the other direction, assume that G is an antiderivative of f. Define a function H such that H͑x͒ ϭ G(x͒ Ϫ F͑x͒. For any two points a and b ͑a < b͒ in the interval,H is continuous on ͓a, b͔ and differentiable on ͑a, b͒. By the Mean Value Theorem, H͑b͒ Ϫ H͑a͒ HЈ͑c͒ ϭ b Ϫ a for some c in ͑a, b͒. However,HЈ͑c͒ ϭ 0, so H͑a͒ ϭ H͑b͒. Because a and b are arbitrary points in the interval, you know that H is a constant function C. So, G͑x͒ Ϫ F͑x͒ ϭ C and it follows that G͑x͒ ϭ F͑x͒ ϩ C. See LarsonCalculus.com for Bruce Edwards’s video of this proof. Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 4.1 Antiderivatives and Indefinite Integration 245 Using Theorem 4.1, you can represent the entire family of antiderivatives of a function by adding a constant to a known antiderivative. For example, knowing that ͓ 2͔ ϭ Dx x 2x you can represent the family of all antiderivatives of f ͑x͒ ϭ 2x by G͑x͒ ϭ x2 ϩ C Family of all antiderivatives of f (x͒ ϭ 2x where C is a constant. The constant C is called the constant of integration. The family of functions represented by G is the general antiderivative of f, and G(x͒ ϭ x2 ϩ C is the general solution of the differential equation GЈ͑x͒ ϭ 2x. Differential equation A differential equation in x and y is an equation that involves x, y, and derivatives of y. For instance, yЈ ϭ 3x and yЈ ϭ x2 ϩ 1 y are examples of differential equations. 2 C = 0 Solving a Differential Equation C = 2 1 Find the general solution of the differential equation yЈ ϭ 2. C = −1 Solution To begin, you need to find a function whose derivative is 2. One such x −2 1 2 function is ϭ −1 y 2x. 2x is an antiderivative of 2. Now, you can use Theorem 4.1 to conclude that the general solution of the differential equation is y ϭ 2x ϩ C. General solution Functions of the form y ϭ 2x ϩ C Figure 4.1 The graphs of several functions of the form y ϭ 2x ϩ C are shown in Figure 4.1. When solving a differential equation of the form dy ϭ f ͑x͒ dx it is convenient to write it in the equivalent differential form dy ϭ f ͑x͒ dx. The operation of finding all solutions of this equation is called antidifferentiation (or indefinite integration) and is denoted by an integral sign ͐. The general solution is denoted by Variable of Constant of integration integration y ϭ ͵ f ͑x͒ dx ϭ F͑x͒ ϩ C. REMARK In this text, the notation ͐ f͑x͒ dx ϭ F͑x͒ ϩ C Integrand An antiderivative means that F is an antiderivative of f͑x͒ of f on an interval. The expression ͐f ͑x͒dx is read as the antiderivative of f with respect to x. So, the differential dx serves to identify x as the variable of integration. The term indefinite integral is a synonym for antiderivative. Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 246 Chapter 4 Integration Basic Integration Rules The inverse nature of integration and differentiation can be verified by substituting FЈ͑x͒ for f ͑x͒ in the indefinite integration definition to obtain ͵FЈ͑x͒ dx ϭ F͑x͒ ϩ C. Integration is the “inverse” of differentiation. Moreover, if ͐ f͑x͒ dx ϭ F͑x͒ ϩ C, then d ΄͵ f ͑x͒ dx΅ ϭ f ͑x͒. Differentiation is the “inverse” of integration. dx These two equations allow you to obtain integration formulas directly from differentiation formulas, as shown in the following summary. Basic Integration Rules Differentiation Formula Integration Formula d ͓C͔ ϭ 0 ͵0 dx ϭ C dx d ͓kx͔ ϭ k ͵k dx ϭ kx ϩ C dx d ͓kf͑x͔͒ ϭ kfЈ͑x͒ ͵kf͑x͒ dx ϭ k͵f ͑x͒ dx dx d ͓f ͑x͒ ± g͑x͔͒ ϭ fЈ͑x͒ ± gЈ͑x͒ ͵͓ f ͑x͒ ± g͑x͔͒ dx ϭ ͵f ͑x͒ dx ± ͵g͑x͒ dx dx nϩ1 d Ϫ x ͓xn͔ ϭ nxn 1 ͵xn dx ϭ ϩ C, n Ϫ1 Power Rule dx n ϩ 1 d ͓sin x͔ ϭ cos x ͵cos x dx ϭ sin x ϩ C dx d ͓cos x͔ ϭϪsin x ͵sin x dx ϭϪcos x ϩ C dx d ͓tan x͔ ϭ sec2 x ͵sec2 x dx ϭ tan x ϩ C dx d ͓sec x͔ ϭ sec x tan x ͵sec x tan x dx ϭ sec x ϩ C dx d ͓cot x͔ ϭϪcsc2 x ͵csc2 x dx ϭϪcot x ϩ C dx d ͓csc x͔ ϭϪcsc x cot x ͵csc x cot x dx ϭϪcsc x ϩ C dx Note that the Power Rule for Integration has the restriction that n Ϫ1. The evaluation of 1 ͵ dx x must wait until the introduction of the natural logarithmic function in Chapter 5. Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 4.1 Antiderivatives and Indefinite Integration 247 Describing Antiderivatives REMARK In Example 2, note that the general pattern of ͵3x dx ϭ 3͵x dx Constant Multiple Rule integration is similar to that of differentiation. ϭ 3͵x1 dx Rewrite x as x1. Original integral x2 ϭ 3΂ ΃ ϩ C Power Rule ͑n ϭ 1͒ 2 3 ϭ x2 ϩ C Simplify. Rewrite 2 3 2 ϩ The antiderivatives of 3x are of the form 2x C, where C is any constant. Integrate When indefinite integrals are evaluated, a strict application of the basic integration rules tends to produce complicated constants of integration. For instance, in Example 2, the solution could have been written as Simplify x2 3 ͵3x dx ϭ 3͵x dx ϭ 3΂ ϩ C΃ ϭ x2 ϩ 3C. 2 2 Because C represents any constant, it is both cumbersome and unnecessary to write 3C 3 2 ϩ 3 2 ϩ as the constant of integration. So,2 x 3C is written in the simpler form 2x C. TECHNOLOGY Some Rewriting Before Integrating software programs, such as Maple and Mathematica, See LarsonCalculus.com for an interactive version of this type of example. are capable of performing Original Integral Rewrite Integrate Simplify integration symbolically. If Ϫ2 1 Ϫ x 1 you have access to such a a. ͵ dx ͵x 3 dx ϩ C Ϫ ϩ C 3 Ϫ 2 symbolic integration utility, try x 2 2x using it to evaluate the indefinite x3͞2 2 b.