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SalasSV_08_02_ex_ans - ANSWERS TO ODD-NUMBERED EXERCISES...

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Unformatted text preview: ANSWERS TO ODD-NUMBERED EXERCISES A-61 CHAPTER 8 SECTION 8.1 1. −e2−x + C 3. 2/π 5. − tan (1 − x) + C √ 2 17. 3 3 tan θ + 1 + C 19. (1/a) ln |a ex − b| + C 25. tan−1 (x + 3) + C 39. (Formula 99) 27. − 1 cos x2 + C 2 7. 1 2 ln 3 1 2 √ 9. − 1 − x2 + C 2 1 2 11. 0 ( 1 [x 2 + 1]) + C x )2 + C 13. e − 23. √ 1 2 e sin 15. π/4c −1 2 21. ln [(x + 1) + 4] − 31. 3/2 1 [ sin 2t 2 tan −1 x +C 37. √ 2 29. tan x − x + C 33. − 1 3 1 ( sin−1 2 35. ln |ln x| + C 43. (Formula 108) 1 3 √ x√ 2 x − 4 − 2 ln |x + x2 − 4| + C 2 √ x2 + 9 45. (Formula 81) − + ln |x + x2 + 9| + C x π 41. (Formula 18) sin3 2t ] + C ln x +C 2x + 3 47. (Formula 11) x4 π 0 ln x 1 − +C 4 16 √ 49. 2 2 51. (a) 0 π sin2 nx dx = 0 π cos 2nx 1 − 2 2 1 n 0 0 dx = sin 2nx 1 x− 2 4n = π 2 π /n (b) 0 sin nx cos nx dx = u du = 0 1 4 u = sin nx, du = 1 2 1 cos nx dx n (c) 1 6 (c) 0 sin nx cos nx dx = tan4 x + 1 2 1 n 0 0 u du = 0. 53. (a) (d) 1 2 tan2 x − ln | sec x| + C tan2k +1 x dx = √ 2, (b) tan4 x − tan2 x + ln | sec x| + C tan6 x − 1 4 tan2 x − ln| sec x| + C 55. (b) A = 1 tan2k x − tan2k −1 x dx 2k √ √ π 2 2+1 B= (c ) ln √ 57. (b) −0. 80, 5. 80 4 2 2−1 (c) 27.60 SECTION 8.2 1. −x e−x − e−x + C 9. 34 e 8 + 1 8 5. 2 − 5 e−1 7. −2x2 (1 − x)1/2 − 8 x(1 − x)3/2 − 3 √ √ 2 11. 2 x + 1 ln (x + 1) − 4 x + 1 + C 13. x(ln x) − 2x ln x + 2x + C +C 17. 1 x (x 15 3. − 1 e−x + C 3 3 16 (1 15 − x)5/2 + C 15. 3x 21. 3x2 x3 6x 6 − + − ln 3 (ln 3)2 (ln 3)3 (ln 3)4 + 1)10 − 1 x (x 55 + 5)15 − 1 (x 240 + 5)16 + C 19. 1 1 −2 2π π π −2 2 π 1 − ln 2 8 4 x [ sin (ln x) − cos (ln x)] + C 2 12 x (x 10 + 1)11 + 1 (x 660 + 1)12 + C 1 2 23. 1x e ( sin x 2 − cos x) + C 31. π + 24 √ 3−2 4 25. ln 2 + 33. 39. 27. 35. x n +1 x n +1 ln x − +C n+1 (n + 1)2 12 x 2 29. − 1 x2 cos x2 + 2 1 4 sin x2 + C sinh 2x − 1 x cosh 2x + 2 sinh 2x + C 37. ln x sin−1 (ln x) + 1 − (ln x)2 + C 41. Set u = ln x, dv = dx and integrate by parts. 45. Integrate by parts twice and solve for 51. (a) 1 (b) x = e2 1 e + , y = −1 4 4 2 43. Set u = ln x, dv = xk dx and integrate by parts. √ π 3−2 eax sin bx dx. + 47. π 49. 12 2 π2 (e + 1) 2 53. x = 1/(e − 1), y = (e + 1)/4 59. V = 4 − 8/π 61. V = 2π (e − 2) (c) x-axis: π (e − 2), y-axis: 55. x = 1 π , y = 1 π 2 8 57. (a) M = (ek − 1)/k 65. area = sinh1 = (b) xM = [(k − 1)ek + 1]/[k (ek − 1)] 63. x = (e2 + 1)/[2(e2 − 1)] e2 − 1 2 e4 + 4 e2 − 1 1 ; x= , y= 67. Let u = xn , dv = eax dx. Then du = nxn−1 dx, v = eax . 2e e+1 8 e(e2 − 1) a 73. ex [x3 − 3x2 + 6x − 6] + C 69. ( 1 x3 − 3 x2 + 3 x − 3 ) e2x + C 2 4 4 8 75. (a) (x2 − 5x + 6) ex + C 71. x[(ln x)3 − 3(ln x)2 + 6 ln x − 6] + C (b) (x3 − 3x2 + 4x − 4) ex + C 77. Let u = f (x), dv = g (x) dx. Then du = f (x) dx, v = g (x), and b a f (x)g (x) dx = [ f (x)g (x)]b − a b f (x)g (x) dx. a Now let u = f (x), dv = g (x) dx and integrate by parts again. The result follows. A-62 79. (a) π (b) 3π (c) 5π ANSWERS TO ODD-NUMBERED EXERCISES 81. (a) π − 2 ∼ 1. 1416 = ∼ (b) π 3 − 2π 2 = 11. 2671 (c) ( 1 π , 0. 31202) 2 (d) (2n + 1)π , n = 0, 1, 2, . . . SECTION 8.3 1. 11. 19. 27. 37. 1 3 1 2 1 2 cos3 x − cos x + C tan2 x + ln | cos x| + C sin4 x + C sin 3x − tan7 x + 1 14 3. π 12 13. 3 π 8 5. − 1 cos5 x + 5 15. 3 64 1 2 1 7 cos7 x + C 1 10 7. 1 4 sin4 x − 17. 1 3 1 6 sin6 x + C 9. (1/π ) tan π x + C cos x − 1 48 cos 5x + C 23. 31. 41. 1 12 tan3 x + C π 3 1 6 21. 5 x 16 − 1 4 sin 2x + 2 7 sin 4x + 2 11 sin3 2x + C √ 3− 25. − 1 csc5 x + 5 tan2 3x + 1 3 1 3 csc3 x + C 33. 2 105π 35. −1/6 1 6 sin 7x + C 29. 39. 1 3 sin7/2 x − sin11/2 x + C tan4 3x − ln| sec 3x| + C 1 7 1 5 tan5 x + C 45. π/2 cos ( 3 x) − 2 1 5 cos ( 5 x) + C 2 π2 −π 2 1 4 π + ln 2 4 43. √ π 3 − 2 6 47. 3π 2 8 49. 51. π 1 − 53. sin mx sin nx = 1 [ cos (m − n)x − cos (m + n)x], m = n 2 1 − cos 2mx , m=n 2 57. Let u = cosn−1 x, dv = cos x dx. Then du = (n − 1) cosn−2 x(− sin x) dx, v = sin x. sin mx sin nx = sin2 mx = cosn x dx = cosn−1 x cos x dx = cosn−1 x sin x + (n − 1) = cosn−1 x sin x + (n − 1) Now solve for 59. 63. 16 35 cosn−2 x sin2 x dx ( cosn−2 x − cosn x) dx cosn x dx. cotn x dx = cotn−2 x ( csc2 x − 1) dx = − cotn−1 x − n−1 cotn−2 x dx 61. cscn x dx = 1 2 cscn−2 x csc2 x dx. Now let u = cscn−2 x, dv = csc2 x dx and use integration by parts. 67. (a) A = 1 π 2 ∼ 4. 9348 = 2 65. (a) sin2 x + C (b) − 1 cos2 x + C 2 (c) − 1 cos 2x + C 4 (d) The results differ by a constant. SECTION 8.4 x 1. sin−1 +C a 3. 1 x 2 √ x2 − 1 − 1 2 ln |x + √ x 2 − 1| + C 5. 2 sin−1 x 1 − x 4 − x2 + C 2 2 √ x +C 15. ln ( 8 + x2 + x) − √ 8 + x2 23. 1 10 √ 1 2 3−π 625π 7. √ +C 9. 13. 11. − 1 (4 − x2 )3/2 + C 3 6 16 1 − x2 √ √ 1 a − a2 − x2 1 17. ln +C 19. 18 − 9 2 21. − 2 a2 + x2 + C a x ax √ 27. 1 e−x e2x − 9 + C 9 − 29. 1 + C, x > 2 2(x − 2)2 1 + C, x < 2 2(2 − x)2 +C 25. 1 a2 x x 2 − a2 + C 31. − 1 (6x − x2 − 8)3/2 + 3 3 2 √ sin−1 (x − 3) + 3 (x − 3) 6x − x2 − 8 + C 2 33. 8(x2 x2 + x 1 tan−1 − 16 + 2x + 5) x+1 2 35. Let u = sec−1 x, dv = dx and integrate by parts. ...
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