Chapter 7

Printed Page AN-32

Section 7.1

1. True

2. uv − ∫ v du

3. \(\dfrac{1}{2}xe^{2x} - \dfrac{1}{4}e^{2x} +C\)

5. x sin x + cos x + C

7. \(\dfrac{2}{3}x^{{3}/{2}}\ \ln \,x - \dfrac{4}{9}x^{{3}/{2}}+C\)

9.

11. x(ln x)2 − 2x ln x + 2x + C

13. \(-x^2 \,\cos x + 2x \,\sin x + 2 \cos x + C\)

15. \(\dfrac{1}{2} x \cos x \sin x + \dfrac{1}{4} x^2 - \dfrac{1}{4} \sin^2 x + C\)

17. x cosh x − sinh x + C

19.

21. \(\dfrac{x}{2}[\sin (\ln x) - \cos (\ln x)]+C\)

23. x(ln x)3 − 3x(ln x)2 + 6x ln x − 6x + C

25. \(\dfrac{1}{3} x^3 (\ln x)^2 - \dfrac{2}{9} x^3 \ln x + \dfrac{2}{27} x^3 + C\)

27. \(\dfrac{1}{3} x^3 \tan^{-1} x - \dfrac{1}{6} x^2 + \dfrac{1}{6} \ln(1 + x^2) + C\)

29. \(\dfrac{7^x x}{\ln 7} - \dfrac{7^x}{(\ln 7)^2} + C\)

31. \(-\dfrac{e^{\pi} + 1}{2}\)

33. \(\dfrac{2}{27} - \dfrac{50}{27} e^{-6}\)

35. \(9 \,\ln 3 - 4\)

37. \(e-2\)

39. \(\dfrac{9}{2} \ln 3 - 4\)

41. \(\dfrac{e^{\pi} + 1}{2}\)

43. 2π

45. eπ − 2π

47. (a)

(b) \(\dfrac{\pi}{2} + \dfrac{1}{2} \tan^{-1} \left(-\dfrac{3}{2}\right)\)

(c) \(\approx 1.890\)

49.

51. (sin x) ln(sin x) − sin x + C

53. \(\dfrac{1}{2} e^{2x} \sin e^{2x} + \dfrac{1}{2} \cos e^{2x} + C\)

55. \(\dfrac{1}{2} x^2 e^{x^2} - \dfrac{1}{2} e^{x^2} + C \)

57. \(\dfrac{1}{2} e^x (x \sin x + x \cos x - \sin x) + C\)

59. See the Student Solutions Manual.

61. See the Student Solutions Manual.

63. (a) \(\dfrac{1}{5} x^2 e^{5x} - \dfrac{2}{25} x e^{5x} + \dfrac{2}{125} e^{5x} + C\)

(b) \(\int{x^{n} e^{kx}}\,dx = \dfrac{1}{k} x^n e^{kx} - \dfrac{n}{k} \int{x^{n-1} e^{kx}}\,dx\)

65. (a) \(f(x) = \dfrac{1}{2} \sin^2 x\)

(b) \(g(x) = -\dfrac{1}{2} \cos^2 x\)

(c) \(h(x) = -\dfrac{1}{4} \cos(2x)\)

(d) \(C_2 = \dfrac{1}{2} + C_1\)

(e) \(C_3 = \dfrac{1}{4} + C_1\)

67. See the Student Solutions Manual.

69. (a) \(\dfrac{5\pi}{32}\)

(b) \(\dfrac{8}{15}\)

(c) \(\dfrac{35\pi}{256}\)

(d) \(\dfrac{5\pi}{32}\)

71. See the Student Solutions Manual.

73. See the Student Solutions Manual.

75. See the Student Solutions Manual.

Section 7.2

1. True

2. True

3.

5. \(\dfrac{5}{16} x - \dfrac{1}{4} \sin(2x) + \dfrac{3}{64} \sin(4x) + \dfrac{1}{48}\sin^3(2x) + C\)

7. \(\dfrac{x}{2} - \dfrac{\sin (2 \pi x)}{4\pi}+C\)

9. 0

11.

13. \(\dfrac{x}{8}-\dfrac{\sin (4x)}{32}+C\)

15.

17. \(\dfrac{2}{3}\sin^{3}\dfrac{x}{2} - \dfrac{2}{5}\sin^{5}\dfrac{x}{2} + C\)

19. \(\dfrac{1}{4} \tan^{4}x+C\)

21.

23. \(\dfrac{1}{4} \sec^3 x \tan x - \dfrac{1}{8} \sec x \tan x - \dfrac{1}{8} \ln|\sec x + \tan x| + C\)

25. \(\csc x - \dfrac{\csc^3 x}{3}+C\)

27. \(-\dfrac{1}{8}\cos (4x) -\dfrac{1}{4}\cos (2x) + C\)

29. \(\dfrac{1}{8}\sin (4x) + \dfrac{1}{4}\sin (2x) + C\)

31. \(\dfrac{1}{4}\sin (2x) - \dfrac{1}{12}\sin (6x) + C\)

33.

35.

37. \(\sec x + C\)

39. \(\dfrac{1}{3} \sin (3x) - \dfrac{1}{9} \sin^3 (3x) + C\)

41. 0

43. \(\ln|\cos x| + \dfrac{1}{2} \sec^2 x + C\)

45. \(-\dfrac{1}{2} \cot^2 x + 2 \ln|\tan x| + \dfrac{1}{2} \tan^2 x + C\)

47.

49.

51. \(\dfrac{7\sqrt{2}}{48} + \dfrac{\ln\,\big(1+\sqrt{2}\big)}{16}\)

53. \(-\dfrac{1}{4} \cos (2x) + \dfrac{1}{2}\cos x + C\)

55. \(\dfrac{1}{2} \sin x - \dfrac{1}{4} \sin (2x) +C\)

57. \(\dfrac{\pi^2}{2}\)

59. (a) \(\dfrac{2}{5\pi}\)

(b) Answers will vary.

(c)

61. (a) \(\ln(\sqrt{2} + 1) + \sqrt{2} - 2\)

(b) \(\dfrac{4\pi-\pi^2}{2}\)

63. (a) \(\dfrac{3x}{8}-\dfrac{1}{4}\sin (2x) + \dfrac{1}{32}\sin (4x) +C\)

(b)

(c) See the Student Solutions Manual.

(d) same as (a)

65. \(\dfrac{1}{2(m-n)}\sin [(m-n)x] - \dfrac{1}{2(m+n)} \sin [(m+n)x] +C\)

67. \(\dfrac{1}{2(m-n)}\sin [(m-n)x] + \dfrac{1}{2(m+n)} \sin [(m+n)x] +C\)

69. See the Student Solutions Manual.

Section 7.3

1. True

2. (d)

3. (c)

4. (b)

5.

7.

9.

11. \(\dfrac{x(x^2-2)}{4}\sqrt{4-x^2}+2\,\sin^{-1}\dfrac{x}{2}+C\)

13. \(\dfrac{1}{4} \dfrac{x}{\sqrt{4-x^2}}+C\)

15. \(\dfrac{1}{2}x\sqrt{4+x^2} + 2\ln\left|\dfrac{\sqrt{4+x^2}+x}{2}\right|+C\)

17. \(\ln\left| \dfrac{\sqrt{x^2+16}+x}{4}\right|+C\)

19. \(\dfrac{x}{2}\sqrt{1+9x^2}+\dfrac{1}{6}\ln\left| \sqrt{1+9x^2}+3x \right|+C\)

21. \(\dfrac{1}{18}x\sqrt{4+9x^2} - \dfrac{2}{27}\ln \left| \dfrac{\sqrt{4 + 9x^2} + 3x}{2} \right| +C\)

23. \(-\dfrac{\sqrt{x^2+4}}{4x}+C\)

25. \(\dfrac{x}{4\sqrt{x^2+4}}+C\)

27. \(\dfrac{x}{2}\sqrt{x^2-25}+\dfrac{25}{2}\ln \left| \dfrac{x+\sqrt{x^2-25}}{5} \right| + C\)

29. \(\sqrt{x^2-1}-\sec^{-1}x + C\)

31. \(\dfrac{\sqrt{x^2-36}}{36x}+C\)

33. \(\dfrac{1}{2} \ln \left| \dfrac{2x+\sqrt{4x^2-9}}{3} \right| + C\)

35. \(-\dfrac{1}{9} \dfrac{x}{\sqrt{x^2-9}}+C\)

37. \(-\dfrac{x}{\sqrt{x^2-9}}+\ln \left| \dfrac{x+\sqrt{x^2-9}}{3} \right| +C\)

39. \(x-4\,\tan^{-1}\dfrac{x}{4}+C\)

41. \(\dfrac{x}{2}\sqrt{4-25x^2}+\dfrac{2}{5}\sin^{-1}\dfrac{5x}{2}+C\)

43. \(\dfrac{x}{4\sqrt{4-25x^2}}+C\)

45. \(\dfrac{1}{2}x \sqrt{4+25x^2} + \dfrac{2}{5} \ln \left| \dfrac{\sqrt{4+25x^2} + 5x}{2} \right| +C\)

47. \(\dfrac{1}{128}\sec^{-1}\dfrac{x}{4} + \dfrac{\sqrt{x^2-16}}{32x^2} +C\)

49. \(\dfrac{\pi}{4}\)

51. \(\dfrac{\sqrt{2}+\ln\big(1+\sqrt{2}\big)}{2}\)

53. \(10-2\sqrt{7}+9\ln3-\dfrac{9}{2}\ln \big(4+\sqrt{7}\big)\)

55. \(\dfrac{\sqrt{3}}{3} - \dfrac{\pi}{6}\)

57. \(3-\dfrac{3\pi}{4}\)

59. \(\pi ab\)

61. \(\dfrac{1}{2}\sin^{-1}\dfrac{2}{3}\)

63. \(18-\dfrac{22\sqrt{5}}{3}\)

65. \(\dfrac{1}{2}\ln\big(2\sqrt{6}+5\big) - \dfrac{1}{2}\ln\big(2\sqrt{2} + 3\big) - 3\sqrt{2} + 5\sqrt{6} \)

67.

  1. (a) \(4\sin^{-1}\dfrac{\sqrt{15}}{8}+\sin^{-1}\dfrac{\sqrt{15}}{4} - \dfrac{\sqrt{15}}{2}\)
  2. (b) \(\pi - 4\sin^{-1}\dfrac{\sqrt{15}}{8}-\sin^{-1}\dfrac{\sqrt{15}}{4} + \dfrac{\sqrt{15}}{2} \)

69. \(\dfrac{1}{4}\ln\big(\sqrt{26} + 5\big) - \dfrac{1}{4}\ln\big(\sqrt{26}-5\big)+\dfrac{5}{2}\sqrt{26}\)

71. \(\dfrac{\pi}{40}+\dfrac{\pi}{16}\tan^{-1}\dfrac{1}{2}\)

73. \(\sin^{-1}(x-2)+C\)

75. \(\ln\left|x-1+\sqrt{(x-1)^2-4}\right|+C\)

77. \(\dfrac{25}{2}\sin^{-1}\dfrac{e^x}{5} + \dfrac{1}{2}e^x\sqrt{25-e^{2x}}+C\)

79. \(\dfrac{1}{2}x^{2}\sin^{-1}x - \dfrac{1}{4}\sin^{-1}x + \dfrac{1}{4}x\sqrt{1-x^2} + C\)

81.

  1. (a) \(\dfrac{x}{2}\sqrt{x^2+a^2}+\dfrac{a^2}{2}\ln \left|\dfrac{x+\sqrt{x^2+a^2}}{a}\right|+C\)
  2. (b) \(\dfrac{x}{2}\sqrt{x^2+a^2}+\dfrac{a^2}{2}\sinh^{-1}\dfrac{x}{a}+C\)

83. See the Student Solutions Manual.

85. See the Student Solutions Manual.

87. \(\sin^{-1}\dfrac{2x-3}{3}+C\)

89. \(\ln\dfrac{x+\sqrt{x^2+a^2}}{a} + C\)

Section 7.4

1. \(\tan^{-1}(x+2)+C\)

3. \(\dfrac{1}{2}\tan^{-1}\dfrac{x+2}{2}+C\)

5. \(\dfrac{2\sqrt{5}}{5}\tan^{-1}\dfrac{2x+1}{\sqrt{5}}+C\)

7. \(\dfrac{1}{4}\ln(2x^2+2x+3)-\dfrac{\sqrt{5}}{10}\tan^{-1}\dfrac{2x+1}{\sqrt{5}}+C\)

9. \(\sin^{-1}\dfrac{x-1}{3}+C\)

11. \(\sin^{-1} \dfrac{x-2}{2} + C\)

13. \(\ln\left|\dfrac{\sqrt{x^{2}+2x+2}-1}{x+1}\right|+C\)

15. \(\sin^{-1}\dfrac{x+1}{5}+C\)

17. \(\sqrt{x^{2}-2x+5}-4\ln\left|x-1+\sqrt{x^{2}-2x+5}\right|+C\)

19. \(\ln\left(\sqrt{2}+1\right)\)

21. \(\ln\left(e^{x} + \dfrac{1}{2} + \sqrt{e^{2x}+e^{x}+1}\right)+C\)

23. \(-2\sqrt{4x-x^2-3}+\sin^{-1}(x-2)+C\)

25. \(\dfrac{x-1}{9\sqrt{x^2-2x+10}}+C\)

27. \(\ln\left|x+1+\sqrt{x^{2}+2x-3}\right|+C\)

29. \(\sqrt{5+4x-x^2}-3\ln\left|3+\sqrt{5+4x-x^{2}}\right| + 3\ln\left|x-2\right|+C\)

31. \(\sqrt{x^{2}+2x-3}-\ln\left|x+1+\sqrt{x^{2}+2x-3}\right|+C\)

33. See the Student Solutions Manual.

35. \(a\sin^{-1}\dfrac{x}{a}-\sqrt{a^{2}-x^{2}}+C\)

Section 7.5

1. (a)

2. True

3. True

4. False

5. \(\dfrac{x^2}{2}-x+2\ln\left|x+1\right|+C\)

7. \(\dfrac{x^3}{3}+x^2+7x+10\ln\left|x-2\right|+C\)

9. \(\dfrac{1}{3}\ln\left|x-2\right| - \dfrac{1}{3}\ln\left|x+1\right| + C\)

11. \(-\ln\left|x-1\right| + 2\ln\left|x-2\right| + C\)

13. \(\dfrac{2}{21}\ln\left|3x-2\right|+\dfrac{1}{14}\ln\left|2x+1\right|+C\)

15. \(-5\ln\left|x+2\right| + 5\ln\left|x+1\right| + \dfrac{4}{x+1}+C\)

17. \(\dfrac{1}{4}\ln\left|x+1\right| + \dfrac{3}{4}\ln\left|x-1\right| - \dfrac{1}{2(x-1)} + C\)

19. \(\ln\left|x\right| -\dfrac{1}{2}\ln(x^{2}+1)+C\)

21. \(\dfrac{1}{6}\ln(x^2+2x+4)+\dfrac{2}{3}\ln\left|x+1\right|+C\)

23. \(-\dfrac{1}{x^{2}+16} + \dfrac{1}{128}\tan^{-1}\dfrac{x}{4} + \dfrac{x}{32(x^{2}+16)} + C\)

25. \(-\dfrac{1}{2(x^{2}+16)} + \dfrac{4}{(x^{2}+ 16)^{2}} + C\)

27. \(\dfrac{1}{4}\ln\left|x-1\right| + \dfrac{3}{4}\ln\left|x+3\right| + C\)

29. \(\dfrac{14}{3}\ln\left|x-1\right| - \dfrac{4}{x-1} - \dfrac{7}{3}\ln(x^{2}+2) + \dfrac{2\sqrt{2}}{3}\tan^{-1}\dfrac{\sqrt{2}x}{2} + C\)

31. \(-\ln\left|x\right| - \ln\left|x+1\right| + 2\ln\left|x-3\right| + C\)

33. \(4\ln\left|x-2\right| - 3\ln\left|x-1\right| + \dfrac{1}{x-1} + C\)

35. \(\ln\left|x-1\right| - \dfrac{1}{2}\ln(x^{2}+x+1) + \dfrac{\sqrt{3}}{3}\tan^{-1}\dfrac{\sqrt{3}(2x+1)}{3} + C\)

37. \(\dfrac{1}{6}\ln\dfrac{1}{2}\)

39. \(\dfrac{1}{8}\ln21\)

41. \(\dfrac{1}{5}\ln(2-\sin\theta) - \dfrac{1}{5}\ln(\sin\theta + 3) +C\)

43. \(-\ln\left|\cos\theta\right| + \dfrac{1}{2}\ln(\cos^{2}\theta + 1) + C\)

45. \(\dfrac{1}{3}\ln\left|e^{t}-1\right| - \dfrac{1}{3}\ln(e^{t}+2) + C\)

47. \(\dfrac{1}{2}\ln\left|e^{x}-1\right| - \dfrac{1}{2}\ln(e^{x}+1) + C\)

49. \(t-\dfrac{1}{2}\ln(e^{2t}+1)+C\)

51. \(\ln\left|\sin x - 1\right| - \dfrac{1}{\sin x - 1}+C\)

53. \(\dfrac{1}{54}\tan^{-1}\dfrac{\sin x}{3} + \dfrac{1}{18}\dfrac{\sin x}{\sin^{2}x + 9} + C\)

55. \(\ln\dfrac{15}{7}\)

57. \(\dfrac{4\pi\sqrt{3}}{3}+\dfrac{4}{3}\ln3\)

59.

  1. (a) \(-4, -2, 3\)
  2. (b) \((x+4)(x+2)(x-3)\)
  3. (c) \(\dfrac{13}{10}\ln\left|x+2\right| + \dfrac{2}{35}\ln\left|x-3\right| - \dfrac{19}{14}\ln\left|x+4\right|+C\)

61. \(2\sqrt{x}-4\,\ln\big(\sqrt{x}+2\big)+C\)

63. \(\dfrac{3}{5}x^{5/3} + \dfrac{3}{4}x^{4/3}+\dfrac{3}{2}x^{2/3}+x+3x^{1/3} + 3\ln\left|x^{1/3}-1\right|+C\)

65. \(\dfrac{2}{3}x^{1/2} + \dfrac{1}{3}x^{1/3} + \dfrac{2}{9}x^{1/6} + \dfrac{2}{27} \ln\left|3x^{1/6}-1\right|+C\)

67. \(\dfrac{2}{3}(2x+1)^{3/4}+C\)

69. \(3(1+x)^{1/3}+C\)

71. \(-\dfrac{3x^{1/6}}{x^{1/3}+1}+3\tan^{-1}x^{1/6}+C\)

73. \(-\dfrac{2}{1+\tan \dfrac{x}{2}}+C\)

75. \(\dfrac{2\sqrt{5}}{5}\tan^{-1}\left(\dfrac{\sqrt{5}}{5}\tan\dfrac{x}{2}\right)+C\)

77. \(-\ln\left|1-\tan\dfrac{x}{2}\right|+C\)

79. \(\dfrac{1}{2}\ln\left|\tan^{2}\dfrac{x}{2}+2\tan\dfrac{x}{2}-1\right| - \dfrac{1}{2}\ln\left(\tan^{2}\dfrac{x}{2}+1\right) - \dfrac{x}{2} + C\)

81. \(\dfrac{\sqrt{5}}{5}\ln\left|\tan\dfrac{x}{2}-\dfrac{\sqrt{5}}{2}+\dfrac{1}{2}\right| - \dfrac{\sqrt{5}}{2}\ln\left|\tan\dfrac{x}{2}+\dfrac{\sqrt{5}}{2}+\dfrac{1}{2}\right| + C\)

83. \(-\dfrac{1}{2}\ln\left|\tan\dfrac{x}{2}-1\right| + \dfrac{1}{2}\ln\left|\tan\dfrac{x}{2}+1\right| + \dfrac{1}{\tan\dfrac{x}{2}+1} - \dfrac{1}{\left(\tan\dfrac{x}{2}+1\right)^{2}} + C\)

85. \(\dfrac{1}{3}\ln\dfrac{\sqrt{3}}{3} - \dfrac{1}{3}\ln(\sqrt{2}-1) - \dfrac{4}{15}\ln(\sqrt{3}+1) - \dfrac{4}{15}\ln(4-\sqrt{2}) + \dfrac{4}{15}\ln(3\sqrt{2}-2) + \dfrac{4}{15}\ln\left(3-\dfrac{\sqrt{3}}{3}\right)\)

87. \(\dfrac{\pi}{2}+\ln2\)

89. See the Student Solutions Manual.

91. See the Student Solutions Manual.

Section 7.6

1. True

2. True

3. 22 and 20

5. 26 and \(\dfrac{58}{3}\)

7.

  1. (a) \(0.483\)
  2. (b) \(0.007\)
  3. (c) \(26\)

9.

  1. (a) \(0.743\)
  2. (b) \(\dfrac{1}{96}\)
  3. (c) \(41\)

11.

  1. (a) \(0.907\)
  2. (b) \(\dfrac{1}{192}\)
  3. (c) \(29\)

13.

  1. (a) \(3.059\)
  2. (b) \(5.309\times10^{-4}\)
  3. (c) \(8\)

15.

  1. (a) \(0.747\)
  2. (b) \(2.604\times10^{-4}\)
  3. (c) \(6\)

17.

  1. (a) \(0.910\)
  2. (b) \(3.255\times10^{-4}\)
  3. (c) \(6\)

19.

  1. (a) See the Student Solutions Manual.
  2. (b) \(0.6956\)
  3. (c) \(0.6932\)

21.

  1. (a) \(1.910\)
  2. (b) \(1.910\)

23. \(\approx 62.983\) in lb

25. \(16,787.5\) \({\rm m}^{3}\)

27. T: \(131,787.5\) \({\rm m}^3\), S: \(132,625\) \({\rm m}^3\)

29. \(\dfrac{105}{2}\pi\approx 164.934\)

31.

  1. (a) Using the disk method: \(V\approx1.6095\), Using the shell method: \(V\approx1.4709\)
  2. (b) Using the disk method: \(V\approx1.9705\), Using the shell method: \(V\approx1.3228\)

33.

  1. (a) \(1.845\)
  2. (b) \(1.852\)

35. See the Student Solutions Manual.

Section 7.7

1. \(\dfrac{1}{5}e^{2x}(2\cos x + \sin x) + C\)

3. \(\dfrac{1}{20}(2x-1)(3+4x)^{3/2}+C\)

5. \(\dfrac{6x+5}{60}(4x+5)^{3/2}+C\)

7. \(\dfrac{\sqrt{6}}{6}\ln\left|\dfrac{\sqrt{6+4x}-\sqrt{6}}{\sqrt{6+4x}+\sqrt{6}}\right|+C\)

9. \(2\sqrt{6+4x}+\sqrt{6}\ln \left| \dfrac{\sqrt{6+4x}-\sqrt{6}}{\sqrt{6+4x}+\sqrt{6}} \right|+C\)

11. \(\dfrac{x^{4}(\ln x)^{2}}{4} - \dfrac{x^{4}\ln x}{8} + \dfrac{x^{4}}{32} + C\)

13. \(x\sin^{-1}(2x) + \dfrac{1}{2}\sqrt{1-4x^{2}} + C\)

15. \(\dfrac{135}{8}\sin^{-1}\dfrac{1}{3} - \dfrac{9\sqrt{2}}{4}\)

17.

  1. (a) \(\dfrac{e^{2x}}{5}(\sin x + 2\cos x)+C\)
  2. (b) See Student Solutions Manual.
  3. (c) See Student Solutions Manual.

19.

  1. (a) \(\dfrac{2x-1}{20}(4x+3)^{3/2}+C\)
  2. (b) See Student Solutions Manual.
  3. (c) See Student Solutions Manual.

21.

  1. (a) \(\dfrac{6x+5}{60}(4x+5)^{3/2}+C\)
  2. (b) See Student Solutions Manual.
  3. (c) See Student Solutions Manual.

23.

  1. (a) \(-\dfrac{1}{3}\sqrt{6}\tanh^{-1}\left(\dfrac{\sqrt{4x+6}}{6}\sqrt{6}\right)+C\)
  2. (b) See Student Solutions Manual.
  3. (c) See Student Solutions Manual.

25.

  1. (a) \(2\sqrt{4x+6}-2\sqrt{6}\tanh^{-1}\sqrt{\dfrac{2x}{3}+1}+C\)
  2. (b) See Student Solutions Manual.
  3. (c) See Student Solutions Manual.

27.

  1. (a) \(\dfrac{1}{32}x^{4}(8\ln^{2}x - 4\ln x + 1)+C\)
  2. (b) See Student Solutions Manual.
  3. (c) See Student Solutions Manual.

29.

  1. (a) \(\dfrac{1}{2}\sqrt{1-4x^2}+x\sin^{-1}(2x)+C\)
  2. (b) See Student Solutions Manual.
  3. (c) See Student Solutions Manual.

31.

  1. (a) \(-\dfrac{9\sqrt{2}}{4}+\dfrac{135}{8}\sin^{-1}\dfrac{1}{3}\)
  2. (b) See Student Solutions Manual.
  3. (c) See Student Solutions Manual.

33. No

35. No

37. No

Section 7.8

1. (c)

2. (b)

3. False

4. True

5. False

6. \(\lim_{t\rightarrow b^{-}}\int_a^tf(x)dx\)

7. Yes, \(\infty\) is a limit of integration.

9. No

11. Yes, the function is undefined at \(0\).

13. Yes, the function is undefined at \(1\).

15. Converges to \(\dfrac{1}{2}\).

17. Diverges

19. Diverges

21. Converges to \(\dfrac{1}{24}\).

23. Converges to \(\dfrac{\pi}{2}\).

25. Diverges

27. Diverges

29. Converges to \(4\).

31. Converges to 0.

33. Diverges

35. Converges to 1.

37. Diverges

39. Diverges

41. Diverges

43. Diverges

45. Diverges

47. Converges to \(\pi\).

49. Diverges

51. Converges to \(3\sqrt[3]{18}\).

53. Diverges

55. Converges to \(0\).

57. Converges to \(4\).

59. Diverges

61. Converges to \(\dfrac{1}{2}\).

63. Diverges

65. Diverges

67.

  1. (a) Converges
  2. (b) \(\approx 0.673\)

69.

  1. (a) Converges
  2. (b) \(\dfrac{\pi}{2}\)

71. \(\ln 2\)

73. \(\dfrac{\pi}{2}\)

75. \(4\pi a^2\)

77.

  1. (a) $1250
  2. (b) $30,612

79. \(\dfrac{\pi N I}{5r}\left(1-\dfrac{x}{\sqrt{r^2+x^2}}\right)\)

81. \(GmM\)

83. \(1\)

85. \(\dfrac{1}{2}\)

87. See the Student Solutions Manual.

89. See the Student Solutions Manual.

91. See the Student Solutions Manual.

93. See the Student Solutions Manual.

95. See the Student Solutions Manual.

97. \(\dfrac{1}{s^2}\)

99. \(\dfrac{1}{s^{2}+1}\)

101. \(\dfrac{1}{s-a}\)

103. \(2\)

105. \(1\)

107. See the Student Solutions Manual.

109. \(\dfrac{a+b}{2}\)

111. \(\sigma^2=\dfrac{(b-a)^2}{12}\), \(\sigma=\dfrac{b-a}{2\sqrt{3}}\)

Review Exercises

1.

3.

5.

7.

9. ln(sin x) − x cot x + C

11.

13. et + 2 ln(2 − et) + C

15.

17.

19. x tan x + ln | cos x| + C

21. (y − 1) ln(1 − y) − y + C

23.

25.

27.

29.

31.

33.

35. 1

37. See the Student Solutions Manual.

39. Converges to 2e−1 ≈ 0.736

41. Converges to 1

43. See the Student Solutions Manual.

45. Diverges

47. f (x) = x2 sin x

49. (a) 1.910

(b) 1.910

51. 3