The Natural Logarithmic Function: Integration. Log Rule for Integration


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1 6_5.qd // :58 PM Page CHAPTER 5 Logarithmic, Eponential, and Other Transcendental Functions Section 5. EXPLORATION Integrating Rational Functions Earl in Chapter, ou learned rules that allowed ou to integrate an polnomial function. The Log Rule presented in this section goes a long wa toward enabling ou to integrate rational functions. For instance, each of the following functions can be integrated with the Log Rule. Eample Eample Eample Eample (a) Eample (c) Eample (d) Eample 5 Eample 6 There are still some rational functions that cannot be integrated using the Log Rule. Give eamples of these functions, and eplain our reasoning. The Natural Logarithmic Function: Integration Use the Log Rule for Integration to integrate a rational function. Integrate trigonometric functions. Log Rule for Integration The differentiation rules d d ln and that ou studied in the preceding section produce the following integration rule. Because du u d, the second formula can also be written as EXAMPLE d d Alternative form of Log Rule Using the Log Rule for Integration Constant Multiple Rule ln C Log Rule for Integration ln C Propert of logarithms Because cannot be negative, the absolute value is unnecessar in the final form of the antiderivative. EXAMPLE u u d ln u C. d d ln u u u THEOREM 5.5 Log Rule for Integration Let u be a differentiable function of. d ln C.. u du ln u C Using the Log Rule with a Change of Variables Find d. Solution If ou let u, then du d. Multipl and divide b. d d u du Substitute: u. ln u C Appl Log Rule. ln C
2 6_5.qd // :58 PM Page SECTION 5. The Natural Logarithmic Function: Integration Eample uses the alternative form of the Log Rule. To appl this rule, look for quotients in which the numerator is the derivative of the denominator. EXAMPLE Finding Area with the Log Rule Find the area of the region bounded b the graph of the ais, and the line = + Area d The area of the region bounded b the graph of, the ais, and is ln. Figure 5.8 Solution From Figure 5.8, ou can see that the area of the region is given b the definite integral If ou let u, then To appl the Log Rule, multipl and divide b as shown. Multipl and divide b. d d d. ln u ln ln ln.5 u. u d ln u C ln EXAMPLE Recognizing Quotient Forms of the Log Rule a. b. c. d. d ln C sec tan d ln tan C d d ln C d d ln C u u tan u u With antiderivatives involving logarithms, it is eas to obtain forms that look quite different but are still equivalent. For instance, which of the following are equivalent to the antiderivative listed in Eample (d)? ln C, C, ln C ln
3 6_5.qd // :58 PM Page CHAPTER 5 Logarithmic, Eponential, and Other Transcendental Functions Integrals to which the Log Rule can be applied often appear in disguised form. For instance, if a rational function has a numerator of degree greater than or equal to that of the denominator, division ma reveal a form to which ou can appl the Log Rule. This is shown in Eample 5. EXAMPLE 5 Using Long Division Before Integrating Find d. Solution Begin b using long division to rewrite the integrand. Now, ou can integrate to obtain d ) d d d Rewrite using long division. Rewrite as two integrals. ln C. Integrate. Check this result b differentiating to obtain the original integrand. The net eample gives another instance in which the use of the Log Rule is disguised. In this case, a change of variables helps ou recognize the Log Rule. EXAMPLE 6 Change of Variables with the Log Rule Find d. Solution If ou let u, then du d and u. Substitute. d u du u TECHNOLOGY If ou have access to a computer algebra sstem, use it to find the indefinite integrals in Eamples 5 and 6. How does the form of the antiderivative that it gives ou compare with that given in Eamples 5 and 6? u u u du du u u du ln u u C ln u u C ln C Rewrite as two fractions. Rewrite as two integrals. Integrate. Simplif. Check this result b differentiating to obtain the original integrand.
4 6_5.qd // :58 PM Page 5 SECTION 5. The Natural Logarithmic Function: Integration 5 As ou stud the methods shown in Eamples 5 and 6, be aware that both methods involve rewriting a disguised integrand so that it fits one or more of the basic integration formulas. Throughout the remaining sections of Chapter 5 and in Chapter 8, much time will be devoted to integration techniques. To master these techniques, ou must recognize the formfitting nature of integration. In this sense, integration is not nearl as straightforward as differentiation. Differentiation takes the form Here is the question; what is the answer? Integration is more like Here is the answer; what is the question? The following are guidelines ou can use for integration. STUDY TIP Keep in mind that ou can check our answer to an integration problem b differentiating the answer. For instance, in Eample 7, the derivative of lnln C is ln. Guidelines for Integration. Learn a basic list of integration formulas. (Including those given in this section, ou now have formulas: the Power Rule, the Log Rule, and ten trigonometric rules. B the end of Section 5.7, this list will have epanded to basic rules.). Find an integration formula that resembles all or part of the integrand, and, b trial and error, find a choice of u that will make the integrand conform to the formula.. If ou cannot find a usubstitution that works, tr altering the integrand. You might tr a trigonometric identit, multiplication and division b the same quantit, or addition and subtraction of the same quantit. Be creative.. If ou have access to computer software that will find antiderivatives smbolicall, use it. EXAMPLE 7 usubstitution and the Log Rule Solve the differential equation Solution The solution can be written as an indefinite integral. ln d Because the integrand is a quotient whose denominator is raised to the first power, ou should tr the Log Rule. There are three basic choices for u. The choices u and u ln fail to fit the uu form of the Log Rule. However, the third choice does fit. Letting u ln produces and ou obtain the following. ln d ln d u u d u, ln ln u C ln C So, the solution is ln ln C. d d ln. Divide numerator and denominator b. Substitute: u ln. Appl Log Rule.
5 6_5.qd // :58 PM Page 6 6 CHAPTER 5 Logarithmic, Eponential, and Other Transcendental Functions Integrals of Trigonometric Functions In Section., ou looked at si trigonometric integration rules the si that correspond directl to differentiation rules. With the Log Rule, ou can now complete the set of basic trigonometric integration formulas. EXAMPLE 8 Using a Trigonometric Identit Find tan d. Solution This integral does not seem to fit an formulas on our basic list. However, b using a trigonometric identit, ou obtain sin tan d cos d. Knowing that D cos sin, ou can let u cos and write sin tan d cos d u u d ln ln u C cos C. Trigonometric identit Substitute: u cos. Appl Log Rule. Eample 8 uses a trigonometric identit to derive an integration rule for the tangent function. The net eample takes a rather unusual step (multipling and dividing b the same quantit) to derive an integration rule for the secant function. EXAMPLE 9 Derivation of the Secant Formula Find sec d. Solution Consider the following procedure. sec d sec sec tan sec tan d Letting u be the denominator of this quotient produces u sec tan So, ou can conclude that sec sec tan sec tan sec d sec sec tan d sec tan u u d ln ln u C sec tan C. d u sec tan sec. Rewrite integrand. Substitute: u sec tan. Appl Log Rule.
6 6_5.qd // :58 PM Page 7 SECTION 5. The Natural Logarithmic Function: Integration 7 With the results of Eamples 8 and 9, ou now have integration formulas for sin, cos, tan, and sec. All si trigonometric rules are summarized below. NOTE Using trigonometric identities and properties of logarithms, ou could rewrite these si integration rules in other forms. For instance, ou could write csc u du lncsc u cot u C. (See Eercises 8 86.) Integrals of the Si Basic Trigonometric Functions sin u du cos u C tan u du lncos u C sec u du lnsec u tan u C cos u du sin u C cot u du ln sin u C csc u du lncsc u cot u C EXAMPLE Integrating Trigonometric Functions Evaluate tan d. Solution Using tan ou can write tan d sec, sec d sec d ln sec tan ln ln.88. sec for. EXAMPLE Finding an Average Value Figure 5.9 f() = tan Average value. Find the average value of Solution Average value ln cos ln. f tan on the interval, tan d ln ln The average value is about., as shown in Figure 5.9. tan d. Average value Simplif. Integrate. b b a a f d
7 6_5.qd // :58 PM Page 8 8 CHAPTER 5 Logarithmic, Eponential, and Other Transcendental Functions Eercises for Section 5. In Eercises, find the indefinite integral.. 5. d d.. d 5 d d d d 9. d. d 9 d d 9 d d d d d d d ln 9.. d ln.. d d. d. d d In Eercises 5 8, find the indefinite integral b usubstitution. ( Hint: Let u be the denominator of the integrand.) d d d d In Eercises 9 6, find the indefinite integral. cos 9.. tan 5 d sin d. csc d. sec d cos t. sin t dt. csc t cot t dt sec tan 5. sec d 6. sec t tan t dt In Eercises 7, solve the differential equation. Use a graphing utilit to graph three solutions, one of which passes through the given point. d 7., 8. d, 9... Determine the function f if f f,,. Determine the function f if Slope Fields In Eercises 6, a differential equation, a point, and a slope field are given. (a) Sketch two approimate solutions of the differential equation on the slope field, one of which passes through the given point. (b) Use integration to find the particular solution of the differential equation and use a graphing utilit to graph the solution. Compare the result with the sketches in part (a). To print an enlarged cop of the graph, go to the website d d ln.,., d, d, d 5., 6. d, dr dt sec t tan t, ds tan, d See for workedout solutions to oddnumbered eercises. f, >. f, >.,, 6 f, d d 9, d sec, d,, f, 5
8 6_5.qd // :58 PM Page 9 SECTION 5. The Natural Logarithmic Function: Integration 9 In Eercises 7 5, evaluate the definite integral. Use a graphing utilit to verif our result d e ln 9. d d cos In Eercises 55 6, use a computer algebra sstem to find or evaluate the integral d d d d sin 59. csc sin d 6. In Eercises 6 6, find F. 6. F 6. F t dt F F t dt Approimation In Eercises 65 and 66, determine which value best approimates the area of the region between the ais and the graph of the function over the given interval. (Make our selection on the basis of a sketch of the region and not b performing an calculations.) 65. f sec,, (a) 6 (b) 6 (c) (d).5 (e) 66. f,, (a) (b) 7 (c) (d) 5 (e) Area In Eercises 67 7, find the area of the given region. Use a graphing utilit to verif our result e e ln.. d ln d d csc cot d d sin cos d cos tan t dt t dt 69. tan 7. sin cos Area In Eercises 7 7, find the area of the region bounded b the graphs of the equations. Use a graphing utilit to verif our result Numerical Integration In Eercises 75 78, use the Trapezoidal Rule and Simpson s Rule to approimate the value of the definite integral. Let n and round our answer to four decimal places. Use a graphing utilit to verif our result ,,,,,, sec,, 6, tan.,,, d 77. ln d 78. sec d d Writing About Concepts In Eercises 79 8, state the integration formula ou would use to perform the integration. Do not integrate d d d 8. sec tan d
9 6_5.qd // :58 PM Page CHAPTER 5 Logarithmic, Eponential, and Other Transcendental Functions In Eercises 8 86, show that the two formulas are equivalent. 8. tan d lncos C 8. cot d lnsin C tan d lnsec C cot d lncsc C sec d lnsec tan C sec d lnsec tan C csc d lncsc cot C csc d lncsc cot C In Eercises 87 9, find the average value of the function over the given interval. 87. f 8, 88. f, ln 89. f, e 9., 9. Population Growth A population of bacteria is changing at a rate of where t is the time in das. The initial population (when t ) is. Write an equation that gives the population at an time t, and find the population when t das. 9. Heat Transfer Find the time required for an object to cool from F to 5F b evaluating t ln T dt 5 dp dt.5t Average value where t is time in minutes., f sec 6,, Average value, 9. Average Price The demand equation for a product is Find the average price p on the interval Sales The rate of change in sales S is inversel proportional to time t t > measured in weeks. Find S as a function of t if sales after and weeks are units and units, respectivel. 95. Orthogonal Trajector (a) Use a graphing utilit to graph the equation 8. (b) Evaluate the integral to find in terms of. e d For a particular value of the constant of integration, graph the result in the same viewing window used in part (a). (c) Verif that the tangents to the graphs of parts (a) and (b) are perpendicular at the points of intersection. 96. Graph the function for k,.5, and. on,. Find lim f k. k True or False? In Eercises 97, determine whether the statement is true or false. If it is false, eplain wh or give an eample that shows it is false. 97. ln ln p 9,. f k k k ln d C d ln c, d ln ln ln ln. Graph the function f on the interval,. (a) Find the area bounded b the graph of f and the line. (b) Determine the values of the slope m such that the line m and the graph of f enclose a finite region. (c) Calculate the area of this region as a function of m.. Prove that the function F t dt c is constant on the interval,.
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