6_8.qd // : PM Page 8 8 CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals Section 8. Basic Integration Rules Review proceres for fitting an integrand to one of the basic integration rules. Fitting Integrands to Basic Rules In this chapter, ou will stud several integration techniques that greatl epand the set of integrals to which the basic integration rules can be applied. These rules are reviewed on page. A major step in solving an integration problem is recognizing which basic integration rule to use. As shown in Eample, slight differences in the integrand can lead to ver different solution techniques. EXAMPLE A Comparison of Three Similar Integrals EXPLORATION A Comparison of Three Similar Integrals Which, if an, of the following integrals can be evaluated using the basic integration rules? For an that can be evaluated, do so. For an that can t, eplain wh. a. d b. d c. d Find each integral. a. b. c. 9 d 9 d Solution a. Use the Arctangent Rule and let u and a. 9 d d arctan C arctan C Constant Multiple Rule Arctangent Rule Simplif. b. Here the Arctangent Rule does not appl because the numerator contains a factor of. Consider the Log Rule and let u 9. Then d, and ou have Constant Multiple Rule 9 d d 9 9 d u Substitution: u 9 NOTE Notice in Eample (c) that some preliminar algebra is required before appling the rules for integration, and that subsequentl more than one rule is needed to evaluate the resulting integral. Log Rule c. Because the degree of the numerator is equal to the degree of the denominator, ou should first use division to rewrite the improper rational function as the sum of a polnomial and a proper rational function. ln u C ln 9 C. 9 d 6 9 d d 6 9 d 6 arctan C arctan C Rewrite using long division. Write as two integrals. Simplif. indicates that in the HM mathspace CD-ROM and the online Espace sstem for this tet, ou will find an Open Eploration, which further eplores this eample using the computer algebra sstems Maple, Mathcad, Mathematica, and Derive.
6_8.qd // : PM Page 9 SECTION 8. Basic Integration Rules 9 = + EXAMPLE Evaluate Using Two Basic Rules to Solve a Single Integral d. Solution Begin b writing the integral as the sum of two integrals. Then appl the Power Rule and the Arcsine Rule as follows. d d d d d The area of the region is approimatel.89. Figure 8. See Figure 8.. arcsin.89 TECHNOLOGY Simpson s Rule can be used to give a good approimation of the value of the integral in Eample (for n, the approimation is.89). When using numerical integration, however, ou should be aware that Simpson s Rule does not alwas give good approimations when one or both of the limits of integration are near a vertical asmptote. For instance, using the Fundamental Theorem of Calculus, ou can obtain EXAMPLE Find.99 d 6.. Appling Simpson s Rule (with n ) to this integral proces an approimation of 6.889. 6 6 d. A Substitution Involving a u STUDY TIP Rules 8, 9, and of the basic integration rules on the net page all have epressions involving the sum or difference of two squares: a u a u u a With such an epression, consider the substitution as in Eample. u f, Solution Because the radical in the denominator can be written in the form a u ou can tr the substitution u. Then d, and ou have 6 d 6 arcsin u C d 6 u arcsin C. Rewrite integral. Substitution: u Arcsine Rule Rewrite as a function of.
6_8.qd // : PM Page CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals Surprisingl, two of the most commonl overlooked integration rules are the Log Rule and the Power Rule. Notice in the net two eamples how these two integration rules can be disguised. Review of Basic Integration Rules a >. kfu kf u. f u ± gu. u C.. f u ±gu u n un C, n u ln u C n EXAMPLE Find A Disguised Form of the Log Rule Solution The integral does not appear to fit an of the basic rules. However, the quotient form suggests the Log Rule. If ou let u e, then e d. You can obtain the required b adding and subtracting e in the numerator, as follows. e d. e d e e e d e e e e d e d e ln e C d Add and subtract Rewrite as two fractions. Rewrite as two integrals. e in numerator. 6. 7. 8. 9.. e u e u C a u ln a au C sin u cos u C cos u sin u C tan u ln cos u C NOTE There is usuall more than one wa to solve an integration problem. For instance, in Eample, tr integrating b multipling the numerator and denominator b e to obtain an integral of the form u. See if ou can get the same answer b this procere. (Be careful: the answer will appear in a different form.) EXAMPLE A Disguised Form of the Power Rule Find cot lnsin d.. cot u ln sin u C... sec u tan u C. csc u cot u C 6. sec u tan u sec u C 7. 8. 9. sec u ln sec u tan u C csc u ln csc u cot u C csc u cot u csc u C a u arcsin u a C a u a arctan u a C. uu a a arcsec u a C Solution Again, the integral does not appear to fit an of the basic rules. However, considering the two primar choices for u u cot and u lnsin, ou can see that the second choice is the appropriate one because So, NOTE u lnsin and cot lnsin d u In Eample, tr checking that the derivative of lnsin C u C is the integrand of the original integral. cos sin d lnsin C. cot d. Substitution: u lnsin Rewrite as a function of.
6_8.qd // : PM Page SECTION 8. Basic Integration Rules Trigonometric identities can often be used to fit integrals to one of the basic integration rules. EXAMPLE 6 Using Trigonometric Identities Find tan d. TECHNOLOGY If ou have access to a computer algebra sstem, tr using it to evaluate the integrals in this section. Compare the form of the antiderivative given b the software with the form obtained b hand. Sometimes the forms will be the same, but often the will differ. For instance, wh is the antiderivative ln C equivalent to the antiderivative ln C? Solution Note that tan u is not in the list of basic integration rules. However, sec u is in the list. This suggests the trigonometric identit tan u sec u. If ou let u, then d and tan d tan u sec u sec u tan u u C Substitution: u Trigonometric identit Rewrite as two integrals. tan C. Rewrite as a function of. This section concludes with a summar of the common proceres for fitting integrands to the basic integration rules. Proceres for Fitting Integrands to Basic Rules Technique Epand (numerator). Separate numerator. Complete the square. Divide improper rational function. Add and subtract terms in numerator. Use trigonometric identities. Multipl and divide b Pthagorean conjugate. Eample e e e cot csc sin sin sin sin sin cos sec sin cos sin sin NOTE Remember that ou can separate numerators but not denominators. Watch out for this common error when fitting integrands to basic rules. Do not separate denominators.
6_8.qd // : PM Page CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals Eercises for Section 8. In Eercises, select the correct antiderivative.... d d (a) C (c) C d d (a) ln C (c) arctan C d d (a) ln C (c) arctan C d. d cos (a) sin C (c) sin C (b) C (d) ln C (b) C (d) ln C (b) C (d) ln C (b) sin C (d) sin C In Eercises, select the basic integration formula ou can use to find the integral, and identif u and a when appropriate. t. 6. d t 7. 8. t dt d t 9.. dt t dt. t sin t. d dt sec tan d. cos esin d. d In Eercises, find the indefinite integral.. 6. 6 d t 9 dt 7. 8. t t dt z dz 9.. v d v dv t.. t.. d 9t dt d d e. 6. e d d See www.calcchat.com for worked-out solutions to odd-numbered eercises. 7. 8. d d 9. cos. d sec d sin. csc cot d. cos d.. csc ecot e d d. 6. e e d d ln 7. d 8. tan lncos d sin cos 9. d. cos sin.. cos d sec d.. t dt d e. tant dt 6. t t t dt 7. 8. 9.. Slope Fields In Eercises, 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 www.mathgraphs.com. ds d.. d tan dt t t 6 d 8 d 6 d d, s t, d
6_8.qd // : PM Page SECTION 8. Basic Integration Rules d. sec tan. d, 9 d d, 7. 9 7..8.6. 9 9 Slope Fields In Eercises and 6, use a computer algebra sstem to graph the slope field for the differential equation and graph the solution through the specified initial condition.. 6. In Eercises 7 6, solve the differential equation. d dr 7. 8. dt et d e e t 9. tan sec 6. In Eercises 6 68, evaluate the definite integral. Use the integration capabilities of a graphing utilit to verif our result. 6. cos d 6. 6. e d 6. 6. 66. 67. 9 d 68. 9 d Area 9 d., d d, d In Eercises 69 7, find the area of the region. e sin t cos t dt ln d d d. 7. 7. sin In Eercises 7 78, use a computer algebra sstem to find the integral. Use the computer algebra sstem to graph two antiderivatives. Describe the relationship between the two graphs of the antiderivatives. 7. 76. 77. d d sin d 78. e e d.. Writing About Concepts In Eercises 79 8, state the integration formula ou would use to perform the integration. Eplain wh ou chose that formula. Do not integrate. π 69. 7. 8 9 6 (., ) 79. d 8. 8. 8. d sec tan d d 8. Eplain wh the antiderivative e C is equivalent to the antiderivative Ce. 8. Eplain wh the antiderivative sec C is equivalent to the antiderivative tan C.
6_8.qd // : PM Page CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals 8. Determine the constants a and b such that sin cos a sin b. is it appropriate to substitute u, u, and d to obtain Eplain. Approimation In Eercises 89 and 9, determine which value best approimates the area of the region between the -ais and the function over the given interval. (Make our selection on the basis of a sketch of the region and not b integrating.) 89. f,, (a) (b) (c) 8 (d) 8 (e) 9. f,, (a) (b) (c) (d) (e) Interpreting Integrals In Eercises 9 and 9, (a) sketch the region whose area is given b the integral, (b) sketch the solid whose volume is given b the integral if the disk method is used, and (c) sketch the solid whose volume is given b the integral if the shell method is used. (There is more than one correct answer for each part.) u? d d Use this result to integrate sin cos. 86. Area The graphs of f and g a intersect at the points, and a, a. Find a a > such that the area of the region bounded b the graphs of these two functions is. 87. Think About It Use a graphing utilit to graph the function f Use the graph to determine whether 7. f d is positive or negative. Eplain. 88. Think About It When evaluating d 9. 9. 9. Volume The region bounded b e,,, and b b > is revolved about the -ais. (a) Find the volume of the solid generated if b. (b) Find b such that the volume of the generated solid is cubic units. 9. Arc Length Find the arc length of the graph of lnsin from to. 9. Surface Area Find the area of the surface formed b revolving the graph of on the interval, 9 about the -ais. d u 96. Centroid Find the -coordinate of the centroid of the region bounded b the graphs of, and In Eercises 97 and 98, find the average value of the function over the given interval. 97. f, 98. f sin n, n, n is a positive integer. Arc Length In Eercises 99 and, use the integration capabilities of a graphing utilit to approimate the arc length of the curve over the given interval. 99. tan,,.,, 8. Finding a Pattern (a) Find cos d. (b) Find cos d. (c) Find cos 7 d.,,. (d) Eplain how to find cos d without actuall integrating.. Finding a Pattern (a) Write tan d in terms of tan d. Then find tan d. (b) Write tan d in terms of tan d. (c) Write tan k d, where k is a positive integer, in terms of tan k d. (d) Eplain how to find tan d without actuall integrating.. Methods of Integration Show that the following results are equivalent. Integration b tables: d ln C Integration b computer algebra sstem: d arcsinh C. Evaluate Putnam Eam Challenge ln9 d ln9 ln. This problem was composed b the Committee on the Putnam Prize Competition. The Mathematical Association of America. All rights reserved.