5.7 Inverse Trigonometric Functions: Integration 375

5.7 Inverse Trigonometric Functions: Integration

Integrate functions whose antiderivatives involve inverse trigonometric functions. Use the method of completing the square to integrate a function. Review the basic integration rules involving elementary functions.

Integrals Involving Inverse Trigonometric Functions The derivatives of the six inverse trigonometric functions fall into three pairs. In each pair, the derivative of one function is the negative of the other. For example, d 1 ͓arcsin x͔ dx Ί1 x 2 and d 1 ͓arccos x͔ . dx Ί1 x 2 When listing the antiderivative that corresponds to each of the inverse trigonometric functions, you need to use only one member from each pair. It is conventional to use arcsin x as the antiderivative of 1͞Ί1 x2, rather than arccos x. The next theorem gives one antiderivative formula for each of the three pairs. The proofs of these integration rules are left to you (see Exercises 75–77).

FOR FURTHER INFORMATION THEOREM 5.17 Integrals Involving Inverse Trigonometric For a detailed proof of rule 2 of Functions Theorem 5.17, see the article Let u be a differentiable function of x, and let a > 0. “A Direct Proof of the Integral du u du 1 u Formula for Arctangent” by 1.͵ arcsin C 2. ͵ arctan C Arnold J. Insel in The College Ίa2 u2 a a2 u2 a a Mathematics Journal. To view this du 1 ԽuԽ 3. ͵ arcsec C article, go to MathArticles.com. uΊu2 a2 a a

Integration with Inverse Trigonometric Functions

dx x a. ͵ arcsin C Ί4 x2 2 dx 1 3 dx b. ͵ ͵ u 3x, a Ί2 2 9x2 3 ͑Ί2͒2 ͑3x͒2 1 3x arctan C 3Ί2 Ί2 dx 2 dx c. ͵ ͵ u 2x, a 3 xΊ4x2 9 2xΊ͑2x͒2 32 1 Խ2xԽ arcsec C 3 3

The integrals in Example 1 are fairly straightforward applications of integration formulas. Unfortunately, this is not typical. The integration formulas for inverse trigonometric functions can be disguised in many ways.

Integration by Substitution

dx Find ͵ . Ίe2x 1 Solution As it stands, this integral doesn’t fit any of the three inverse trigonometric formulas. Using the substitution u ex, however, produces du du u ex du ex dx dx . ex u With this substitution, you can integrate as shown. dx dx ͵ ͵ Write e2x as ͑e x͒2. Ίe2x 1 Ί͑ex͒2 1 du͞u ͵ Substitute. Ίu2 1 du ͵ Rewrite to fit Arcsecant Rule. uΊu2 1 ԽuԽ arcsec C Apply Arcsecant Rule. 1 arcsec ex C Back-substitute.

TECHNOLOGY PITFALL A symbolic integration utility can be useful for integrating functions such as the one in Example 2. In some cases, however, the utility may fail to find an antiderivative for two reasons. First, some elementary functions do not have antiderivatives that are elementary functions. Second, every utility has limitations—you might have entered a function that the utility was not programmed to handle. You should also remember that antiderivatives involving trigonometric functions or logarithmic functions can be written in many different forms. For instance, one utility found the integral in Example 2 to be dx ͵ arctan Ίe2x 1 C. Ίe2x 1 Try showing that this antiderivative is equivalent to the one found in Example 2.

Rewriting as the Sum of Two Quotients

x 2 Find ͵ dx. Ί4 x2 Solution This integral does not appear to fit any of the basic integration formulas. By splitting the integrand into two parts, however, you can see that the first part can be found with the Power Rule and the second part yields an inverse sine function. x 2 x 2 ͵ dx ͵ dx ͵ dx Ί4 x2 Ί4 x2 Ί4 x2 1 1 ͵͑4 x2͒1͞2͑2x͒ dx 2 ͵ dx 2 Ί4 x2 1 ͑4 x2͒1͞2 x ΄ ΅ 2 arcsin C 2 1͞2 2 x Ί4 x2 2 arcsin C 2

Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 5.7 Inverse Trigonometric Functions: Integration 377 Completing the Square Completing the square helps when quadratic functions are involved in the integrand. For example, the quadratic x2 bx c can be written as the difference of two squares by adding and subtracting ͑b͞2͒2. b 2 b 2 b 2 b 2 x2 bx c x2 bx ΂ ΃ ΂ ΃ c ΂x ΃ ΂ ΃ c 2 2 2 2

Completing the Square

See LarsonCalculus.com for an interactive version of this type of example. dx Find ͵ . x2 4x 7 Solution You can write the denominator as the sum of two squares, as shown. x2 4x 7 ͑x2 4x 4͒ 4 7 ͑x 2͒2 3 u2 a2 Now, in this completed square form, let u x 2 and a Ί3. dx dx 1 x 2 ͵ ͵ arctan C x2 4x 7 ͑x 2͒2 3 Ί3 Ί3

y When the leading coefficient is not 1, it helps to factor before completing the square. For instance, you can complete the square of 2x2 8x 10 by factoring first. 1 f(x) = 2 2 3 3x − x2 2 x 8x 10 2͑x 4x 5͒ 2͑x2 4x 4 4 5͒ 2͓͑x 2͒2 1͔ 2 To complete the square when the coefficient of x2 is negative, use the same factoring process shown above. For instance, you can complete the square for 3x x2 as shown. 1 2 ͑ 2 ͒ ͓ 2 ͑3͒2 ͑3͒2͔ ͑3͒2 ͑ 3͒2 3x x x 3x x 3x 2 2 2 x 2

x 1 x = 3 23x = 9 2 4 Completing the Square The area of the region bounded by the 3 9 Find the area of the region bounded by the graph of graph of f, the x -axis,x 2, and x 4 is ͞6. 1 f͑x͒ Figure 5.28 Ί3x x2 3 9 the x- axis, and the lines x 2 and x 4. TECHNOLOGY With definite integrals such as the one Solution In Figure 5.28, you can see that the area is ͞ given in Example 5, remember 9 4 1 Area ͵ dx that you can resort to a numerical Ί 2 3͞2 3x x solution. For instance, applying 9͞4 ) dx Simpson’s Rule (with n 12 ͵ Use completed square form derived above. Ί 2 2 to the integral in the example, 3͞2 ͑3͞2͒ ͓x ͑3͞2͔͒ you obtain x ͑3͞2͒ 9͞4 arcsin 9͞4 ͞ ΅ 1 3 2 3͞2 ͵ dx Ϸ 0.523599. Ί 2 1 3͞2 3x x arcsin arcsin 0 2 This differs from the exact value of the integral ͑͞6 Ϸ 0.5235988͒ by less 6 than one-millionth. Ϸ 0.524.

Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 378 Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions Review of Basic Integration Rules You have now completed the introduction of the basic integration rules. To be efficient at applying these rules, you should have practiced enough so that each rule is committed to memory.

BASIC INTEGRATION RULES ͧa > 0ͨ

1.͵k f͑u͒ du k͵f͑u͒ du 2. ͵͓ f͑u͒ ± g͑u͔͒ du ͵f͑u͒ du ± ͵g͑u͒ du

un1 3.͵du u C 4. ͵un du C, n 1 n 1

du 5.͵ lnԽuԽ C 6. ͵eu du eu C u

1 7.͵au du ΂ ΃au C 8. ͵sin u du cos u C ln a

9.͵cos u du sin u C 10. ͵tan u du lnԽcos uԽ C

11.͵cot u du lnԽsin uԽ C 12. ͵sec u du lnԽsec u tan uԽ C

13.͵csc u du lnԽcsc u cot uԽ C 14. ͵sec2 u du tan u C

15.͵csc2 u du cot u C 16. ͵sec u tan u du sec u C

du u 17.͵csc u cot u du csc u C 18. ͵ arcsin C Ίa2 u2 a

du 1 u du 1 ԽuԽ 19.͵ arctan C 20. ͵ arcsec C a2 u2 a a uΊu2 a2 a a

You can learn a lot about the nature of integration by comparing this list with the summary of differentiation rules given in the preceding section. For differentiation, you now have rules that allow you to differentiate any elementary function. For integration, this is far from true. The integration rules listed above are primarily those that were happened on during the development of differentiation rules. So far, you have not learned any rules or techniques for finding the antiderivative of a general product or quotient, the natural logarithmic function, or the inverse trigonometric functions. More important, you cannot apply any of the rules in this list unless you can create the proper du corresponding to the u in the formula. The point is that you need to work more on integration techniques, which you will do in Chapter 8. The next two examples should give you a better feeling for the integration problems that you can and cannot solve with the techniques and rules you now know.

Comparing Integration Problems

Find as many of the following integrals as you can using the formulas and techniques you have studied so far in the text. dx a. ͵ xΊx2 1 x dx b. ͵ Ίx2 1 dx c. ͵ Ίx2 1 Solution a. You can find this integral (it fits the Arcsecant Rule). dx ͵ arcsecԽxԽ C xΊx2 1 b. You can find this integral (it fits the Power Rule). x dx 1 ͵ ͵͑x2 1͒1͞2͑2x͒ dx Ίx2 1 2 1 ͑x2 1͒1͞2 ΄ ΅ C 2 1͞2 Ίx2 1 C c. You cannot find this integral using the techniques you have studied so far. (You should scan the list of basic integration rules to verify this conclusion.)

Comparing Integration Problems

Find as many of the following integrals as you can using the formulas and techniques you have studied so far in the text. dx a. ͵ x ln x ln x dx b. ͵ x

c. ͵ln x dx

Solution a. You can find this integral (it fits the Log Rule). dx 1͞x ͵ ͵ dx x ln x ln x lnԽln xԽ C b. You can find this integral (it fits the Power Rule). ln x dx 1 ͵ ͵΂ ΃͑ln x͒1 dx REMARK Note in x x Examples 6 and 7 that the ͑ln x͒2 simplest functions are the ones C 2 that you cannot yet integrate. c. You cannot find this integral using the techniques you have studied so far.

5.7 Exercises See CalcChat.com for tutorial help and worked-out solutions to odd-numbered exercises.

Finding an Indefinite Integral In Exercises 1–20, find the 3 2x 3 1 39.͵ dx 40. ͵ dx indefinite integral. Ί 2 Ί 2 2 4x x ͑x 1͒ x 2x dx dx x x 1.͵ 2. ͵ 41.͵ dx 42. ͵ dx Ί9 x2 Ί1 4x2 x 4 2x2 2 Ί9 8x2 x 4 1 12 3.͵ dx 4. ͵ dx Integration by Substitution In Exercises 43–46, use the xΊ4x2 1 1 9x2 specified substitution to find or evaluate the integral. 1 1 5.͵ dx 6. ͵ dx Ί1 ͑x 1͒2 4 ͑x 3͒2 43. ͵Ίet 3 dt t 1 7.͵ dt 8. ͵ dx u Ίet 3 Ί1 t4 xΊx 4 4 Ίx 2 t 1 44. ͵ dx 9.͵ dt 10. ͵ dx x 1 t4 25 xΊ1 ͑ln x͒2 u Ίx 2 2x e 2 3 11.͵ dx 12. ͵ dx dx 4 e4x xΊ 2 45. ͵ 9x 25 Ί 1 x͑1 x͒ sec2 x sin x 13.͵ dx 14. ͵ dx u Ίx Ί25 tan2 x 7 cos2 x 1 dx 1 3 46. ͵ 15.͵ dx 16. ͵ dx 2Ί3 xΊx 1 ΊxΊ1 x 2Ίx͑1 x͒ 0 u Ίx 1 x 3 x2 3 17.͵ dx 18. ͵ dx x2 1 xΊx2 4 WRITING ABOUT CONCEPTS x 5 x 2 19.͵ dx 20. ͵ dx Ί9 ͑x 3͒2 ͑x 1͒2 4 Comparing Integration Problems In Exercises 47–50, determine which of the integrals can be found using the Evaluating a Definite Integral In Exercises 21–32, basic integration formulas you have studied so far in the evaluate the definite integral. text.

͞ Ί 1 6 2 1 2 ͵ 3 ͵ 1 47. (a) ͵ dx 48. (a) ͵ex dx 21. dx 22. dx 2 Ί 2 2 Ί1 x 0 1 9x 0 Ί4 x Ί3͞2 3 1 1 x 2 23.͵ dx 24. ͵ dx (b)͵ dx (b) ͵xex dx 2 Ί 2 Ί 2 0 1 4x Ί3 x 4x 9 1 x 6 1 4 1 1 1 25.͵ dx 26. ͵ dx (c)͵ dx (c) ͵ e1͞x dx ͑ ͒2 Ί 2 2 2 3 25 x 3 1 x 16x 5 xΊ1 x x ln 5 ex ln 4 ex 27.͵ dx 28. ͵ dx Ί 1 2x Ί 2x 49. (a) ͵ x 1 dx 50. (a) ͵ dx 0 1 e ln 2 1 e 1 x4 ͞2 sin x cos x x 29.͵ dx 30. ͵ dx ͵ Ί ͵ 2 2 (b)x x 1 dx (b) 4 dx ͞2 1 cos x 0 1 sin x 1 x ͞Ί ͞Ί 1 2 arcsin x 1 2 arccos x x x3 31.͵ dx 32. ͵ dx (c)͵ dx (c) ͵ dx 2 2 Ί 4 0 Ί1 x 0 Ί1 x x 1 1 x

Completing the Square In Exercises 33 –42, find or 51. Finding an Integral Decide whether you can find the evaluate the integral by completing the square. integral 2 dx 2 dx 2 dx 33.͵ 34. ͵ ͵ 2 2 Ί 2 0 x 2x 2 2 x 4x 13 x 4 2x 2x 5 35.͵ dx 36. ͵ dx using the formulas and techniques you have studied so far. 2 2 x 6x 13 x 2x 2 Explain your reasoning. 1 2 37.͵ dx 38. ͵ dx Ίx2 4x Ίx2 4x

Area In Exercises 61–66, find the area of the region. 52. HOW DO YOU SEE IT? Using the graph, 2 1 which value best approximates the area of the 61.y 62. y Ί 2 Ί 2 region between the x -axis and the function over 4 x x x 1 ͓1 1 ͔ y y the interval 2, 2 ? Explain. y 3 2 2 2

1 3 2 2 x = x 2 − − f(x) = 1 2121 x − 2 −1 1 1 x 12 2

x 1 2 63.y 64. y −1 − 1 1 1 2 2 2 2 x 2x 5 x 4x 8 y y 1 (a) 3 (b) 2 (c) 1 (d) 2 (e) 4 0.4 0.5 0.3 Differential Equation In Exercises 53 and 54, use the 0.2 differential equation and the specified initial condition to find y. 0.2 x 0.1 dy 1 dy 1 −21234−1 53. 54. x dx Ί4 x2 dx 4 x2 −5 −4 −3 −2 −1 1 −0.2 y͑0͒ y͑2͒ 3 cos x 4ex 65.y 66. y Slope Field In Exercises 55 and 56, a differential equation, 1 sin2 x 1 e2x a point, and a slope field are given. (a) Sketch two approximate y y solutions of the differential equation on the slope field, one of which passes through the given point. (b) Use integration to 3 3 find the particular solution of the differential equation and use x = ln 3 a graphing utility to graph the solution. Compare the result 1 with the sketches in part (a). To print an enlarged copy of the x 1 π π π graph, go to MathGraphs.com. − 4 4 2 x dy 2 dy 2 −2 −21−1 2 55. , ͑0, 2͒ 56. , ͑5, ͒ − dx 9 x2 dx Ί25 x2 −3 1 y y 67. Area 5 5 (a) Sketch the region whose area is represented by

1 ͵ arcsin x dx. x 0 x −5 5 −4 4 (b) Use the integration capabilities of a graphing utility to approximate the area. − 3 −5 (c) Find the exact area analytically. 68. Approximating Pi Slope Field In Exercises 57–60, use a computer algebra system to graph the slope field for the differential equation and (a) Show that graph the solution satisfying the specified initial condition. 1 4 ͵ dx . 2 dy 10 dy 1 0 1 x 57. 58. Ί 2 2 dx x x 1 dx 12 x (b) Approximate the number using Simpson’s Rule (with y ͑3͒ 0 y ͑4͒ 2 n 6) and the integral in part (a). dy 2y dy Ίy (c) Approximate the number by using the integration 59. 60. dx Ί16 x2 dx 1 x2 capabilities of a graphing utility. y͑0͒ 2 y͑0͒ 4

69. Investigation Consider the function 79. Numerical Integration 1 x 2 2 (a) Write an integral that represents the area of the region in F͑x͒ ͵ dt. 2 the figure. 2 x t 1 (b) Use the Trapezoidal Rule with n 8 to estimate the area (a) Write a short paragraph giving a geometric interpretation of the region. of the function F͑x͒ relative to the function (c) Explain how you can use the results of parts (a) and (b) to 2 estimate . f͑x͒ . x2 1 y Use what you have written to guess the value of x that will 2 make F maximum. 3 (b) Perform the specified integration to find an alternative 2 1 y = 2 form of F͑x͒. Use calculus to locate the value of x that will 1 + x make F maximum and compare the result with your guess 1 in part (a). 2 70. Comparing Integrals Consider the integral x −21−1 2 1 ͵ dx. 80. Vertical Motion An object is projected upward from Ί6x x2 ground level with an initial velocity of 500 feet per second. In (a) Find the integral by completing the square of the radicand. this exercise, the goal is to analyze the motion of the object during its upward flight. (b) Find the integral by making the substitution u Ίx. (a) If air resistance is neglected, find the velocity of the object (c) The antiderivatives in parts (a) and (b) appear to be as a function of time. Use a graphing utility to graph this significantly different. Use a graphing utility to graph each function. antiderivative in the same viewing window and determine the relationship between them. Find the domain of each. (b) Use the result of part (a) to find the position function and determine the maximum height attained by the object. True or False? In Exercises 71–74, determine whether the (c) If the air resistance is proportional to the square of the statement is true or false. If it is false, explain why or give an velocity, you obtain the equation example that shows it is false. dv dx 1 3x ͑32 kv2͒ 71. ͵ arcsec C dt 3xΊ9x2 16 4 4 dx 1 x where 32 feet per second per second is the acceleration 72. ͵ arctan C 25 x2 25 25 due to gravity and k is a constant. Find the velocity as a function of time by solving the equation dx x 73. ͵ arccos C Ί4 x2 2 dv ͵ ͵dt. 2e2x 32 kv2 74. One way to find ͵ dx is to use the Arcsine Rule. Ί 2x 9 e (d) Use a graphing utility to graph the velocity function v͑t͒ in part (c) for k 0.001. Use the graph to approximate the Verifying an Integration Rule In Exercises 75–77, verify time t at which the object reaches its maximum height. the rule by differentiating. Let a > 0. 0 (e) Use the integration capabilities of a graphing utility to du u 75. ͵ arcsin C approximate the integral Ίa2 u2 a t0 du 1 u ͵ v͑t͒ dt 76. ͵ arctan C a2 u2 a a 0 du 1 ԽuԽ where v͑t͒ and t are those found in part (d). This is the 77. ͵ arcsec C 0 uΊu2 a2 a a approximation of the maximum height of the object. (f) Explain the difference between the results in parts (b) and 78. Proof Graph (e). x y , y arctan x, and y x FOR FURTHER INFORMATION For more information 1 2 2 3 1 x on this topic, see the article “What Goes Up Must Come Down; on ͓0, 10͔. Prove that Will Air Resistance Make It Return Sooner, or Later?” by John Lekner in Mathematics Magazine. To view this article, go to x MathArticles.com. < arctan x < x for x > 0. 1 x2

y y 73. 75. 43. 2Ίet 3 2Ί3 arctan͑Ίet 3͞Ί3͒ C 45. ͞6 π − 1 , π ( 2 ( 47. a and b 49. a, b, and c π π 2, 51. No. This integral does not correspond to any of the basic 2 ) 2) (1, 0) x π integration rules. −1123 2 ͑ ͞ ͒ π 53. y arcsin x 2 − π 1 2 − , 0 0, ( 2 ( x ) 2) y 2 −π x 55. (a) (b) y arctan 2 −2 −112 4 3 3 5 1 Maximum: ΂2, ΃ Maximum: ΂ , ΃ 2 2 x 1 4 Minimum: ΂0, ΃ Minimum: ΂ , 0΃ − 2 2 4 4 −1 Point of inflection: ͑1, 0͒ Asymptote: y −4 2 77. y 2x͑͞ 8͒ 1 2͑͞2 16͒ 57. 4 59. 3 79. y x Ί2 −612 81. If the domains were not restricted, then the trigonometric functions would have no inverses, because they would not be −3 3 one-to-one. − − 83. (a) arcsin͑arcsin 0.5͒ Ϸ 0.551 8 1 arcsin͑arcsin 1͒ does not exist. 61.͞3 63.͞8 65. 3͞2 (b) sin͑1͒ x sin͑1͒ 67. (a) y (b) 0.5708 85. False. The range of arccos is ͓0, ͔. (c) ͑ 2͒͞2 2 87. True 89. True 91. (a) arccot͑x͞5͒ (b) x 10: 16 rad͞h; x 3: 58.824 rad͞h 1 93. (a) h͑t͒ 16t 2 256; t 4 sec x (b) t 1: 0.0520 rad͞sec; t 2: 0.1116 rad͞sec 12 95. 50Ί2 Ϸ 70.71 ft 97. (a) and (b) Proofs 99. (a) y 69. (a)F͑x͒ represents the average value of f͑x͒ over the interval (b) The graph is a horizontal ͓x, x 2͔. Maximum at x 1 2 (b) Maximum at x 1 line at . dx 1 Խ3xԽ 2 71. False. ͵ arcsec C 1 (c) Proof 3xΊ9x2 16 12 4 73. True 75–77. Proofs x 1 1 −11 79. (a) ͵ dx (b) About 0.7847 2 0 1 x 101.c 2 103. Proof 1 1 (c) Because ͵ dx , you can use the Trapezoidal 2 Section 5.7 (page 380) 0 1 x 4 Rule to approximate. Multiplying the result by 4 gives 4 x an estimation of . 1. arcsin C 3. arcsecԽ2xԽ C 3 ͑ ͒ 1 2 5. arcsin x 1 C 7. 2 arcsin t C 1 t2 1 9. arctan C 11. arctan ͑e2x͞2͒ C 10 5 4 tan x 13. arcsin΂ ΃ C 15. 2 arcsinΊx C 5 1 ͑ 2 ͒ 17. 2 ln x 1 3 arctan x C 19. 8 arcsin͓͑x 3͒͞3͔ Ί6x x2 C 21. ͞6 ͞ 1 3 Ϸ 23.6 25. 5 arctan 5 0.108 ͞ Ϸ ͞ 1 2 Ϸ 27. arctan 5 4 0.588 29.4 31. 32 0.308 33.͞2 35. lnԽx2 6x 13Խ 3 arctan͓͑x 3͒͞2͔ C ͓͑ ͒͞ ͔ Ί 1 Ϸ 37. arcsin x 2 2 C 39. 4 2 3 6 1.059 1 ͑ 2 ͒ 41. 2 arctan x 1 C