Review Exercises for Chapter 7

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Εἰλείθυια Αργυριάδης
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1 8 Chapter 7 Integration Techniques, L Hôpital s Rule, and Improper Integrals 8. For n, I d b For n >, I n n u n, du n n d, dv (a) d b 6 b 6 (b) (c) n d 5 d b n n b n n n d, v d d 5 5 b d 6. b 6 b 6 n n n n d n I n n n 85. False. is continuous on but f,,, Diverges d b ln b. 87. True Review Eercises for Chapter 7. d d. C C d d ln C 5. ln ln 6 d C 7. 6 d 6 arcsin C 9. e sin d e cos e cos d e cos e sin e sin d 9 e sin d e cos 9 e sin e sin d e sin cos C () dv sin d v cos () dv cos d v sin u e du e d u e du e d

2 Review Eercises for Chapter 7 9. u, du d, dv 5 d, v 5. sin d cos cos d 5 d 5 5 d cos sin sin d C C C 5 5 C () () dv sin d u du d dv cos d cos sin cos C v sin u du d v cos 5. arcsin d arcsin d arcsin 8 d arcsin 8 arcsin C 6 8 arcsin C (by Formula of Integration Tables) dv d v u arcsin du d 7. cos d sin cos d sin sin C sin sin C sin cos C sin cos C 9. sec d tan sec d tan sec d sec d tan tan C tan tan C. sin d sin d d sec sec tan tan sec C cos

3 Chapter 7 Integration Techniques, L Hôpital s Rule, and Improper Integrals. d cos d sin cos csc d θ cot C C sin, d cos d, cos 5. tan d sec d + sec d 8 tan sec d sec θ 8tan sec d 8sec tan sec d 8 sec sec C 8 C C 8 C 8 C 7. d cos cos d cos d θ sin C sin cos C arcsin C arcsin C sin, d cos d, cos

4 Review Eercises for Chapter 7 9. (a) d 8 sin cos d 8cos cos sin d (b) d u du u u C 8 sec sec C u u C tan, 8 C d sec d u, u du d 8 C (c) d d dv d C v u du d 8 C. 8 6 A B 8 A B B5 B 6 5 A5 A d 5 6 d 5 ln 6 ln C. A B C A B C Let : A A Let : A C C Let : 8 5A B C B d d d d d d ln ln arctan C 6 ln ln 6 arctan C

5 Chapter 7 Integration Techniques, L Hôpital s Rule, and Improper Integrals A B 5 5 A 5 B Let : 9 8A A 9 8 Let 5: 5 8B B d d d 8 5 d 9 8 ln 5 8 ln 5 C 7. d 9 ln C 9. sin d sin u du u (Formula ) tan u sec u C (Formula 56) tan sec C. 8 d ln 8 8 d ln 8 ln 8 arctan C 6 arctan 6 C (Formula 5) (Formula ). d sin cos ln tan C u sin cos d5. (Formula 58) dv d v u ln n du nln n d ln n d ln n nln n d 7. sin cos d sin d cos cos d cos 8 sin C 8 sin cos C dv sin d v cos u du d

6 9. d uu u du 5. u u du u u arctan u C sin cos d cos d cos C u cos, du sin d Review Eercises for Chapter 7 cos sin d arctan C y, u, d u du cos lnsin d sin lnsin cos d 55. y 9 9 d ln C sin lnsin sin C (by Formula of Integration Tables) dv cos d v sin u lnsin du cos sin d y ln d ln d ln d ln d d ln ln C d dv d v u ln du d ln d ln ln ln.96 A d u u du u 5 u u, u, d u du 69. s cos d.8 u uu du 6. sin d cos sin 67. By symmetry,, A. y, y, ln 7. d ln e 7. e e 75. y ln ln y lnln Since ln y, y. ln

7 Chapter 7 Integration Techniques, L Hôpital s Rule, and Improper Integrals n n n.9 n n n Let.9 y n n n. ln y n ln.9 n n n Thus, ln y.9 y e.9 6 d b 6 b Converges t 5, 5,e.5t dt.5 t e.5t (a) t : $6,,5.59 (b) t : $,, ln.9 n n n.9n.9n.9.9 n n.9 n and.9 n n n e , e.5.5t,, e.5t 8. Diverges ln d b 9 ln b 85. (a) P <.95 e.9.95 d.58 (b) P5 <.95 e.9.95 d.5 Problem Solving for Chapter 7. (a) d d (b) Let sin u, d cos u du, sin u cos u. n d cos u n cos u du n n! n! 5 d cos n u du n n 6... n 5... nn n n! n! (Wallis s Formula) 6 5

8 Problem Solving for Chapter 7 5. ln c ln c ln 9 c c c ln 9 c c 9 ln c ln 9 c c c ln 9 c c ln 9 c ln 9 c ln c ln 5. sin PB OP PB, cos OB AQ AP The triangles AQR and BPR are similar: AR BR AQ BP OR sin cos cos cos sin cos sin sin BR OR OB OR cos OR sin OR sin OR cos sin sin OR cos cos cos cos OR cos sin cos sin sin sin cos y Q P θ R O B A (, )

9 6 Chapter 7 Integration Techniques, L Hôpital s Rule, and Improper Integrals 7. (a). Area.986 (b) Let (c) tan, d sec d, 9 9 sec. 9 d 9 tan 9 sec sec d Area A tan sec sin cos d d cos cos ln sec tan sin C 9 d ln sec tan sin tan sin 9 d ln ln 9 d ln 5 5 ln 5 sinh sinh tanh u du u tanh u sinh sinh tanh sinh ln 5 tanh ln 5 ln tanhln sech u du 9 sinh u, d cosh u du, 9 9 sinh u 9 9 cosh u 9 sinh u 9 cosh cosh u du u θ + 9 ln 6 9 tanh ln 6 9 ln 5

10 Problem Solving for Chapter y ln, y y Arc length y d d d d ln ln ln ln ln ln ln ln Consider ln d. Let Then u ln, du d d, eu. If were elementary, then eu would be too, which is false. ln d u du Hence, is not elementary. ln d ln d u eu du eu u du.. a b c d a c ac b d ad bc bd a c, b d, a d A B d arctan arctan 8 ln ln d d C D d arctan arctan ln ln 8 8

11 8 Chapter 7 Integration Techniques, L Hôpital s Rule, and Improper Integrals 5. Using a graphing utility, (a) (b) (c) Analytically, (a) (b) (c) cot cot cot cot. cot cot Now, Thus, The form cot cos sin cos sin cos cot cot cot sin cos sin is indeterminant. cos sin sin sin cos. cos cos sin cot cot cot cot csc cot cot csc cos sin cos sin sin sin cos sin sin cos sin cos cos sin sin cos sin cos sin cos. cot cot. sin sin cos

12 Problem Solving for Chapter P P P P c, c, c, c N D 6 P N D P N D P N D P N D Thus,. 9. By parts, b b f g d a f g fg d a b fg d a fg b b a a g f d b a fg d.

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