y sin n x dx cos x sin n 1 x n 1 y sin n 2 x cos 2 x dx y sin n xdx cos x sin n 1 x n 1 y sin n 2 x dx n 1 y sin n x dx

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1 SECTION 7. INTEGRATION BY PARTS 57 EXAPLE 6 Prove the reduction formula N Equation 7 is called a reduction formula because the eponent n has been reduced to n and n. 7 sin n n cos sinn n n sin n where n is an integer. SOLUTION Let Then u sin n du n sin n cos dv sin v cos so integration b parts gives sin n cos sin n n sin n cos Since cos sin, we have sin n cos sin n n sin n n sin n As in Eample, we solve this equation for the desired integral b taking the last term on the right side to the left side. Thus we have n sin n cos sin n n sin n or sin n n cos sinn n n sin n The reduction formula (7) is useful because b using it repeatedl we could eventuall epress sin n in terms of sin (if n is odd) or sin (if n is even). 7. EXERCISES Evaluate the integral using integration b parts with the indicated choices of u and dv.. arctan t dt. p 5 ln p dp. ln ; u ln, dv 3. t sec t dt. s s ds. cos d; u, dv cos d 5. ln 6. t sinh mt dt 3 3 Evaluate the integral. 7. e sin 3 d 8. e cos d 3. cos 5 5. re r dr 6. t sin t dt 7. sin 8. cos m 9. ln. sin. e 9. t sin 3t dt.. t cosh t dt. e 3. ln. 3 cos 9 ln s d

2 58 CHAPTER 7 TECHNIQUES OF INTEGRATION 5. d 6. e 7. cos cos lnsin 3. s3 arctan ln 3 r 3 s r dr (b) Use part (a) to evaluate sin 3 and sin 5. (c) Use part (a) to show that, for odd powers of sine, sin n 6. Prove that, for even powers of sine, 6 n n 3. ln 3. t e s sint s ds sin n 3 5 n 6 n First make a substitution and then use integration b parts to evaluate the integral. 33. cos s 3. t 3 e t dt 35. s s 3 cos d ; 39 Evaluate the indefinite integral. Illustrate, and check that our answer is reasonable, b graphing both the function and its antiderivative (take C ) ln 38. sinln 39. 3e. 3 ln. 3 s. sin 3. (a) Use the reduction formula in Eample 6 to show that (b) Use part (a) and the reduction formula to evaluate sin.. (a) Prove the reduction formula (b) Use part (a) to evaluate cos. (c) Use parts (a) and (b) to evaluate cos. 5. (a) Use the reduction formula in Eample 6 to show that sin cos n n cosn sin n n sin n n n where n is an integer. sin e cos t sin t dt C sin n cos n 7 5 Use integration b parts to prove the reduction formula Use Eercise 7 to find ln Use Eercise 8 to find e Find the area of the region bounded b the given curves. 53. e.,, 5. ln n ln n n ln n n e n e n n e tan n tann n tan n sec n tan secn n 5 ln, ; Use a graph to find approimate -coordinates of the points of intersection of the given curves. Then find (approimatel) the area of the region bounded b the curves. 55. sin, 56. arctan 3, ln 5 n n n sec n 57 6 Use the method of clindrical shells to find the volume generated b rotating the region bounded b the given curves about the specified ais. 57. cos,, ; about the -ais 58. e, e, ; about the -ais 59. e,,, ; about 6. e,, ; about the -ais n

3 SECTION 7. TRIGONOETRIC INTEGRALS 65 Then sec 3 sec tan sec tan sec tan sec sec sec tan sec 3 sec Using Formula and solving for the required integral, we get sec 3 (sec tan ln sec tan ) C Integrals such as the one in the preceding eample ma seem ver special but the occur frequentl in applications of integration, as we will see in Chapter 8. Integrals of the form cot m csc n can be found b similar methods because of the identit cot csc. Finall, we can make use of another set of trigonometric identities: N These product identities are discussed in Appendi D. To evaluate the integrals (a) sin m cos n, (b) sin m sin n, or (c) cos m cos n, use the corresponding identit: (a) (b) sin A cos B sina B sina B sin A sin B cosa B cosa B (c) cos A cos B cosa B cosa B EXAPLE 9 Evaluate sin cos 5. SOLUTION This integral could be evaluated using integration b parts, but it s easier to use the identit in Equation (a) as follows: sin cos 5 sin sin 9 sin sin 9 (cos 9 cos 9 C 7. EXERCISES 9 Evaluate the integral.. sin 3 cos. sin 6 cos sin cos sin 5 cos 3 cos d. 8. cos 5 sin3 (s ) s sin d 9.. sin 3t dt. cos d. cos 3. sin cos 5. cos5 6. cos cos 5 sin d ssin d. cos6 d sin t cos t dt

4 66 CHAPTER 7 TECHNIQUES OF INTEGRATION 7. cos tan cos sin sin.. sec tan. 3. tan. 5. sec 6 t dt tan 5 sec tan 3 sec tan tan3 3. cos d cot 5 sin d cos sin 3 sec t dt tan tan sec tan d tan 3 sec 5 tan 5 sec 6 tan 6 a d tan sec 53. sin 3 sin 6 5. sec 55. Find the average value of the function f sin cos 3 on the interval,. 56. Evaluate sin cos b four methods: (a) the substitution u cos (b) the substitution u sin (c) the identit sin sin cos (d) integration b parts Eplain the different appearances of the answers Find the area of the region bounded b the given curves. 57. sin, cos, 58. sin 3, cos 3, 5 ; 59 6 Use a graph of the integrand to guess the value of the integral. Then use the methods of this section to prove that our guess is correct. 35. sec tan 36. sin cos 3 d 59. cos 3 6. sin cos cot cot 3 csc 3 d.. csc. 3. sin 8 cos 5. cot 3 csc cot csc3 cos cos 6 6 Find the volume obtained b rotating the region bounded b the given curves about the specified ais. 6. sin,, ; about the -ais 6. sin,, ; about the -ais 63. sin, cos, ; about 6. sec, cos, 3; about 5. sin 5 sin d tan 8. sec 9. t sec t tan t dt 5. If tan 6 sec I, epress the value of tan 8 sec in terms of I. ; 5 5 Evaluate the indefinite integral. Illustrate, and check that our answer is reasonable, b graphing both the integrand and its antiderivative (taking C. 5. sin 5. sin 3 cos cos sin sin cos 65. A particle moves on a straight line with velocit function vt sin t cos t. Find its position function s f t if f. 66. Household electricit is supplied in the form of alternating current that varies from 55 V to 55 V with a frequenc of 6 ccles per second (Hz). The voltage is thus given b the equation Et 55 sint where t is the time in seconds. Voltmeters read the RS (root-mean-square) voltage, which is the square root of the average value of Et over one ccle. (a) Calculate the RS voltage of household current. (b) an electric stoves require an RS voltage of V. Find the corresponding amplitude A needed for the voltage Et A sint.

5 7 CHAPTER 7 TECHNIQUES OF INTEGRATION N Figure 5 shows the graphs of the integrand in Eample 7 and its indefinite integral (with C ). Which is which? 3 We now substitute u sin, giving du cos d and s u cos, so sin cos d s3 cos _ sin d cos C s u sin u C FIGURE 5 _5 s3 sin C 7.3 EXERCISES 3 Evaluate the integral using the indicated trigonometric substitution. Sketch and label the associated right triangle.. ; s 9. 3 s9 ; 3. 3 ; s 9 3 Evaluate the integral. 3 sec 3 sin 3 tan..6 s s s 7. s s 3. s dt st 6t cos t s sin t dt. s3 3 s s t 3 st dt s 3. (a) Use trigonometric substitution to show that s a ln( s a ) C 7. s5 9.. s s t 5 st dt (b) Use the hperbolic substitution a sinh t to show that s a sinh a C. s sa s 9 3 a s 7 8. s 9.. s 3 s3 du us5 u 5 s9 a b 3 t s5 t dt 3. Evaluate These formulas are connected b Formula a 3 (a) b trigonometric substitution. (b) b the hperbolic substitution a sinh t. 33. Find the average value of f s, Find the area of the region bounded b the hperbola 9 36 and the line 3.

6 SECTION 7. INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS 8 could be evaluated b the method of Case III, it s much easier to observe that if u 3 3 3, then du 3 3 and so 3 3 ln 3 3 C RATIONALIZING SUBSTITUTIONS Some nonrational functions can be changed into rational functions b means of appropriate substitutions. In particular, when an integrand contains an epression of the form s n t, then the substitution u s n t ma be effective. Other instances appear in the eercises. s EXAPLE 9 Evaluate. SOLUTION Let u s. Then u, so u and u du. Therefore s u u u du u du u u du We can evaluate this integral either b factoring u as u u and using partial fractions or b using Formula 6 with a : s du 8 u 8 du u s ln ln u u C s s C 7. EXERCISES 6 Write out the form of the partial fraction decomposition of the function (as in Eample 7). Do not determine the numerical values of the coefficients.. (a) (b) 33. (a) (b) 3. (a) (b) (a) (b) (a) (b) 6. (a) (b) Evaluate the integral t t t t 6 3 r r dr t t dt

7 8 CHAPTER 7 TECHNIQUES OF INTEGRATION.. a 3.. b ake a substitution to epress the integrand as a rational function and then evaluate the integral s. 6 s [Hint: Substitute u s 6.] s s d s 3 s s s a b ds s s s 3 s 5 6 opcional Use integration b parts, together with the techniques of this section, to evaluate the integral. 5. ln 5. tan ; 53. Use a graph of f 3 to decide whether f is positive or negative. Use the graph to give a rough estimate of the value of the integral and then use partial fractions to find the eact value. ; 5. Graph both 3 and an antiderivative on the same screen Evaluate the integral b completing the square and using Formula The German mathematician Karl Weierstrass (85 897) noticed that the substitution t tan will convert an rational function of sin and cos into an ordinar rational function of t. (a) If t tan,, sketch a right triangle or use trigonometric identities to show that cos and sin t s t s t (b) Show that (c) Show that opcional 58 6 Use the substitution in Eercise 57 to transform the integrand into a rational function of t and then evaluate the integral. 58. e e 3e cos sin sin sec t tan t 3 tan t dt e e e 3 5 sin 56. cos t t and sin t t t dt sin cos 7 3 sin cos

8 APPENDIX I ANSWERS TO ODD-NUBERED EXERCISES A93 EXERCISES 6.5 N PAGE e (a) (b), (c) 5. (a) (b) (c)., cos sin cos 3 C 9. ln C. t arctan t 8 ln 6t C 3. t tan t lnsec t C 5. ln ln C 7. 3 e sin 3 3 cos 3 C e ln e 7. 6 ( 6 3s3) 9. sin ln sin C ln 6 5 ln s sin s cos s C ln 3 C 39. e C 7 ƒ F F 59F 9. 6 kgm. 5. L CHAPTER 6 REVIEW N PAGE 6 Eercises ( )(cos 3 ah h 3 3 ) 5. (a) 5 (b) 6 (c) (a).38 (b) Solid obtained b rotating the region cos, about the -ais. Solid obtained b rotating the region, sin about the -ais s3 m J 9. (a) ft-lb (b). ft 3. f PROBLES PLUS N PAGE s 6.5 min. (a) f t 3t (b) f s (b).6 (c).6736 m (d) (i) 5.3 ins (ii) (a) V h f d (c) f skac Advantage: the markings on the container are equall spaced. 3. b a 5. B 6A CHAPTER 7 EXERCISES 7. N PAGE ln 9 3 C r e r C sin 5 5 cos 5 C. 3. (b) cos sin sin C 5. (b) 3, ln 3 3ln 6 ln 6 C e ,.873; e 6. ln e t t t m 65. EXERCISES 7. N PAGE C. 5 cos 5 3 cos 3 C sin 3 sin 5 sin 7 C ssin 5 8 sin 5 sin C 7. cos ln cos C 9. ln sin sin C. tan C 3. tan C tan 5 t 3 tan 3 t tan t C sec 3 sec C 3. sec tan ln sec C tan 6 tan C 35. sec ln sec tan C 37. s ln csc cot C 3 csc 3 5 csc 5 C sin 6 cos 3 6 cos 3 C sin 6 C 7. sin C 9. tan 5 t C 3 _ 5 F f _ 38 7 sin sin C

9 A9 APPENDIX I ANSWERS TO ODD-NUBERED EXERCISES 5. sin cos C s cos 3t3 EXERCISES 7.3 N PAGE 7. s 99 C s 9 C 5. s38 7. s5 5 C 9. ln(s 6 ) C. sin s C 3. 6 sec 3 s 9 C 5. 6a ln. 9 s 7 C (s ) s C 3. sin 3 s5 C 5. s ln(s ) C 7. s ln s C 9. sin s C (s8 sec 7) 37..8, ;.. rsr r r R arcsinrr 3. EXERCISES 7. N PAGE 8 A. (a) (b) 3 B 3 A 3. (a) B C D E 3 A (b) 3 B 3 5. (a) A B C D At B Ct D (b) t t Et F t 7. 6 ln 6 C 9. ln 5 ln C ln ln 9 5 ln 3 (or 9 5 ln 8 3) _π a ln b C 36 ln 5 6 ln tan C ln 3 ln C ln ln 9 3 tan 3 C ln 5 3 tan 3 ln 6 ln tan s3 ln 8 3 F f π _π A C 3 B D ln C C ln (s ) tan (s ) C π 6 ln 3 ln 6 sin 3 8 sin 9 C F _ ln 3 ƒ (s 5 ) 5 Rr C C s3 8 C s tan s ln s s C 3. ln C 5. s 3s 7. ln 6s e 6 ln s 6 C C e 9. ln tan t ln tan t C 5. ( ) ln s7 tan C s7 53. ln 3.55 tan 55. ln ln tan C C 6. ln ln 65. t ln P 9 ln.9p 9 C, where C.3, (a) ,55 3 7,98 8,935 6,5 5 (b) ln ln,9 6,5 ln 5 s9 The CAS omits the absolute value signs and the constant of integration. EXERCISES 7.5 N PAGE 88 sin 3 sin 3 C sin ln csc cot C ln 9 7. e e ln 3 5. ln 5 tan C 3. 8 cos 8 6 cos 6 C (or sin 3 sin 6 8 sin 8 C) 5. s C 7. sin cos sin C (or sin 8 cos C) 9. e e C. arctan s s C ln 5 3 ln C 5 7. ln e C ln 7 3. sin s 33. sin C s3 C ln sec ln sec C. C 3. 3 e 3 C e 3 C 7. ln 3 s ln s ln C s C C 53. m coshm m sinhm 3 coshm C m tan ln sec C 75,77 tan 6,5s9 36 8,55 ln 3 7 C

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