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Calculus

# Calculus - Chapter 4 Integration Chapter 4 Section 4.1 3 x5...

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Chapter 4 Integration 124 Chapter 4 Section 4.1 5. 5 4 3 3 5 x xd x c =+ 7. () 44 52 33 3 3 xx d x x d d x c −= =−+ ∫∫ 9. 3/2 32 xdx x c 11. 3 4 4 1 3 3 x dx dx x dx x c x  − = + +   13. 1/3 2/3 3 3 3 9 2 x dx x dx x dx x c −− =− + 15. 2 sin cos 2 sin cos 2 cos sin d x x d x x d x c += + + + 17. 2 sec tan 2 sec d x x c 19. 2 5 sec 5 tan x c 21. 3 2 32 x ed x e d x d x e x c + 23. 11 3 cos 3 cos 3 sin ln x dx x dx dx c + 25. 2 2 4 2 ln 4 4 x dx x c x + + 27. 2 3 55 3 5 3 2 x x x x x d x e d x e x ec =+ + 29. 5 5 sin 2 cos 2 2 + 31. 3 x x ex d x c + 33. 3 3 sec 2 tan 2 sec 2 2 d x x c 35. ln 3 3 x x x e dx e c e + + 37. 3 3 3 x x x x e dx dx e dx e xe c + + 39. 1/4 5/4 5/2 21 6 x x dx x dx x dx c + 41. 3 4:N / A x + 43. 2 2 2 1 34 x dx dx x dx x c + 45. 2 sec : N/A 47. 1 2 sin 4 cos 4 2 + 49. 2 :N /A x x 51. 2 1 1 dx x dx dx x x c x = + 53. Example 1.12b. ln sec tan c ++ Example 1.12f. sin 2 cos2 42 x c −+ (The CAS does not show the absolute value bars or the constant c . 55. 2 () 4 1 , ( 0 ) 2 fx x f = ± 23 3 3 4 4 1 3 4 (0) 0 0 2 3 2 4 2 3 x d x x x c fc c x x = + =⋅ −+= = +

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Chapter 4 Integration 125 57. () 3 , ( 0 ) 4 x fx e x f =+ = ± () 2 2 0 2 3 3 2 0 (0) 3 4 2 1 1 2 xx x x e xd x e c fe c c x e = + + + = = + 59. () 1 2 0 ) 2 0 ) 3 f f == = ²± 1 11 2 2 2 22 2 ( ) 12 12 (0) 12 0 2, so 2 and ( ) 12 2 1 2 2 6 2 (0) 6 0 2 0 3, so 3 () 6 2 3 d x xc fc c x x d x x x c c x x + =⋅+= = = + + =⋅ +⋅+ = = + ± ± 61. 2 s i n 4 x x ² 23 1 3 1 4 12 4 3 s i n 4 3 c o s 3 4 3c o s 3 1 3sin 3 x x d x x x x c d x c x c = + +  =− + +   + + + ± 63. 3 3 2 () 4 4 2 x x ′′′ 32 1 2 1 21 2 1 4 4 2 1 2 2 ln x d x xx c cd x c x c x cx c d x x c x c x c −− = ++ + + + + + + + + ² ± 65. 2 2 2 () 3 12, (0 3 12 3 6 (0) 3 0 6 0 3 3 6 3 vt t s st vtd t t d t t t c sc c t t = = + =⋅−⋅ += = =− + ∫∫ 67. ( ) 3 sin 1, (0) 0, (0) 4 at t v s =+== 1 3 sin 1 3 cos atd t td t ttc = + + 1 (0) 3 cos 0 0 0 3 3cos 3 vc c t t + + = = 2 2 3cos 3 1 3 2 t tt d t tt t c = + + + 2 2 2 2 1 (0) 3 sin 0 0 3 0 4 2 4 1 3s in 3 4 2 c t t t + + ⋅ + = = + + + 69. 1 hr 1 (0) 30 mph 30 miles/sec 3600 sec 120 v = 2 72 120 1 1 hr 1 (4) 50 mph 50 miles/sec 3600 sec 72 (4) (0) 1 = miles/sec 4-0 4 720 1 720 720 v vv a t t d t c = = + 1 1 2 2 10 (0) 120 720 1 120 1 720 120 1 720 120 1440 120 c t t t dt c ==+ = = + 2 2 2 (0) 0, so 0 1440 120 44 1 4 4 1 2 (4) 1440 120 90 120 360 2 miles 45 s + = + = =
Chapter 4 Integration 126 71. y 1 x 0 0.5 73. All functions that have the derivative shown in Exercise 71 are vertical translations of the graph given as the answer for Exercise 71. 75. 2 ( ) 16 100 100,000 0 75.99 76 sec (76) 32(76) 100 2532 ft/s yt t t t v =− + = ≈≈ 77. Use derivative formulas that you know. Section 4.2 3. 50 2 1 50(50 1)(2 50 1) 42,925 6 i i = +⋅ + == 5. 10 1 1234567891 0 22.47 i i = =+ ++ ++ + ++ + 7. 6 2222222 1 33 1 3 2 3 3 3 4 3 5 3 6 31 22 7 4 87 51 0 8 273 i i = =⋅ +⋅ +⋅ +⋅ =+ + + + + = 9. 7 22222 2 3 ( ) (3 3) (4 4) (5 5) (6 6) (7 7) 12 20 30 42 56 160 i ii = += ++ + + + =++++ = 11. 7 0 (3 1) (3 0 1) (3 1 1) (3 2 1) (3 3 1) (3 4 1) (3 5 1) (3 6 1) (3 7 1) 1 2 5 8 11 14 17 20 76 i i = −=⋅−+⋅−+⋅−+⋅−+⋅−+⋅−+⋅−+⋅− =− + + + + + + + = 13.

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Calculus - Chapter 4 Integration Chapter 4 Section 4.1 3 x5...

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