Integration by Tables and Other Integration Techniques. Integration by Tables

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1 3346_86.qd //4 3:8 PM Page 56 SECTION 8.6 and Other Integration Techniques 56 Section 8.6 and Other Integration Techniques Evaluate an indefinite integral using a table of integrals. Evaluate an indefinite integral using rection formulas. Evaluate an indefinite integral involving rational functions of sine and cosine. So far in this chapter you have studied several integration techniques that can be used with the basic integration rules. But merely 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, you must recognize which rule or technique to apply to obtain an antiderivative. Frequently, a slight alteration of an integrand will require a different integration technique (or proce a function whose antiderivative is not an elementary function), as shown below. TECHNOLOGY A computer algebra system consists, in part, of a database of integration formulas. The primary difference between using a computer algebra system and using tables of integrals is that with a computer algebra system the computer searches through the database to find a fit. With integration tables, you must do the searching. ln d ln 4 C ln ln d C Integration by parts Power Rule Log Rule Not an elementary function Many 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 by tables is not a cure-all for all of the difficulties that can accompany 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 try to verify several of the formulas. For instance, Formula 4 ln d ln ln C d? ln u a bu b a a bu ln a bu C Formula 4 can be verified using the method of partial fractions, and Formula 9 a bu a bu a Formula 9 u ua bu can be verified using integration by parts. Note that the integrals in Appendi B are classified according to forms involving the following. u ± a u n a bu a bu cu a bu a ± u a u Trigonometric functions Inverse trigonometric functions Eponential functions Logarithmic functions

2 3346_86.qd //4 3:9 PM Page 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 you do this, you should obtain d u. Does this proce the same result as that obtained in Eample? EXAMPLE Solution Because the epression inside the radical is linear, you should consider forms involving a bu. Formula 7 a < ua bu a arctan a bu C a Let a, b, and u. Then d, and you can write d. d arctan C. EXAMPLE 4 9 d. Solution Because the radical has the form u a, you should consider Formula 6. u a uu a a ln u u a C Let u and a 3. Then d, and you have 4 9 d 3 d ln 4 9 C. EXAMPLE 3 Solution Of the forms involving e u, consider the following formula. e d. e u u ln eu C Let u. Then d, and you have d e d e ln e C ln e C. Formula 84 TECHNOLOGY Eample 3 shows the importance of having several solution techniques at your disposal. This integral is not difficult to solve with a table, but when it was entered into a well-known computer algebra system, the utility was unable to find the antiderivative.

3 3346_86.qd //4 3:9 PM Page 563 SECTION 8.6 and Other Integration Techniques 563 Rection Formulas Several of the integrals in the integration tables have the form f d g h d. Such integration formulas are called rection formulas because they rece a given integral to the sum of a function and a simpler integral. EXAMPLE 4 Using a Rection Formula 3 sin d. Solution Consider the following three formulas. u sin u sin u u cos u C Formula 5 u n sin u u n cos u n u n cos u u n cos u u n sin u n u n sin u Using Formula 54, Formula 55, and then Formula 5 proces Formula 54 Formula 55 TECHNOLOGY Sometimes when you use computer algebra systems you obtain results that look very different, but are actually equivalent. Here is how several different systems evaluated the integral in Eample 5. Maple arctanh Derive ln 3 5 Mathematica Sqrt3 5 Sqrt3 5 Sqrt3 ArcTanh Sqrt3 Mathcad ln Notice that computer algebra systems do not include a constant of integration. EXAMPLE 5 Using a Rection Formula 3 5 d. Solution 3 sin d 3 cos 3 cos d Consider the following two formulas. ua bu a ln a bu a a bu C a bu a bu a u ua bu Using Formula 9, with a 3, b 5, and u, proces cos 3 sin sin d 3 cos 3 sin 6 cos 6 sin C. d d 3 5 d 3 5. Formula 7 a > Formula 9 Using Formula 7, with a 3, b 5, and u, proces 3 5 d ln C ln C. 3 5

4 3346_86.qd //4 3:9 PM Page CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals Rational Functions of Sine and Cosine EXAMPLE 6 Solution Substituting sin cos for sin proces 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, you can consider forms involving a bu. For eample, sin cos d. u a bu b bu a ln a bu C. Formula 3 Let a, b, and u cos. Then sin d, and you have sin cos cos d cos sin d cos cos ln cos 4 ln cos C cos C. Eample 6 involves a rational epression of sin and cos. If you are unable to find an integral of this form in the integration tables, try 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 yields u sin cos tan u cos u, sin u u, and d u. Proof From the substitution for u, it follows that sin u cos cos cos cos cos. Solving for cos proces cos u u. To find sin, write u sin cos as sin u cos u u u u u. Finally, to find d, consider u tan. Then you have arctan u and d u.

5 3346_86.qd //4 3:9 PM Page 565 SECTION 8.6 and Other Integration Techniques 565 Eercises for Section 8.6 In Eercises and, use a table of integrals with forms involving a bu to find the integral... d In Eercises 3 and 4, use a table of integrals with forms involving u ± a to find the integral e e d d 3 In Eercises 5 and 6, use a table of integrals with forms involving a u to find the integral d d 4 In Eercises 7, use a table of integrals with forms involving the trigonometric functions to find the integral. cos sin 4 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.. d. e sin d e In Eercises 3 and 4, use a table of integrals with forms involving ln u to find the integral ln d 4. ln 3 d In Eercises 5 8, find the indefinite integral (a) using integration tables and (b) using the given method. Integral Method 5. e d Integration by parts 6. 4 ln d Integration by parts 7. Partial fractions 8. d Partial fractions 75 d In Eercises 9 4, use integration tables to find the integral. 9. arcsec. d arcsec d 3 5 d d 4 d 3 d sin d e 5. 3 e 6. arccos e d tan e d sec d t ln t dt cos 9 d 3 sin sin arctan 3 d ln e 33. d 9 d 34. d e ln d d cos sin d d d 4. e 4. 3 tan 3 d e d 3 In Eercises 43 5, use integration tables to evaluate the integral. 43. e d 44. d ln d sin d 4 cos sin d 3 5 d 49. t 3 cos t dt 5. In Eercises 5 56, verify the integration formula See for worked-out solutions to odd-numbered eercises. 3 d 3 3 d 3 u b bu a a bu 3 u n a bu u ± a ±u 3 a u ± a C u n cos u u n sin u n u n sin u a bu a ln a bu C n b un a bu na un 55. arctan u u arctan u ln u C 3 d a bu

6 3346_86.qd //4 3:9 PM Page CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals 56. In Eercises 57 6, use a computer algebra system to determine the antiderivative that passes through the given point. Use the system to graph the resulting antiderivative , 5 d,, d, 3, 6 d, 6. d,, ln u n uln u n n ln u n d, sin tan sin cos sin d, In Eercises 63 7, find or evaluate the integral. sin sin cos sin cos 3 cos sin cos cos cos cos sec tan d d d 4,, d Area In Eercises 7 and 7, find the area of the region bounded by the graphs of the equations. 7. y, y, 8 7. y, y, e Writing About Concepts In Eercises 73 78, state (if possible) the method or integration formula you would use to find the antiderivative. Eplain why you chose that method or formula. Do not integrate e e e e d d 75. e d 76. e d 77. e d 78. e e d 79. (a) Evaluate n ln d for n,, and 3. Describe any patterns you notice. (b) Write a general rule for evaluating the integral in part (a), for an integer n. 8. Describe what is meant by a rection formula. Give an eample. d d d d True or False? In Eercises 8 and 8, determine whether the statement is true or false. If it is false, eplain why or give an eample that shows it is false. 8. To use a table of integrals, the integral you are evaluating must appear in the table. 8. When using a table of integrals, you may have to make substitutions to rewrite your integral in the form in which it appears in the table. 83. Work A hydraulic cylinder on an instrial machine pushes a steel block a distance of feet 5, where the variable force required is F e pounds. the work done in pushing the block the full 5 feet through the machine. 84. Work Repeat Eercise 83, using F 5 pounds Building Design The cross section of a precast concrete beam for a building is bounded by the graphs of the equations, y, and y 3 where and y are measured in feet. The length of the beam is feet (see figure). (a) the volume V and the weight W of the beam. Assume the concrete weighs 48 pounds per cubic foot. (b) Then find the centroid of a cross section of the beam. 86. Population A population is growing according to the logistic 5 model N where t is the time in days. the e 4.8.9t average population over the interval,. In Eercises 87 and 88, use a graphing utility to (a) solve the integral equation for the constant k and (b) graph the region whose area is given by the integral. 4 k k 87. d e d 5 3 y, Evaluate y 4 y ft Putnam Eam Challenge d tan. This problem was composed by the Committee on the Putnam Prize Competition. The Mathematical Association of America. All rights reserved. 3