Chapter 7

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.

19.

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.

13.

15.

17.

19.

21.

23.

25.

27.

29.

31.

33.

35.

37.

39.

41.

43.

45.

47.

49.

51.

53.

55.

57.

59.

61.

63.

65.

67.

69.

71.

73.

75.

77.

79.

81.

83. See the Student Solutions Manual.

85. See the Student Solutions Manual.

87.

89.

Section 7.4

1.

3.

5.

7.

9.

11.

13.

15.

17.

19.

21.

23.

25.

27.

29.

31.

33. See the Student Solutions Manual.

35.

Section 7.5

1. (a)

2. True

3. True

4. False

5.

7.

9.

11.

13.

15.

17.

19.

21.

23.

25.

27.

29.

31.

33.

35.

37.

39.

41.

43.

45.

47.

49.

51.

53.

55.

57.

59.

61.

63.

65.

67.

69.

71.

73.

75.

77.

79.

81.

83.

85.

87.

89. See the Student Solutions Manual.

91. See the Student Solutions Manual.

Section 7.6

1.

2.

3.

5.

7.

9.

11.

13.

15.

17.

19.

21.

23.

25.

27.

29.

31.

33.

35.

Section 7.7

1.

3.

5.

7.

9.

11.

13.

15.

17.

19.

21.

23.

25.

27.

29.

31.

33.

35.

37.

Section 7.8

1.

2.

3.

4.

5.

6.

7.

9.

11.

13.

15.

17.

19.

21.

23.

25.

27.

29.

31.

33.

35.

37.

39.

41.

43.

45.

47.

49.

51.

53.

55.

57.

59.

61.

63.

65.

67.

69.

71.

73.

75.

77.

79.

81.

83.

85.

87.

89.

91.

93.

95.

97.

99.

101.

103.

105.

107.

109.

111.

Review Exercises

1.

3.

5.

7.

9. ln(sin x) − x cot x + C

11.

13. e^t + 2 ln(2 − e^t) + 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