Integration Techniques, L Hôpital s Rule, and Improper Integrals

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1 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals In previous chapters, ou studied several basic techniques for evaluating simple integrals. In this chapter, ou will stud other integration techniques, such as integration b parts, that are used to evaluate more complicated integrals. You will also learn an important rule for evaluating its called L Hôpital s Rule. This rule can also help ou evaluate improper integrals. In this chapter, ou should learn the following. How to fit an integrand to one of the basic integration rules. (8.) How to find an antiderivative using integration b parts. (8.) How to evaluate trigonometric integrals. (8.) How to use trigonometric substitution to evaluate an integral. (8.) How to use partial fraction decomposition to integrate rational functions. (8.5) How to evaluate an indefinite integral using a table of integrals and using reduction formulas. (8.6) How to appl L Hôpital s Rule to evaluate a it. (8.7) How to evaluate an improper integral. (8.8) AP Photo/Topeka Capital-Journal, Anthon S. Bush/Wide World Partial fraction decomposition is an integration technique that can be used to evaluate integrals involving rational functions. How can partial fraction decomposition be used to evaluate an integral that gives the average cost of removing a certain percent of a chemical from a compan s waste water? (See Section 8.5, Eercise 6.) d = d = d = From our studies of calculus thus far, ou know that a definite integral has finite its of integration and a continuous integrand. In Section 8.8, ou will stud improper integrals. Improper integrals have at least one infinite it of integration or have an integrand with an infinite discontinuit. You will see that improper integrals either converge or diverge. 59

2 5 Chapter 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals 8. Basic Integration Rules Review procedures for fitting an integrand to one of the basic integration rules. Fitting Integrands to Basic Integration 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 5. 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 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 du d, and ou have Constant Multiple Rule 9 d d 9 du u Substitution: u 9 NOTE Notice in Eample (c) that some preinar 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. Integrate. Simplif. The icon indicates that ou will find a CAS Investigation on the book s website. The CAS Investigation is a collaborative eploration of this eample using the computer algebra sstems Maple and Mathematica.

3 8. Basic Integration Rules 5 = + EXAMPLE Using Two Basic Rules to Solve a Single Integral Evaluate 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 its of integration are near a vertical asmptote. For instance, using the Fundamental Theorem of Calculus, ou can obtain.99 d 6.. Appling Simpson s Rule (with n ) to this integral produces an approimation of 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 These epressions are often apparent after a u-substitution, as shown in Eample. EXAMPLE A Substitution Involving a u Find 6 d. 6 Solution Because the radical in the denominator can be written in the form a u ou can tr the substitution u. Then du d, and ou have 6 d 6 arcsin u C d 6 du u arcsin C. Rewrite integral. Substitution: u Arcsine Rule Rewrite as a function of.

4 5 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 du kf u du. f u ± gu du. du u C f u du ±gu du. u n du un C, n n 5. du u ln u C EXAMPLE A Disguised Form of the Log Rule Find e d. 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 du e d. You can obtain the required du b adding and subtracting e in the numerator, as follows. 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. Integrate. e in numerator e u du e u C a u du ln a au C sin u du cos u C cos u du sin 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 duu. See if ou can get the same answer b this procedure. (Be careful: the answer will appear in a different form.) EXAMPLE 5 A Disguised Form of the Power Rule. tan u du ln cos u C. cot u du ln sin u C... sec u du tan u C 5. csc u du cot u C 6. sec u tan u du sec u C sec u du ln sec u tan u C csc u du ln csc u cot u C csc u cot u du csc u C du a u arcsin u a C du a u a arctan u a C. du uu a a arcsec u a C Find cot lnsin d. 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 du du cos sin d u C lnsin C. cot d. In Eample 5, tr checking that the derivative of lnsin C is the integrand of the original integral. Substitution: u lnsin Integrate. Rewrite as a function of.

5 8. Basic Integration Rules 5 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 forms of the antiderivatives given b the software with the forms 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 du d and tan d tan u du sec u du sec u du du Substitution: u Trigonometric identit Rewrite as two integrals. tan u u C Integrate. tan C. Rewrite as a function of. This section concludes with a summar of the common procedures for fitting integrands to the basic integration rules. PROCEDURES FOR FITTING INTEGRANDS TO BASIC INTEGRATION 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 5 Chapter 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals 8. Eercises See for worked-out solutions to odd-numbered eercises. In Eercises, select the correct antiderivative... d d (a) C (c) C d d (b) C (d) ln C (a) C (b) C (c) arctan C (d) ln C. d d (a) C (b) C (c) arctan C (d) ln C. d d cos (a) sin C (b) sin C (c) sin C (d) sin C In Eercises 5, select the basic integration formula ou can use to find the integral, and identif u and a when appropriate. t d t t dt d t 9.. dt t dt. t sin t. d dt sec 5 tan 5 d. cos esin d. d In Eercises 5 5, find the indefinite integral d 6. t 8 dt t t dt z dz v 5 d v dv t.. t.. d 9t dt d 8 d e e d 7 d d d 9. cos. d sec d sin. csc cot d. cos d.. csc ecot e d d e e d d ln d tan lncos d sin cos 9. d. cos sin.. cos d sec d.. t dt 9 5 d e 5. tant dt 6. t t t dt 6 7. d d d 65 d d 8 d Slope Fields In Eercises 5 56, 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 ds d d tan dt t t, s t, d

7 8. Basic Integration Rules 55 d d d sec tan d, 9, CAS Slope Fields In Eercises 57 and 58, use a computer algebra sstem to graph the slope field for the differential equation and graph the solution through the specified initial condition sin. d 57..8, 58. d In Eercises 59 6, solve the differential equation. 59. d d 6. d e d e 5 6. dr dr 6. dt et dt et e t e t 6. tan sec 6. In Eercises 65 7, evaluate the definite integral. Use the integration capabilities of a graphing utilit to verif our result. 65. cos d e d d 7. 9 d Area In Eercises 7 78, find the area of the region (.5, ) d 5, d e 7 ln d d sin t cos t dt d CAS In Eercises 79 8, use a computer algebra sstem to find the integral. Use the computer algebra sstem to graph two antiderivatives. Describe the relationship between the graphs of the two antiderivatives d 8. sin d 87. Determine the constants a and b such that sin cos a sin b. d Use this result to integrate sin cos. 88. Show that sec sin Then use this identit to cos cos sin..5 WRITING ABOUT CONCEPTS derive the basic integration rule sec d lnsec tan C. π d e e d In Eercises 8 86, state the integration formula ou would use to perform the integration. Eplain wh ou chose that formula. Do not integrate. 8. d 8. sec tan d d d

8 56 Chapter 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals 89. 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 CAPSTONE 9. (a) Eplain wh the antiderivative e C is equivalent to the antiderivative Ce. (b) Eplain wh the antiderivative sec C is equivalent to the antiderivative tan C. 9. Think About It Use a graphing utilit to graph the function f 5 7. Use the graph to determine whether 5 f d is positive or negative. Eplain. 9. Think About It When evaluating d, is it appropriate to substitute u u, and d du, to obtain u u du? Eplain. Approimation In Eercises 9 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.) 9. f,, (a) (b) (c) 8 (d) 8 (e) 9. f,, (a) (b) (c) (d) (e) Interpreting Integrals In Eercises 95 and 96, (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.) d 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. 98. Volume Consider the region bounded b the graphs of, cos, sin, and. Find the volume of the solid generated b revolving the region about the -ais. 99. Arc Length Find the arc length of the graph of lnsin from to.. Arc Length Find the arc length of the graph of lncos from to. d.. Surface Area Find the area of the surface formed b revolving the graph of on the interval, 9 about the -ais.. Centroid Find the -coordinate of the centroid of the region bounded b the graphs of 5 5, and In Eercises and, find the average value of the function over the given interval.. f,. f sin n, n, n is a positive integer. Arc Length In Eercises 5 and 6, use the integration capabilities of a graphing utilit to approimate the arc length of the curve over the given interval. 5. tan,, 6.,, 8 7. Finding a Pattern (a) Find cos d. (c) Find cos 7 d.,,. (b) Find cos 5 d. (d) Eplain how to find cos 5 d without actuall integrating. 8. Finding a Pattern (a) Write tan d in terms of tan d. Then find tan d. (b) Write tan 5 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 5 d without actuall integrating. 9. Methods of Integration Show that the following results are equivalent. Integration b tables: d ln C Integration b computer algebra sstem: d arcsinh C PUTNAM EXAM CHALLENGE. Evaluate ln9 d ln9 ln. This problem was composed b the Committee on the Putnam Prize Competition. The Mathematical Association of America. All rights reserved.

9 8. Integration b Parts Integration b Parts Find an antiderivative using integration b parts. Use a tabular method to perform integration b parts. EXPLORATION Proof Without Words Here is a different approach to proving the formula for integration b parts. Eercise taken from Proof Without Words: Integration b Parts b Roger B. Nelsen, Mathematics Magazine, 6, No., April 99, p., b permission of the author. s = g(b) r = g(a) s r r s v u = f() p = f(a) u dv uv q,s p p,r v = g() p u dv v du q uv q,s p,r Eplain how this graph proves the theorem. Which notation in this proof is unfamiliar? What do ou think it means? q u q = f(b) Area Area qs pr v du Integration b Parts In this section ou will stud an important integration technique called integration b parts. This technique can be applied to a wide variet of functions and is particularl useful for integrands involving products of algebraic and transcendental functions. For instance, integration b parts works well with integrals such as ln d, e d, and Integration b parts is based on the formula for the derivative of a product d dv du uv u v d d d uv vu where both u and v are differentiable functions of. If and are continuous, ou can integrate both sides of this equation to obtain uv uv d vu d u dv v du. e sin d. B rewriting this equation, ou obtain the following theorem. THEOREM 8. INTEGRATION BY PARTS If u and v are functions of and have continuous derivatives, then u dv uv v du. This formula epresses the original integral in terms of another integral. Depending on the choices of u and dv, it ma be easier to evaluate the second integral than the original one. Because the choices of u and dv are critical in the integration b parts process, the following guidelines are provided. u v GUIDELINES FOR INTEGRATION BY PARTS. Tr letting dv be the most complicated portion of the integrand that fits a basic integration rule. Then u will be the remaining factor(s) of the integrand.. Tr letting u be the portion of the integrand whose derivative is a function simpler than u. Then dv will be the remaining factor(s) of the integrand. Note that dv alwas includes the d of the original integrand.

10 58 Chapter 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals EXAMPLE Integration b Parts Find e d. Solution To appl integration b parts, ou need to write the integral in the form u dv. There are several was to do this. e d, e d, e d, e d u dv u dv u dv u dv The guidelines on page 57 suggest the first option because the derivative of u is simpler than, and dv e d is the most complicated portion of the integrand that fits a basic integration formula. dv e d v dv e d e NOTE In Eample, note that it is not necessar to include a constant of integration when solving v e d e C. To illustrate this, replace v e b v e C and appl integration b parts to see that ou obtain the same result. u du d Now, integration b parts produces u dv uv v du Integration b parts formula e d e e d Substitute. e e C. Integrate. To check this, differentiate e e C to see that ou obtain the original integrand. EXAMPLE Integration b Parts Find ln d. Solution In this case, is more easil integrated than ln. Furthermore, the derivative of ln is simpler than ln. So, ou should let dv d. dv d u ln v d du d Integration b parts produces u dv uv v du Integration b parts formula TECHNOLOGY ln d Tr graphing and ln 9 on our graphing utilit. Do ou get the same graph? (This will take a while, so be patient.) ln d ln d ln d ln C. 9 You can check this result b differentiating. Substitute. Simplif. Integrate. d d ln 9 ln ln

11 8. Integration b Parts 59 FOR FURTHER INFORMATION To see how integration b parts is used to prove Stirling s approimation lnn! n ln n n see the article The Validit of Stirling s Approimation: A Phsical Chemistr Project b A. S. Wallner and K. A. Brandt in Journal of Chemical Education. One surprising application of integration b parts involves integrands consisting of single terms, such as ln d or arcsin d. In these cases, tr letting dv d, as shown in the net eample. EXAMPLE An Integrand with a Single Term Evaluate arcsin d. Solution Let dv d. dv d v d u arcsin du d Integration b parts now produces π = arcsin ) π, ) u dv uv v du arcsin d arcsin d arcsin d Integration b parts formula Substitute. Rewrite. arcsin C. Integrate. The area of the region is approimatel.57. Figure 8. Using this antiderivative, ou can evaluate the definite integral as follows. arcsin d arcsin.57 The area represented b this definite integral is shown in Figure 8.. TECHNOLOGY Remember that there are two was to use technolog to evaluate a definite integral: () ou can use a numerical approimation such as the Trapezoidal Rule or Simpson s Rule, or () ou can use a computer algebra sstem to find the antiderivative and then appl the Fundamental Theorem of Calculus. Both methods have shortcomings. To find the possible error when using a numerical method, the integrand must have a second derivative (Trapezoidal Rule) or a fourth derivative (Simpson s Rule) in the interval of integration: the integrand in Eample fails to meet either of these requirements. To appl the Fundamental Theorem of Calculus, the smbolic integration utilit must be able to find the antiderivative. Which method would ou use to evaluate arctan d? Which method would ou use to evaluate arctan d?

12 5 Chapter 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals Some integrals require repeated use of the integration b parts formula. EXAMPLE Repeated Use of Integration b Parts Find sin d. Solution The factors and sin are equall eas to integrate. However, the derivative of becomes simpler, whereas the derivative of sin does not. So, ou should let u. dv sin d v sin d cos u du d Now, integration b parts produces sin d cos cos d. First use of integration b parts This first use of integration b parts has succeeded in simplifing the original integral, but the integral on the right still doesn t fit a basic integration rule. To evaluate that integral, ou can appl integration b parts again. This time, let u. dv cos d v cos d sin u du d Now, integration b parts produces cos d sin sin d Second use of integration b parts sin cos C. Combining these two results, ou can write sin d cos sin cos C. When making repeated applications of integration b parts, ou need to be careful not to interchange the substitutions in successive applications. For instance, in Eample, the first substitution was u and dv sin d. If, in the second application, ou had switched the substitution to u cos and dv, ou would have obtained EXPLORATION Tr to find e cos d b letting u cos and dv e d in the first substitution. For the second substitution, let u sin and dv e d. sin d cos cos d cos cos sin d sin d thereb undoing the previous integration and returning to the original integral. When making repeated applications of integration b parts, ou should also watch for the appearance of a constant multiple of the original integral. For instance, this occurs when ou use integration b parts to evaluate e cos d, and also occurs in Eample 5 on the net page. The integral in Eample 5 is an important one. In Section 8. (Eample 5), ou will see that it is used to find the arc length of a parabolic segment.

13 8. Integration b Parts 5 EXAMPLE 5 Integration b Parts Find sec d. Solution The most complicated portion of the integrand that can be easil integrated is sec, so ou should let dv sec d and u sec. dv sec d u sec Integration b parts produces v sec d tan du sec tan d u dv uv v du sec d sec tan sec tan d sec d sec tan sec sec d Integration b parts formula Substitute. Trigonometric identit STUDY TIP identities sin cos The trigonometric cos cos pla an important role in this chapter. sec d sec tan sec d sec d sec d sec tan sec d sec d sec tan ln sec tan C sec d sec tan ln sec tan C. Rewrite. Collect like integrals. Integrate. Divide b. EXAMPLE 6 Finding a Centroid = sin π, ) A machine part is modeled b the region bounded b the graph of sin and the -ais,, as shown in Figure 8.. Find the centroid of this region. Solution Begin b finding the area of the region. A sin d cos Figure 8. sin Δ π ) Now, ou can find the coordinates of the centroid as follows. sin d cos d sin A sin You can evaluate the integral for, A sin d, with integration b parts. To do this, let dv sin d and u. This produces v cos and du d, and ou can write sin d cos cos d 8 Finall, ou can determine to be A sin d cos sin. So, the centroid of the region is, 8. cos sin C.

14 5 Chapter 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals As ou gain eperience in using integration b parts, our skill in determining u and dv will increase. The following summar lists several common integrals with suggestions for the choices of u and dv. STUDY TIP You can use the acronm LIATE as a guideline for choosing u in integration b parts. In order, check the integrand for the following. Is there a Logarithmic part? Is there an Inverse trigonometric part? Is there an Algebraic part? Is there a Trigonometric part? Is there an Eponential part? SUMMARY OF COMMON INTEGRALS USING INTEGRATION BY PARTS. For integrals of the form or n e a d, n sin a d, n cos a d let u n and let dv e a d, sin a d, or cos a d.. For integrals of the form or n ln d, n arcsin a d, n arctan a d let u ln, arcsin a, or arctan a and let dv n d.. For integrals of the form ea sin b d or e a cos b d let u sin b or cos b and let dv e a d. Tabular Method In problems involving repeated applications of integration b parts, a tabular method, illustrated in Eample 7, can help to organize the work. This method works well for integrals of the form n sin a d, n cos a d, and n e a d. EXAMPLE 7 Using the Tabular Method Find sin d. FOR FURTHER INFORMATION For more information on the tabular method, see the article Tabular Integration b Parts b David Horowitz in The College Mathematics Journal, and the article More on Tabular Integration b Parts b Leonard Gillman in The College Mathematics Journal. To view these articles, go to the website Solution Begin as usual b letting u and dv v d sin d. Net, create a table consisting of three columns, as shown. Alternate Signs u and Its Derivatives 6 sin 6 cos Differentiate until ou obtain as a derivative. v and Its Antiderivatives sin The solution is obtained b adding the signed products of the diagonal entries: sin d cos sin cos C. 8 cos

15 8. Integration b Parts 5 8. Eercises See for worked-out solutions to odd-numbered eercises. In Eercises 6, identif u and dv for finding the integral using integration b parts. (Do not evaluate the integral.). e d. e d. ln d. ln 5 d 5. sec d 6. cos d In Eercises 7, evaluate the integral using integration b parts with the given choices of u and dv. 7. ln d; u ln, dv d )e d; u 7, dv e d sin d; u, dv sin d cos d; u, dv cos d In Eercises 8, find the integral. (Note: Solve b the simplest method not all require integration b parts.) In Eercises 9, solve the differential equation. e 9... d dt t 5t.. cos. Slope Fields In Eercises 5 and 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 5. cos,, 6., 8 d 7 e sin, d ln d d arctan 5. e d.. e d. e d e d et t dt t lnt dt 8. ln 9. d. e.. d. e d d cos d sin d.. t csc t cot t dt.. arctan d. 5. e sin d e cos d 8. ln d ln d ln d ln d e d 5 d sin d cos d sec tan d arccos d e sin 5 d e cos d CAS Slope Fields In Eercises 7 and 8, use a computer algebra sstem to graph the slope field for the differential equation and graph the solution through the specified initial condition. d d d d e8 sin In Eercises 9 6, evaluate the definite integral. Use a graphing utilit to confirm our result. 9. e d cos d arccos d e sin d ln d 58. ln d arcsec d 6. sec d 5 e d sin d arcsin d e cos d

16 5 Chapter 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals CAS In Eercises 6 66, use the tabular method to find the integral e d e d sin d cos d 65. sec 66. d d In Eercises 67 7, find or evaluate the integral using substitution first, then using integration b parts. 67. sin d 68. cos d 69. d 7. cos d CAS e d e d 7. cosln d 7. ln d WRITING ABOUT CONCEPTS 75. Integration b parts is based on what differentiation rule? Eplain. 76. In our own words, state how ou determine which parts of the integrand should be u and dv. 77. When evaluating sin d, eplain how letting u sin and dv d makes the solution more difficult to find. CAPSTONE 78. State whether ou would use integration b parts to evaluate each integral. If so, identif what ou would use for u and dv. Eplain our reasoning. ln (a) (b) ln d (c) e d d (d) e (e) (f) d d d In Eercises 79 8, use a computer algebra sstem to (a) find or evaluate the integral and (b) graph two antiderivatives. (c) Describe the relationship between the graphs of the antiderivatives. 79. t 8. e t dt sin d 8. e sin d Integrate d (a) b parts, letting dv d. (b) b substitution, letting u. 5 5 d 8. Integrate 9 d (a) b parts, letting dv 9 d. (b) b substitution, letting u Integrate d (a) b parts, letting dv d. (b) b substitution, letting u. 86. Integrate d (a) b parts, letting dv d. (b) b substitution, letting u. In Eercises 87 and 88, use a computer algebra sstem to find the integrals for n,,, and. Use the result to obtain a general rule for the integrals for an positive integer n and test our results for n n ln d n e d In Eercises 89 9, use integration b parts to prove the formula. (For Eercises 89 9, assume that n is a positive integer.) n sin d n cos n n cos d n cos d n sin n n sin d n ln d n n n ln C n e n e a d a n a a n e a d e a sin b d ea a sin b b cos b a b e a cos b d ea a cos b b sin b a b In Eercises 95 98, find the integral b using the appropriate formula from Eercises ln d cos d 97. e cos d 98. e d Area In Eercises 99, use a graphing utilit to graph the region bounded b the graphs of the equations, and find the area of the region. 99. e,,. 6 e,,,. e sin,,. sin,, C C

17 8. Integration b Parts 55. Area, Volume, and Centroid Given the region bounded b the graphs of ln,, and e, find (a) the area of the region. (b) the volume of the solid generated b revolving the region about the -ais. (c) the volume of the solid generated b revolving the region about the -ais. (d) the centroid of the region.. Volume and Centroid Given the region bounded b the graphs of sin,,, and, find (a) the volume of the solid generated b revolving the region about the -ais. (b) the volume of the solid generated b revolving the region about the -ais. (c) the centroid of the region. 5. Centroid Find the centroid of the region bounded b the graphs of arcsin,, and. How is this problem related to Eample 6 in this section? 6. Centroid Find the centroid of the region bounded b the graphs of f, g,, and. 7. Average Displacement A damping force affects the vibration of a spring so that the displacement of the spring is given b e t cos t 5 sin t. Find the average value of on the interval from t to t. 8. Memor Model A model for the abilit M of a child to memorize, measured on a scale from to, is given b M.6t ln t, < t, where t is the child s age in ears. Find the average value of this model (a) between the child s first and second birthdas. (b) between the child s third and fourth birthdas. Present Value In Eercises 9 and, find the present value P of a continuous income flow of ct dollars per ear if P t where t is the time in ears and r is the annual interest rate compounded continuousl. 9. ct, t, r 5%, t. ct, 5t, r 7%, t 5 Integrals Used to Find Fourier Coefficients In Eercises and, verif the value of the definite integral, where n is a positive integer... cte rt dt, n sin n d, n cos n d n n n is odd n is even. Vibrating String A string stretched between the two points, and, is plucked b displacing the string h units at its midpoint. The motion of the string is modeled b a Fourier Sine Series whose coefficients are given b b n h sin n Find b n.. Find the fallac in the following argument that. dv d u d d d So,. 5. Let f be positive and strictl increasing on the interval < a b. Consider the region R bounded b the graphs of f,, a, and b. If R is revolved about the -ais, show that the disk method and shell method ield the same volume. 6. Euler s Method Consider the differential equation f e with the initial condition f. (a) Use integration to solve the differential equation. (b) Use a graphing utilit to graph the solution of the differential equation. (c) Use Euler s Method with h.5, and the recursive capabilities of a graphing utilit, to generate the first 8 points of the graph of the approimate solution. Use the graphing utilit to plot the points. Compare the result with the graph in part (b). (d) Repeat part (c) using h. and generate the first points. (e) Wh is the result in part (c) a better approimation of the solution than the result in part (d)? Euler s Method In Eercises 7 and 8, consider the differential equation and repeat parts (a) (d) of Eercise f sin 8. f cos f 9. Think About It Give a geometric eplanation of wh sin d d. d h v d du d f Verif the inequalit b evaluating the integrals.. Finding a Pattern Find the area bounded b the graphs of sin and over each interval. (a), (b), (c), Describe an patterns that ou notice. What is the area between the graphs of sin and over the interval n, n, where n is an nonnegative integer? Eplain. sin n d.

18 56 Chapter 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals 8. Trigonometric Integrals Solve trigonometric integrals involving powers of sine and cosine. Solve trigonometric integrals involving powers of secant and tangent. Solve trigonometric integrals involving sine-cosine products with different angles. SHEILA SCOTT MACINTYRE (9 96) Sheila Scott Macintre published her first paper on the asmptotic periods of integral functions in 95. She completed her doctorate work at Aberdeen Universit, where she taught. In 958 she accepted a visiting research fellowship at the Universit of Cincinnati. Integrals Involving Powers of Sine and Cosine In this section ou will stud techniques for evaluating integrals of the form sin m cos n d and where either m or n is a positive integer. To find antiderivatives for these forms, tr to break them into combinations of trigonometric integrals to which ou can appl the Power Rule. For instance, ou can evaluate sin 5 cos d with the Power Rule b letting u sin. Then, du cos d and ou have sin 5 cos d u 5 du u6 6 C sin6 C. 6 To break up sinm cos n d into forms to which ou can appl the Power Rule, use the following identities. sin cos sin cos cos cos sec m tan n d Pthagorean identit Half-angle identit for sin Half-angle identit for cos GUIDELINES FOR EVALUATING INTEGRALS INVOLVING POWERS OF SINE AND COSINE. If the power of the sine is odd and positive, save one sine factor and convert the remaining factors to cosines. Then, epand and integrate. Odd Convert to cosines Save for du sin k cos n d sin k cosn sin d cos k cos n sin d. If the power of the cosine is odd and positive, save one cosine factor and convert the remaining factors to sines. Then, epand and integrate. Odd Convert to sines Save for du sin m cos k d sin m cos k cos d sin m sin k cos d. If the powers of both the sine and cosine are even and nonnegative, make repeated use of the identities sin cos and cos cos to convert the integrand to odd powers of the cosine. Then proceed as in guideline.

19 8. Trigonometric Integrals 57 TECHNOLOGY A computer algebra sstem used to find the integral in Eample ielded the following. sin cos d cos 5 7 sin 5 C Is this equivalent to the result obtained in Eample? EXAMPLE Power of Sine Is Odd and Positive Find sin cos d. Solution Because ou epect to use the Power Rule with u cos, save one sine factor to form du and convert the remaining sine factors to cosines. sin cos d sin cos sin d cos cos sin d Rewrite. Trigonometric identit cos cos 6 sin d Multipl. cos sin d cos 6 sin d Rewrite. cos sin d cos 6 sin d cos5 5 cos7 7 C Integrate. In Eample, both of the powers m and n happened to be positive integers. However, the same strateg will work as long as either m or n is odd and positive. For instance, in the net eample the power of the cosine is, but the power of the sine is π 6 = cos sin π The area of the region is approimatel.9. Figure 8. EXAMPLE Power of Cosine Is Odd and Positive cos Evaluate 6 sin d. Solution Because ou epect to use the Power Rule with u sin, save one cosine factor to form du and convert the remaining cosine factors to sines. cos sin d cos cos d sin sin.9 sin cos sin sin cos sin cos d sin Figure 8. shows the region whose area is represented b this integral. d

20 58 Chapter 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals EXAMPLE Power of Cosine Is Even and Nonnegative Find cos d. Solution Because m and n are both even and nonnegative m, ou can replace cos b cos. cos cos d d cos cos 8d cos d cos d 8 sin cos d cos d sin C Half-angle identit Epand. Half-angle identit Rewrite. Integrate. Use a smbolic differentiation utilit to verif this. Can ou simplif the derivative to obtain the original integrand? Bettmann/Corbis JOHN WALLIS (66 7) Wallis did much of his work in calculus prior to Newton and Leibniz, and he influenced the thinking of both of these men. Wallis is also credited with introducing the present smbol ( ) for infinit. In Eample, if ou were to evaluate the definite integral from to, ou would obtain cos d sin sin Note that the onl term that contributes to the solution is 8. This observation is generalized in the following formulas developed b John Wallis. WALLIS S FORMULAS. If n is odd n, then cos n d n n.. If n is even n, then cos n d n n. These formulas are also valid if cos n is replaced b sin n. (You are asked to prove both formulas in Eercise 8.)

21 8. Trigonometric Integrals 59 Integrals Involving Powers of Secant and Tangent The following guidelines can help ou evaluate integrals of the form sec m tan n d. GUIDELINES FOR EVALUATING INTEGRALS INVOLVING POWERS OF SECANT AND TANGENT. If the power of the secant is even and positive, save a secant-squared factor and convert the remaining factors to tangents. Then epand and integrate. Even Convert to tangents Save for du sec k tan n d sec k tan n sec d tan k tan n sec d. If the power of the tangent is odd and positive, save a secant-tangent factor and convert the remaining factors to secants. Then epand and integrate. Odd Convert to secants Save for du sec m tank d secm tan k sec tan d secm sec k sec tan d. If there are no secant factors and the power of the tangent is even and positive, convert a tangent-squared factor to a secant-squared factor, then epand and repeat if necessar. Convert to secants tan n d tan n tan d tan n sec d. If the integral is of the form sec m d, where m is odd and positive, use integration b parts, as illustrated in Eample 5 in the preceding section. 5. If none of the first four guidelines applies, tr converting to sines and cosines. EXAMPLE Power of Tangent Is Odd and Positive Find tan sec d. Solution Because ou epect to use the Power Rule with u sec, save a factor of sec tan to form du and convert the remaining tangent factors to secants. tan sec d sec tan d sec tan sec tan d sec sec sec tan d sec sec sec tan d sec sec C

22 5 Chapter 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals NOTE In Eample 5, the power of the tangent is odd and positive. So, ou could also find the integral using the procedure described in guideline on page 59. In Eercise 89, ou are asked to show that the results obtained b these two procedures differ onl b a constant. EXAMPLE 5 Power of Secant Is Even and Positive Find sec tan d. Solution Let u tan, then du sec d and ou can write sec tan d sec tan sec d tan tan sec d tan tan 5 sec d tan tan tan6 6 C tan6 8 C. EXAMPLE 6 Power of Tangent Is Even Evaluate tan d. Solution Because there are no secant factors, ou can begin b converting a tangentsquared factor to a secant-squared factor. tan d tan tan d. tan sec d = tan ) π, ) tan sec d tan d tan sec d sec d.5 π 8 π The area of the region is approimatel.9. Figure 8.5 tan You can evaluate the definite integral as follows. tan d tan.9 tan C tan The area represented b the definite integral is shown in Figure 8.5. Tr using Simpson s Rule to approimate this integral. With n 8, ou should obtain an approimation that is within. of the actual value.

23 8. Trigonometric Integrals 5 For integrals involving powers of cotangents and cosecants, ou can follow a strateg similar to that used for powers of tangents and secants. Also, when integrating trigonometric functions, remember that it sometimes helps to convert the entire integrand to powers of sines and cosines. EXAMPLE 7 Converting to Sines and Cosines sec Find tan d. Solution Because the first four guidelines on page 59 do not appl, tr converting the integrand to sines and cosines. In this case, ou are able to integrate the resulting powers of sine and cosine as follows. sec tan d cos cos sin d sin cos d sin C csc C FOR FURTHER INFORMATION To learn more about integrals involving sine-cosine products with different angles, see the article Integrals of Products of Sine and Cosine with Different Arguments b Sherrie J. Nicol in The College Mathematics Journal. To view this article, go to the website Integrals Involving Sine-Cosine Products with Different Angles Integrals involving the products of sines and cosines of two different angles occur in man applications. In such instances ou can use the following product-to-sum identities. sin m sin n cosm n cosm n sin m cos n sinm n sinm n cos m cos n cosm n cosm n EXAMPLE 8 Using Product-to-Sum Identities Find sin 5 cos d. Solution Considering the second product-to-sum identit above, ou can write sin 5 cos d sin sin 9 d cos 9 cos 9 C cos cos 9 8 C.

24 5 Chapter 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals 8. Eercises See for worked-out solutions to odd-numbered eercises. In Eercises, use differentiation to match the antiderivative with the correct integral. [Integrals are labeled (a), (b), (c), and (d).] (a) sin tan d (c) sin sec d (b) (d). sec. cos sec. tan tan. sin cos sin cos In Eercises 5 8, find the integral. 8cos d tan d 5. cos 5 sin d 6. cos sin d. sec tan d. tan t sec t dt 5. tan sec d 6. tan 5 sec d 7. sec 6 tan d sec 5 tan d. tan d.. tan tan sec d sec 5 d In Eercises 6, solve the differential equation.. dr ds sin. sin cos d tan sec sec tan d d tan sec 7. sin 7 cos d sin cos d.. sin cos d.. cos d. 5. cos d sin d 8. sin d cos d cos5 t sin t dt sin 5 d sin 6 d sin d Slope Fields In Eercises 7 and 8, 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 d sec tan,, d sin,,.5 In Eercises 9, use Wallis s Formulas to evaluate the integral. 9. cos 7 d.. cos d.. sin 6 d. cos 9 d In Eercises 5, find the integral involving secant and tangent. 5. sec 7 d 6. sec d sin 5 d sin 8 d CAS Slope Fields In Eercises 9 and 5, use a computer algebra sstem to graph the slope field for the differential equation, and graph the solution through the specified initial condition. d sin d 9., 5. d tan, d.5 In Eercises 5 56, find the integral sec 5 d 8. sec 6 d 5. cos cos 6 d sec d.. tan 5. d sin cos d cos cos d tan5 d tan sec d sin cos d 55. sin sin d 56. sin 5 sin d

25 8. Trigonometric Integrals 5 CAS CAS In Eercises 57 66, find the integral. Use a computer algebra sstem to confirm our result. 57. cot 58. tan d sec d 59. csc 6. d cot csc d cot t cot t csc t dt csc t dt sin cos d sec tan d cos sec t 65. tan t sec t dt 66. cos t dt In Eercises 67 7, evaluate the definite integral. 67. sin 68. d tan d tan 7. d sec ttan t dt cos t sin 5 cos d sin t dt 7. cos d 7. In Eercises 75 8, use a computer algebra sstem to find the integral. Graph the antiderivatives for two different values of the constant of integration. 75. cos 76. d 77. sec 5 d 78. tan d 79. sec 5 tan d 8. sec tan d In Eercises 8 8, use a computer algebra sstem to evaluate the definite integral. 8. sin sin d sin d 8. WRITING ABOUT CONCEPTS sin cos d sin d cos d sin d 85. In our own words, describe how ou would integrate sin m cos n d for each condition. (a) m is positive and odd. (b) n is positive and odd. (c) m and n are both positive and even. 86. In our own words, describe how ou would integrate sec m tan n d for each condition. (a) m is positive and even. (b) n is positive and odd. (c) n is positive and even, and there are no secant factors. (d) m is positive and odd, and there are no tangent factors. 87. Evaluate sin cos d using the given method. Eplain how our answers differ for each method. (a) Substitution where u sin (b) Substitution where u cos (c) Integration b parts (d) Using the identit sin sin cos CAPSTONE 88. For each pair of integrals, determine which one is more difficult to evaluate. Eplain our reasoning. (a) sin 7 cos d, sin cos d (b) tan sec d, tan sec d In Eercises 89 and 9, (a) find the indefinite integral in two different was. (b) Use a graphing utilit to graph the antiderivative (without the constant of integration) obtained b each method to show that the results differ onl b a constant. (c) Verif analticall that the results differ onl b a constant. 89. sec tan d 9. sec tan d Area In Eercises 9 9, find the area of the region bounded b the graphs of the equations. 9. sin, sin,, 9. sin,,, 9. cos, sin,, 9. cos, sin cos,, Volume In Eercises 95 and 96, find the volume of the solid generated b revolving the region bounded b the graphs of the equations about the -ais. 95. tan, 96. cos, Volume and Centroid In Eercises 97 and 98, for the region bounded b the graphs of the equations, find (a) the volume of the solid formed b revolving the region about the -ais and (b) the centroid of the region In Eercises 99, use integration b parts to verif the reduction formula , sin,, sin,,, cos,,, sinn d sinn cos n cos n d cosn sin n, n n sin n d n n cos n d cosm sin n d cosm sin n m n n m ncos m sin n d

26 5 Chapter 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals. sec n d n secn tan n n sec n d In Eercises 6, use the results of Eercises 99 to find the integral.. sin 5 d. cos d 5. sec 5 d 6. sin cos d 7. Modeling Data The table shows the normal maimum (high) and minimum (low) temperatures (in degrees Fahrenheit) in Erie, Pennslvania for each month of the ear. (Source: NOAA) Month Jan Feb Mar Apr Ma Jun Ma Min Month Jul Aug Sep Oct Nov Dec Ma Min The maimum and minimum temperatures can be modeled b ft a a cos t6 b sin t6, where t corresponds to Januar and a, a, and b are as follows. a 6 a t ft dt ft cos 6 dt b 6 t ft sin 6 dt (a) Approimate the model Ht for the maimum temperatures. (Hint: Use Simpson s Rule to approimate the integrals and use the Januar data twice.) (b) Repeat part (a) for a model Lt for the minimum temperature data. (c) Use a graphing utilit to graph each model. During what part of the ear is the difference between the maimum and minimum temperatures greatest? 8. Wallis s Formulas Use the result of Eercise to prove the following versions of Wallis s Formulas. (a) If n is odd n, then cos n d n n. (b) If n is even n, then cos n d 9. The inner product of two functions f and g on a, b is given b f, g b a fg d. Two distinct functions f and g are said to be orthogonal if f, g. Show that the following set of functions is orthogonal on,. sin, sin, sin,..., cos, cos, cos,.... Fourier Series The following sum is a finite Fourier series. f N i a i sin i a sin a sin a sin... a N sin N (a) Use Eercise 9 to show that the nth coefficient given b a n f sin n d. (b) Let f. Find a, a, and a n n. a n is SECTION PROJECT Power Lines Power lines are constructed b stringing wire between supports and adjusting the tension on each span. The wire hangs between supports in the shape of a catenar, as shown in the figure. ( L/, ) (, ) Let T be the tension (in pounds) on a span of wire, let u be the densit (in pounds per foot), let g. be the acceleration due to gravit (in feet per second per second), and let L be the distance (in feet) between the supports. Then the equation of the catenar is T ug where and are measured in feet. ug cosh T, (L/, ) (a) Find the length of the wire between two spans. (b) To measure the tension in a span, power line workers use the return wave method. The wire is struck at one support, creating a wave in the line, and the time t (in seconds) it takes for the wave to make a round trip is measured. The velocit v (in feet per second) is given b v Tu. How long does it take the wave to make a round trip between supports? (c) The sag s (in inches) can be obtained b evaluating when L in the equation for the catenar (and multipling b ). In practice, however, power line workers use the lineman s equation given b s.75t. Use the fact that coshuglt to derive this equation. FOR FURTHER INFORMATION To learn more about the mathematics of power lines, see the article Constructing Power Lines b Thomas O Neil in The UMAP Journal.

27 8. Trigonometric Substitution Trigonometric Substitution Use trigonometric substitution to solve an integral. Use integrals to model and solve real-life applications. EXPLORATION Integrating a Radical Function Up to this point in the tet, ou have not evaluated the following integral. d From geometr, ou should be able to find the eact value of this integral what is it? Using numerical integration, with Simpson s Rule or the Trapezoidal Rule, ou can t be sure of the accurac of the approimation. Wh? Tr finding the eact value using the substitution sin and d cos d. Does our answer agree with the value ou obtained using geometr? Trigonometric Substitution Now that ou can evaluate integrals involving powers of trigonometric functions, ou can use trigonometric substitution to evaluate integrals involving the radicals a u, a u, and u a. The objective with trigonometric substitution is to einate the radical in the integrand. You do this b using the Pthagorean identities cos sin, For eample, if a >, let u a sin, where. Then a u a a sin a sin a cos a cos. Note that cos, because sec tan,. TRIGONOMETRIC SUBSTITUTION a >. For integrals involving a u, let u a sin.a Then a u a cos, where.. For integrals involving a u, let u a tan. Then a u a sec, where < <.. For integrals involving u a, let u a sec. and tan sec. θ θ a u a + u u a u u u a Then θ a u a a tan if u > a, where < a tan if u < a, where < NOTE The restrictions on ensure that the function that defines the substitution is one-to-one. In fact, these are the same intervals over which the arcsine, arctangent, and arcsecant are defined.

28 56 Chapter 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals EXAMPLE Trigonometric Substitution: u a sin θ 9 sin cot, Figure Find d 9. Solution First, note that none of the basic integration rules applies. To use trigonometric substitution, ou should observe that 9 is of the form a u. So, ou can use the substitution a sin sin. Using differentiation and the triangle shown in Figure 8.6, ou obtain d cos d, 9 cos, and 9 sin. So, trigonometric substitution ields d 9 cos d 9 sin cos 9 d sin 9csc d Substitute. Simplif. Trigonometric identit Appl Cosecant Rule. Substitute for cot. Note that the triangle in Figure 8.6 can be used to convert the s back to s, as follows. cot adj. opp. 9 cot C 9 9 C 9 C. 9 9 TECHNOLOGY d 9 Use a computer algebra sstem to find each indefinite integral. d d d Then use trigonometric substitution to duplicate the results obtained with the computer algebra sstem. In an earlier chapter, ou saw how the inverse hperbolic functions can be used to evaluate the integrals du u ± a, du a u, and du ua ± u. You can also evaluate these integrals using trigonometric substitution. This is shown in the net eample.

29 8. Trigonometric Substitution 57 EXAMPLE Trigonometric Substitution: u a tan d Find. Solution Let u, a, and tan, as shown in Figure 8.7. Then, θ tan, sec Figure d sec d and Trigonometric substitution produces d sec d sec sec d sec. Substitute. Simplif. ln sec tan C Appl Secant Rule. ln C. Back-substitute. Tr checking this result with a computer algebra sstem. Is the result given in this form or in the form of an inverse hperbolic function? You can etend the use of trigonometric substitution to cover integrals involving epressions such as a u n b writing the epression as a u n a u n. EXAMPLE Trigonometric Substitution: Rational Powers θ + tan, sin Figure 8.8 Find d. Solution Begin b writing as. Then, let a and u tan, as shown in Figure 8.8. Using d sec d and sec ou can appl trigonometric substitution, as follows. d d sec d sec d sec Rewrite denominator. Substitute. Simplif. cos d sin C C Trigonometric identit Appl Cosine Rule. Back-substitute.

30 58 Chapter 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals For definite integrals, it is often convenient to determine the integration its for that avoid converting back to. You might want to review this procedure in Section.5, Eamples 8 and 9. θ tan sec, Figure 8.9 EXAMPLE Converting the Limits of Integration Evaluate d. Solution Because has the form u a, ou can consider u, a, and sec as shown in Figure 8.9. Then, d sec tan d and tan. To determine the upper and lower its of integration, use the substitution sec, as follows. Lower Limit When, sec So, ou have and. Upper Limit When, sec and 6. Integration its for Integration its for 6 tan sec tan d d sec 6 tan d 6 sec d tan In Eample, tr converting back to the variable and evaluating the antiderivative at the original its of integration. You should obtain d arcsec.

31 8. Trigonometric Substitution 59 When using trigonometric substitution to evaluate definite integrals, ou must be careful to check that the values of lie in the intervals discussed at the beginning of this section. For instance, if in Eample ou had been asked to evaluate the definite integral d then using u and a in the interval, would impl that u < a. So, when determining the upper and lower its of integration, ou would have to choose such that <. In this case the integral would be evaluated as follows. d 56 tan d 56 sec d tan 6.9 tan sec tan d sec 5 6 Trigonometric substitution can be used with completing the square. For instance, tr evaluating the following integral. d To begin, ou could complete the square and write the integral as d. 56 Trigonometric substitution can be used to evaluate the three integrals listed in the following theorem. These integrals will be encountered several times in the remainder of the tet. When this happens, we will simpl refer to this theorem. (In Eercise 85, ou are asked to verif the formulas given in the theorem.) 56 THEOREM 8. SPECIAL INTEGRATION FORMULAS a >.. a u du a arcsin u ua a u C u a du uu a a lnu u a C,. u a du uu a a ln u u a C u > a

32 55 Chapter 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals Applications EXAMPLE 5 Finding Arc Length (, ) f() =, The arc length of the curve from, to, Figure 8. ) ) Find the arc length of the graph of f from to (see Figure 8.). Solution Refer to the arc length formula in Section 7.. s f d d sec d sec tan ln sec tan Formula for arc length f Let a and tan. Eample 5, Section 8. ln.8 EXAMPLE 6 Comparing Two Fluid Forces The barrel is not quite full of oil the top. foot of the barrel is empt. Figure 8.. ft Figure 8. + =.8 ft A sealed barrel of oil (weighing 8 pounds per cubic foot) is floating in seawater (weighing 6 pounds per cubic foot), as shown in Figures 8. and 8.. (The barrel is not completel full of oil. With the barrel ling on its side, the top. foot of the barrel is empt.) Compare the fluid forces against one end of the barrel from the inside and from the outside. Solution In Figure 8., locate the coordinate sstem with the origin at the center of the circle given b. To find the fluid force against an end of the barrel from the inside, integrate between and.8 (using a weight of w 8). d F w hl d General equation (see Section 7.7) c.8 F inside 8.8 d d 96 d To find the fluid force from the outside, integrate between and. (using a weight of w 6).. F outside 6. d.. 5. d 8 d The details of integration are left for ou to complete in Eercise 8. Intuitivel, would ou sa that the force from the oil (the inside) or the force from the seawater (the outside) is greater? B evaluating these two integrals, ou can determine that F inside. pounds and F outside 9. pounds.

33 8. Trigonometric Substitution Eercises See for worked-out solutions to odd-numbered eercises. In Eercises, state the trigonometric substitution ou would use to find the integral. Do not integrate.. d. 5 d. 9. d d.. 5 d 6 d In Eercises 5 8, find the indefinite integral using the substitution sin. 5. d d d 6 d In Eercises 9, find the indefinite integral using the substitution 5 sec d. 5 d. 5 d 5 d In Eercises 6, find the indefinite integral using the substitution tan. 9.. d d d d In Eercises 7, use the Special Integration Formulas (Theorem 8.) to find the integral d d d 5 d In Eercises, find the integral... 6 d. 6 d. 6 d 9 d d 6 d t dt t 9.. d 9 d d.. 6 d 9 d CAS 5. e 6. e d d 7. e 8. e d d 9... arcsec d, >. d d arcsin d In Eercises 6, complete the square and find the integral... d d d 6 d In Eercises 7 5, evaluate the integral using (a) the given integration its and (b) the its obtained b trigonometric substitution t d d In Eercises 5 and 5, find the particular solution of the differential equation d d 9, d, d In Eercises 55 58, use a computer algebra sstem to find the integral. Verif the result b differentiation. 6 t dt,, d d 58. d d 5 6 t 5 dt 9 5 d 9 d WRITING ABOUT CONCEPTS 59. State the substitution ou would make if ou used trigonometric substitution and the integral involving the given radical, where a >. Eplain our reasoning. (a) a u (b) a u (c) u a

34 55 Chapter 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals WRITING ABOUT CONCEPTS (continued) In Eercises 6 and 6, state the method of integration ou would use to perform each integration. Eplain wh ou chose that method. Do not integrate. 6. d 6. d CAPSTONE 6. (a) Evaluate the integral using u-substitution. 9 d Then evaluate using trigonometric substitution. Discuss the results. (b) Evaluate the integral algebraicall using 9 d 9 9. Then evaluate using trigonometric substitution. Discuss the results. (c) Evaluate the integral using trigonometric d substitution. Then evaluate using the identit Discuss the results Area Find the area of the shaded region of the circle of radius a, if the chord is h units < h < a from the center of the circle (see figure). 69. Mechanical Design The surface of a machine part is the region between the graphs of and k 5 (see figure). (a) Find k if the circle is tangent to the graph of. (b) Find the area of the surface of the machine (, k) part. (c) Find the area of the surface of the machine part as a function of the radius r of the circle. 7. Volume The ais of a storage tank in the form of a right circular clinder is horizontal (see figure). The radius and length of the tank are meter and meters, respectivel. m m True or False? In Eercises 6 66, determine whether the statement is true or false. If it is false, eplain wh or give an eample that shows it is false. 6. If sin, then d d. 6. If sec, then 65. If tan, then 66. If sin, then 67. Area Find the area enclosed b the ellipse shown in the figure. a b b = b a d sec tan d. d cos d. d a a a a sin cos d. h a (a) Determine the volume of fluid in the tank as a function of its depth d. (b) Use a graphing utilit to graph the function in part (a). (c) Design a dip stick for the tank with markings of,, and. (d) Fluid is entering the tank at a rate of cubic meter per minute. Determine the rate of change of the depth of the fluid as a function of its depth d. (e) Use a graphing utilit to graph the function in part (d). When will the rate of change of the depth be minimum? Does this agree with our intuition? Eplain. Volume of a Torus In Eercises 7 and 7, find the volume of the torus generated b revolving the region bounded b the graph of the circle about the -ais. 7. (see figure) d Circle: ( ) + = = b a a a Figure for 67 Figure for h r, h > r

35 8. Trigonometric Substitution 55 Arc Length In Eercises 7 and 7, find the arc length of the curve over the given interval. 7. ln,, 5 7.,, 75. Arc Length Show that the length of one arch of the sine curve is equal to the length of one arch of the cosine curve. 76. Conjecture (a) Find formulas for the distances between, and a, a along the line between these points and along the parabola. (b) Use the formulas from part (a) to find the distances for a and a. (c) Make a conjecture about the difference between the two distances as a increases. Projectile Motion In Eercises 77 and 78, (a) use a graphing utilit to graph the path of a projectile that follows the path given b the graph of the equation, (b) determine the range of the projectile, and (c) use the integration capabilities of a graphing utilit to determine the distance the projectile travels. 8. Fluid Force Find the fluid force on a circular observation window of radius foot in a vertical wall of a large water-filled tank at a fish hatcher when the center of the window is (a) feet and (b) d feet d > below the water s surface (see figure). Use trigonometric substitution to evaluate the one integral. (Recall that in Section 7.7 in a similar problem, ou evaluated one integral b a geometric formula and the other b observing that the integrand was odd.) 8. Fluid Force Evaluate the following two integrals, which ield the fluid forces given in Eample 6..8 (a) F inside 8.8 d. (b) F outside 6. d 85. Use trigonometric substitution to verif the integration formulas given in Theorem Arc Length Show that the arc length of the graph of sin on the interval, is equal to the circumference of the ellipse (see figure) π π Centroid In Eercises 79 and 8, find the centroid of the region determined b the graphs of the inequalities ,,,, 6, π π π π 5 8. Surface Area Find the surface area of the solid generated b revolving the region bounded b the graphs of,,, and about the -ais. 8. Field Strength The field strength H of a magnet of length L on a particle r units from the center of the magnet is H where ±m are the poles of the magnet (see figure). Find the average field strength as the particle moves from to R units from the center b evaluating the integral R +m R ml r L ml dr. r L + = Figure for 86 Figure for Area of a Lune The crescent-shaped region bounded b two circles forms a lune (see figure). Find the area of the lune given that the radius of the smaller circle is and the radius of the larger circle is Area Two circles of radius, with centers at, and,, intersect as shown in the figure. Find the area of the shaded region. 6 6 L r m Figure for 8 Figure for 8 PUTNAM EXAM CHALLENGE 89. Evaluate ln d. This problem was composed b the Committee on the Putnam Prize Competition. The Mathematical Association of America. All rights reserved.

36 55 Chapter 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals 8.5 Partial Fractions Understand the concept of partial fraction decomposition. Use partial fraction decomposition with linear factors to integrate rational functions. Use partial fraction decomposition with quadratic factors to integrate rational functions. Partial Fractions θ 5 sec 5 Figure This section eamines a procedure for decomposing a rational function into simpler rational functions to which ou can appl the basic integration formulas. This procedure is called the method of partial fractions. To see the benefit of the method of partial fractions, consider the integral 5 6 d. To evaluate this integral without partial fractions, ou can complete the square and use trigonometric substitution (see Figure 8.) to obtain a, 5 sec 5 6 d d 5 sec tan d tan d sec tan d csc d Mar Evans Picture Librar JOHN BERNOULLI (667 78) The method of partial fractions was introduced b John Bernoulli, a Swiss mathematician who was instrumental in the earl development of calculus. John Bernoulli was a professor at the Universit of Basel and taught man outstanding students, the most famous of whom was Leonhard Euler. Now, suppose ou had observed that 5 6. Then ou could evaluate the integral easil, as follows. 5 6 d d Partial fraction decomposition This method is clearl preferable to trigonometric substitution. However, its use depends on the abilit to factor the denominator, 5 6, and to find the partial fractions and ln ln csc cot C 5 ln C ln C ln C ln ln ln ln C. 5 6 C C. In this section, ou will stud techniques for finding partial fraction decompositions.

37 8.5 Partial Fractions 555 STUDY TIP In precalculus ou learned how to combine functions such as 5. The method of partial fractions shows ou how to reverse this process. 5?? Recall from algebra that ever polnomial with real coefficients can be factored into linear and irreducible quadratic factors.* For instance, the polnomial 5 can be written as 5 where is a linear factor, is a repeated linear factor, and is an irreducible quadratic factor. Using this factorization, ou can write the partial fraction decomposition of the rational epression N 5 where N is a polnomial of degree less than 5, as follows. N A B C D E DECOMPOSITION OF N/D INTO PARTIAL FRACTIONS. Divide if improper: If ND is an improper fraction (that is, if the degree of the numerator is greater than or equal to the degree of the denominator), divide the denominator into the numerator to obtain N D a polnomial N D where the degree of N is less than the degree of D. Then appl Steps,, and to the proper rational epression N D.. Factor denominator: Completel factor the denominator into factors of the form p q m and a b c n where a b c is irreducible.. Linear factors: For each factor of the form p q m, the partial fraction decomposition must include the following sum of m fractions. A p q A p q... A m p q m. Quadratic factors: For each factor of the form a b c n, the partial fraction decomposition must include the following sum of n fractions. B C a b c B C a b c... B n C n a b c n * For a review of factorization techniques, see Precalculus, 7th edition, b Larson and Hostetler or Precalculus: A Graphing Approach, 5th edition, b Larson, Hostetler, and Edwards (Boston, Massachusetts: Houghton Mifflin, 7 and 8, respectivel).

38 556 Chapter 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals Linear Factors Algebraic techniques for determining the constants in the numerators of a partial fraction decomposition with linear or repeated linear factors are shown in Eamples and. EXAMPLE Distinct Linear Factors Write the partial fraction decomposition for 5 6. Solution Because 5 6, ou should include one partial fraction for each factor and write 5 6 A where A and B are to be determined. Multipling this equation b the least common denominator ) ields the basic equation A B. B Basic equation Because this equation is to be true for all, ou can substitute an convenient values for to obtain equations in A and B. The most convenient values are the ones that make particular factors equal to. NOTE Note that the substitutions for in Eample are chosen for their convenience in determining values for A and B; is chosen to einate the term A, and is chosen to einate the term B. The goal is to make convenient substitutions whenever possible. To solve for A, let and obtain A B A B A. To solve for B, let and obtain A B A B B. So, the decomposition is 5 6 as shown at the beginning of this section. Let in basic equation. Let in basic equation. FOR FURTHER INFORMATION To learn a different method for finding partial fraction decompositions, called the Heavside Method, see the article Calculus to Algebra Connections in Partial Fraction Decomposition b Joseph Wiener and Will Watkins in The AMATYC Review. Be sure ou see that the method of partial fractions is practical onl for integrals of rational functions whose denominators factor nicel. For instance, if the denominator in Eample were changed to 5 5, its factorization as would be too cumbersome to use with partial fractions. In such cases, ou should use completing the square or a computer algebra sstem to perform the integration. If ou do this, ou should obtain 5 d ln ln 5 5 C.

39 8.5 Partial Fractions 557 EXAMPLE Repeated Linear Factors FOR FURTHER INFORMATION For an alternative approach to using partial fractions, see the article A Shortcut in Partial Fractions b Xun-Cheng Huang in The College Mathematics Journal. TECHNOLOGY Most computer algebra sstems, such as Maple, Mathematica, and the TI-89, can be used to convert a rational function to its partial fraction decomposition. For instance, using Maple, ou obtain the following. > convert 5 6, parfrac, 6 9 Find 5 6 d. Solution Because ( ou should include one fraction for each power of and and write 5 6 A B C. Multipling b the least common denominator ields the basic equation 5 6 A B C. Basic equation To solve for A, let. This einates the B and C terms and ields 6 A A 6. To solve for C, let. This einates the A and B terms and ields 5 6 C The most convenient choices for have been used, so to find the value of B, ou can use an other value of along with the calculated values of A and C. Using, A 6, and C 9 produces 5 6 A B C B So, it follows that C 9. 6 B 9 B. 5 6 d 6 9 d 9 C ln 6 9 C. 6 ln ln Tr checking this result b differentiating. Include algebra in our check, simplifing the derivative until ou have obtained the original integrand. NOTE It is necessar to make as man substitutions for as there are unknowns A, B, C,... to be determined. For instance, in Eample, three substitutions,, and were made to solve for A, B, and C.

40 558 Chapter 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals Quadratic Factors When using the method of partial fractions with linear factors, a convenient choice of immediatel ields a value for one of the coefficients. With quadratic factors, a sstem of linear equations usuall has to be solved, regardless of the choice of. EXAMPLE Distinct Linear and Quadratic Factors Find 8 d. Solution Because ou should include one partial fraction for each factor and write To solve for A, let and obtain To solve for B, let and obtain B5 ields the basic At this point, C and D are et to be determined. You can find these remaining constants b choosing two other values for and solving the resulting sstem of linear equations. If, then, using A and B, ou can write C D If, ou have 8 8 C D 8 C D. Solving the linear sstem b subtracting the first equation from the second C D ields C. Consequentl, D, and it follows that 8 A B C D. Multipling b the least common denominator equation 8 A B C D. 8 A C D. C D 8 8 d A. B. d ln ln ln arctan C.

41 8.5 Partial Fractions 559 In Eamples,, and, the solution of the basic equation began with substituting values of that made the linear factors equal to. This method works well when the partial fraction decomposition involves linear factors. However, if the decomposition involves onl quadratic factors, an alternative procedure is often more convenient. EXAMPLE Repeated Quadratic Factors Find 8 d. Solution Include one partial fraction for each power of and write 8 A B C D. Multipling b the least common denominator ields the basic equation 8 A B C D. Epanding the basic equation and collecting like terms produces 8 A A B B C D 8 A B A C B D. Now, ou can equate the coefficients of like terms on opposite sides of the equation. 8 A B D 8 A B A C B D B A C Using the known values A 8 and B, ou can write A C 8 C B D D Finall, ou can conclude that 8 d 8 ln C D. d C. TECHNOLOGY Use a computer algebra sstem to evaluate the integral in Eample ou might find that the form of the antiderivative is different. For instance, when ou use a computer algebra sstem to work Eample, ou obtain 8 d ln Is this result equivalent to that obtained in Eample? C.

42 56 Chapter 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals When integrating rational epressions, keep in mind that for improper rational epressions such as N D 7 7 ou must first divide to obtain N 5 D. The proper rational epression is then decomposed into its partial fractions b the usual methods. Here are some guidelines for solving the basic equation that is obtained in a partial fraction decomposition. GUIDELINES FOR SOLVING THE BASIC EQUATION Linear Factors. Substitute the roots of the distinct linear factors in the basic equation.. For repeated linear factors, use the coefficients determined in guideline to rewrite the basic equation. Then substitute other convenient values of and solve for the remaining coefficients. Quadratic Factors. Epand the basic equation.. Collect terms according to powers of.. Equate the coefficients of like powers to obtain a sstem of linear equations involving A, B, C, and so on.. Solve the sstem of linear equations. Before concluding this section, here are a few things ou should remember. First, it is not necessar to use the partial fractions technique on all rational functions. For instance, the following integral is evaluated more easil b the Log Rule. d ln C Second, if the integrand is not in reduced form, reducing it ma einate the need for partial fractions, as shown in the following integral. d d d d ln C Finall, partial fractions can be used with some quotients involving transcendental functions. For instance, the substitution u sin allows ou to write cos u sin, du cos d sin sin d du uu.

43 8.5 Partial Fractions Eercises See for worked-out solutions to odd-numbered eercises. In Eercises 6, write the form of the partial fraction decomposition of the rational epression. Do not solve for the constants In Eercises 7 8, use partial fractions to find the integral d 9 d d d 5 d.. d d 5. d 5 5 d d 8 d d d d d d d 8 d d d d d In Eercises 9, evaluate the definite integral. Use a graphing utilit to verif our result d. d CAS In Eercises, use a computer algebra sstem to determine the antiderivative that passes through the given point. Use the sstem to graph the resulting antiderivative ,, 5., 6. d, 5 d,, d, d, d d In Eercises 5, use substitution to find the integral. sin sin.. cos cos d cos cos 5 cos.. d cos sin sin sin sec sec sin d d tan tan tan d 5 tan 6 d d, d, 5 d, d, e 7,,, 6, 7. 8 e 9. e d 5. d In Eercises 5 5, use the method of partial fractions to verif the integration formula. 5. a b d a ln a b C 5. a 5. d a ln a a C a b d 5. a b a b ln a b C a b d a b a ln a b C CAS Slope Fields In Eercises 55 and 56, use a computer algebra sstem to graph the slope field for the differential equation and graph the solution through the given initial condition. d d d d 6 5 WRITING ABOUT CONCEPTS e e e d d 57. What is the first step when integrating Eplain. 5 d? 58. Describe the decomposition of the proper rational function ND (a) if D p q m and (b) if D a b c n, where a b c is irreducible. Eplain wh ou chose that method.

44 56 Chapter 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals 59. Area Find the area of the region bounded b the graphs of 5 6,,, and. 6. Area Find the area of the region bounded b the graphs of 5 7,,, and. 6. Area Find the area of the region bounded b the graphs of 76 and. CAPSTONE 6. State the method ou would use to evaluate each integral. Eplain wh ou chose that method. Do not integrate. (a) (b) 7 (c) 8 d 8 d 5 d 6. Modeling Data The predicted cost C (in hundreds of thousands of dollars) for a compan to remove p% of a chemical from its waste water is shown in the table. p C p A model for the data is given b C p p, p <. Use the model to find the average cost of removing between 75% and 8% of the chemical. 6. Logistic Growth In Chapter 6, the eponential growth equation was derived from the assumption that the rate of growth was proportional to the eisting quantit. In practice, there often eists some upper it L past which growth cannot occur. In such cases, ou assume the rate of growth to be proportional not onl to the eisting quantit, but also to the difference between the eisting quantit and the upper it L. That is, ddt kl. In integral form, ou can write this relationship as d L k dt. (a) A slope field for the differential equation ddt is shown. Draw a possible solution to the differential equation if 5, and another if. To print an enlarged cop of the graph, go to the website 5 (b) Where is greater than, what is the sign of the slope of the solution? (c) For >, find t. t (d) Evaluate the two given integrals and solve for as a function of t, where is the initial quantit. (e) Use the result of part (d) to find and graph the solutions in part (a). Use a graphing utilit to graph the solutions and compare the results with the solutions in part (a). (f) The graph of the function is a logistic curve. Show that the rate of growth is maimum at the point of inflection, and that this occurs when L. 65. Volume and Centroid Consider the region bounded b the graphs of,,, and. Find the volume of the solid generated b revolving the region about the -ais. Find the centroid of the region. 66. Volume Consider the region bounded b the graph of on the interval,. Find the volume of the solid generated b revolving this region about the -ais. 67. Epidemic Model A single infected individual enters a communit of n susceptible individuals. Let be the number of newl infected individuals at time t. The common epidemic model assumes that the disease spreads at a rate proportional to the product of the total number infected and the number not et infected. So, ddt k n and ou obtain Solve for as a function of t. 68. Chemical Reactions In a chemical reaction, one unit of compound Y and one unit of compound Z are converted into a single unit of compound X. is the amount of compound X formed, and the rate of formation of X is proportional to the product of the amounts of unconverted compounds Y and Z. So, ddt k z, where and z are the initial amounts of compounds Y and Z. From this equation ou obtain (a) Perform the two integrations and solve for in terms of t. (b) Use the result of part (a) to find as t if () < z, () > z, and () z. 69. Evaluate n d k dt. z d k dt. d in two different was, one of which is partial fractions. PUTNAM EXAM CHALLENGE 7. Prove 7 d. This problem was composed b the Committee on the Putnam Prize Competition. The Mathematical Association of America. All rights reserved. 5 t

45 8.6 Integration b Tables and Other Integration Techniques Integration b Tables and Other Integration Techniques Evaluate an indefinite integral using a table of integrals. Evaluate an indefinite integral using reduction formulas. Evaluate an indefinite integral involving rational functions of sine and cosine. Integration b Tables So far in this chapter ou have studied several integration techniques that can be used with the basic integration rules. But merel knowing how to use the various techniques is not enough. You also need to know when to use them. Integration is first and foremost a problem of recognition. That is, ou must recognize which rule or technique to appl to obtain an antiderivative. Frequentl, a slight alteration of an integrand will require a different integration technique (or produce a function whose antiderivative is not an elementar function), as shown below. TECHNOLOGY A computer algebra sstem consists, in part, of a database of integration formulas. The primar difference between using a computer algebra sstem and using tables of integrals is that with a computer algebra sstem the computer searches through the database to find a fit. With integration tables, ou must do the searching. ln d ln C ln ln d C Integration b parts Power Rule Log Rule Not an elementar function Man people find tables of integrals to be a valuable supplement to the integration techniques discussed in this chapter. Tables of common integrals can be found in Appendi B. Integration b tables is not a cure-all for all of the difficulties that can accompan integration using tables of integrals requires considerable thought and insight and often involves substitution. Each integration formula in Appendi B can be developed using one or more of the techniques in this chapter. You should tr to verif several of the formulas. For instance, Formula ln d ln ln C d? ln u a bu du b a a bu ln a bu C Formula can be verified using the method of partial fractions, and Formula 9 a bu du du a bu a Formula 9 u ua bu can be verified using integration b parts. Note that the integrals in Appendi B are classified according to forms involving the following. u n a bu a bu cu a bu a ± u u ± a a u Trigonometric functions Inverse trigonometric functions Eponential functions Logarithmic functions

46 56 Chapter 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals EXPLORATION Use the tables of integrals in Appendi B and the substitution u to evaluate the integral in Eample. If ou do this, ou should obtain d du u. Does this produce the same result as that obtained in Eample? EXAMPLE Integration b Tables d Find. Solution Because the epression inside the radical is linear, ou should consider forms involving a bu. du Formula 7 a < ua bu a arctan a bu C a Let a, b, and u. Then du d, and ou can write d arctan C. EXAMPLE Integration b Tables Find 9 d. Solution Because the radical has the form u a, ou should consider Formula 6. u a du uu a a ln u u a C Let u and a. Then du d, and ou have 9 d d 9 9 ln 9 C. EXAMPLE Integration b Tables Find d. e Solution Of the forms involving e u, consider the following formula. du e u u ln eu C Let u. Then du d, and ou have d e d e ln e C Formula 8 ln e C. TECHNOLOGY Eample shows the importance of having several solution techniques at our disposal. This integral is not difficult to solve with a table, but when it was entered into a well-known computer algebra sstem, the utilit was unable to find the antiderivative.

47 8.6 Integration b Tables and Other Integration Techniques 565 Reduction Formulas Several of the integrals in the integration tables have the form f d g h d. Such integration formulas are called reduction formulas because the reduce a given integral to the sum of a function and a simpler integral. EXAMPLE Using a Reduction Formula Find sin d. Solution Consider the following three formulas. u sin u du sin u u cos u C Formula 5 u n sin u du u n cos u n u n cos u du u n cos u du u n sin u n u n sin u du Using Formula 5, Formula 55, and then Formula 5 produces sin d cos cos d cos sin sin d Formula 5 Formula 55 cos sin 6 cos 6 sin C. TECHNOLOGY Sometimes when ou use computer algebra sstems ou obtain results that look ver different, but are actuall equivalent. Here is how two different sstems evaluated the integral in Eample 5. Maple 5 arctanh 5 Mathematica Sqrt 5 Sqrt 5 Sqrt ArcTanh Sqrt Notice that computer algebra sstems do not include a constant of integration. EXAMPLE 5 Using a Reduction Formula 5 Find d. Solution Consider the following two formulas. du ua bu a ln a bu a a bu C a bu du du a bu a u ua bu Using Formula 9, with a, b 5, and u, produces 5 d 5 5 d 5 d 5. Formula 7 a > Formula 9 Using Formula 7, with a, b 5, and u, produces 5 d 5 ln 5 5 C 5 ln 5 C. 5

48 566 Chapter 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals Rational Functions of Sine and Cosine EXAMPLE 6 Integration b Tables sin Find cos d. Solution Substituting sin cos for sin produces sin cos d sin cos cos d. A check of the forms involving sin u or cos u in Appendi B shows that none of those listed applies. So, ou can consider forms involving a bu. For eample, u du a bu b bu a ln a bu C. Formula Let a, b, and u cos. Then du sin d, and ou have sin cos cos d cos sin d cos cos ln cos ln cos C cos C. Eample 6 involves a rational epression of sin and cos. If ou are unable to find an integral of this form in the integration tables, tr using the following special substitution to convert the trigonometric epression to a standard rational epression. SUBSTITUTION FOR RATIONAL FUNCTIONS OF SINE AND COSINE For integrals involving rational functions of sine and cosine, the substitution ields u sin cos tan u cos u, sin u u, and d du u. PROOF From the substitution for u, it follows that sin u cos cos cos cos cos. Solving for cos produces cos u u. To find sin, write u sin cos as sin u cos u u u u u. Finall, to find d, consider u tan. Then ou have arctan u and d du u.

49 8.6 Integration b Tables and Other Integration Techniques Eercises See for worked-out solutions to odd-numbered eercises. In Eercises and, use a table of integrals with forms involving a bu to find the integral... 5 d In Eercises and, use a table of integrals with forms involving u ± a to find the integral... 6 e e d d 6 In Eercises 5 and 6, use a table of integrals with forms involving a u to find the integral d d In Eercises 7, use a table of integrals with forms involving the trigonometric functions to find the integral. sin cos d d 9.. cos d tan 5 d In Eercises and, use a table of integrals with forms involving e u to find the integral... e sin d e d In Eercises and, use a table of integrals with forms involving ln u to find the integral.. 7 ln d. ln d In Eercises 5 8, find the indefinite integral (a) using integration tables and (b) using the given method. Integral 5. e d Integration b parts 6. 6 ln d Integration b parts 7. Partial fractions 8. d Partial fractions 8 d In Eercises 9, use integration tables to find the integral. Method 5 d.. 5 d sin d e 5. e 6. arccos e d tan e d sec d t ln t 9.. dt cos 9 d sin sin.. arctan d ln e. d 9 d. d ln e d d cos sin d 6 5 d 9.. d. e. cot d e d In Eercises 5, use integration tables to evaluate the integral. 9. t cos t dt 5. In Eercises 5 56, verif the integration formula d 7. e d. 9 d ln d cos d 6 cos sin d 7 d d u b bu a du a bu u n du a bu u ± a du ±u a u ± a C u n cos u du u n sin u n u n sin u du d a bu a ln a bu C n b un a bu na un a bu du 9. arccsc d... d arcsec d 8 d 55. arctan u du u arctan u ln u C 56. ln u n du uln u n n ln u n du

50 568 Chapter 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals CAS In Eercises 57 6, use a computer algebra sstem to determine the antiderivative that passes through the given point. Use the sstem to graph the resulting antiderivative. 57., d, d,, 59., 6 d, 6. d,, d, sin tan In Eercises 6 7, find or evaluate the integral. sin sin cos sin cos cos sin cos cos cos sin csc cot d d d, sin cos sin d,, d Area In Eercises 7 and 7, find the area of the region bounded b the graphs of the equations. 7.,, 6 7.,, e WRITING ABOUT CONCEPTS 7. (a) Evaluate n ln d for n,, and. Describe an patterns ou notice. (b) Write a general rule for evaluating the integral in part (a), for an integer n. 7. Describe what is meant b a reduction formula. Give an eample. True or False? In Eercises 75 and 76, determine whether the statement is true or false. If it is false, eplain wh or give an eample that shows it is false. 75. To use a table of integrals, the integral ou are evaluating must appear in the table. 76. When using a table of integrals, ou ma have to make substitutions to rewrite our integral in the form in which it appears in the table. 77. Volume Consider the region bounded b the graphs of 6,,, and. Find the volume of the solid generated b revolving the region about the -ais. d d d d CAPSTONE 78. State (if possible) the method or integration formula ou would use to find the antiderivative. Eplain wh ou chose that method or formula. Do not integrate. (a) (c) (e) e e d e d e d 85. Evaluate (b) (d) PUTNAM EXAM CHALLENGE d tan. e e d e d (f) e e d 79. Work A hdraulic clinder on an industrial machine pushes a steel block a distance of feet 5, where the variable force required is F e pounds. Find the work done in pushing the block the full 5 feet through the machine. 8. Work Repeat Eercise 79, using F 5 pounds Building Design The cross section of a precast concrete beam for a building is bounded b the graphs of the equations,, and where and are measured in feet. The length of the beam is feet (see figure). (a) Find the volume V and the weight W of the beam. Assume the concrete weighs 8 pounds per cubic foot. (b) Then find the centroid of a cross section of the beam. 8. Population A population is growing according to the logistic 5 model N where t is the time in das. Find the e.8.9t, average population over the interval,. In Eercises 8 and 8, use a graphing utilit to (a) solve the integral equation for the constant k and (b) graph the region whose area is given b the integral. k k 8. d 8. 6 e d 5, This problem was composed b the Committee on the Putnam Prize Competition. The Mathematical Association of America. All rights reserved. ft

51 8.7 Indeterminate Forms and L Hôpital s Rule Indeterminate Forms and L Hôpital s Rule Recognize its that produce indeterminate forms. Appl L Hôpital s Rule to evaluate a it. Indeterminate Forms Recall that the forms and are called indeterminate because the do not guarantee that a it eists, nor do the indicate what the it is, if one does eist. When ou encountered one of these indeterminate forms earlier in the tet, ou attempted to rewrite the epression b using various algebraic techniques. Indeterminate Form Limit Algebraic Technique Divide numerator and denominator b. Divide numerator and denominator b = e Occasionall, ou can etend these algebraic techniques to find its of transcendental functions. For instance, the it e e produces the indeterminate form. Factoring and then dividing produces e e e e e e. However, not all indeterminate forms can be evaluated b algebraic manipulation. This is often true when both algebraic and transcendental functions are involved. For instance, the it e produces the indeterminate form. Rewriting the epression to obtain e merel produces another indeterminate form,. Of course, ou could use technolog to estimate the it, as shown in the table and in Figure 8.. From the table and the graph, the it appears to be. (This it will be verified in Eample.) The it as approaches appears to be. Figure e ?

52 57 Chapter 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals The Granger Collection GUILLAUME L HÔPITAL (66 7) L Hôpital s Rule is named after the French mathematician Guillaume François Antoine de L Hôpital. L Hôpital is credited with writing the first tet on differential calculus (in 696) in which the rule publicl appeared. It was recentl discovered that the rule and its proof were written in a letter from John Bernoulli to L Hôpital. I acknowledge that I owe ver much to the bright minds of the Bernoulli brothers. I have made free use of their discoveries, said L Hôpital. L Hôpital s Rule To find the it illustrated in Figure 8., ou can use a theorem called L Hôpital s Rule. This theorem states that under certain conditions the it of the quotient fg is determined b the it of the quotient of the derivatives f g. To prove this theorem, ou can use a more general result called the Etended Mean Value Theorem. THEOREM 8. THE EXTENDED MEAN VALUE THEOREM If f and g are differentiable on an open interval a, b and continuous on a, b such that g for an in a, b, then there eists a point c in a, b such that fc fb fa gc gb ga. NOTE To see wh this is called the Etended Mean Value Theorem, consider the special case in which g. For this case, ou obtain the standard Mean Value Theorem as presented in Section.. The Etended Mean Value Theorem and L Hôpital s Rule are both proved in Appendi A. THEOREM 8. L HÔPITAL S RULE Let f and g be functions that are differentiable on an open interval a, b containing c, ecept possibl at c itself. Assume that g for all in a, b, ecept possibl at c itself. If the it of fg as approaches c produces the indeterminate form, then f f c g c g provided the it on the right eists (or is infinite). This result also applies if the it of fg as approaches c produces an one of the indeterminate forms,,, or. FOR FURTHER INFORMATION To enhance our understanding of the necessit of the restriction that g be nonzero for all in a, b, ecept possibl at c, see the article Countereamples to L Hôpital s Rule b R. P. Boas in The American Mathematical Monthl. To view this article, go to the website NOTE People occasionall use L Hôpital s Rule incorrectl b appling the Quotient Rule to fg. Be sure ou see that the rule involves fg, not the derivative of fg. L Hôpital s Rule can also be applied to one-sided its. For instance, if the it of fg as approaches c from the right produces the indeterminate form, then f c g f c g provided the it eists (or is infinite).