6.E: Applications of Antidifferentiation (Exercises)

6.1: Substitution

Terms and Concepts

1. Substitution "undoes" what derivative Rule?

2. T/F: One can use algebra to rewrite the integrand of an integral to make it easier to evaluate.

Problems

In Exercises 3-14, evaluate the indefinite integrand to develop an understanding of Substitution.

3. \(\int 3x^2 (x^3-5)^7\,dx\)

4. \(\int (2x-5)(x^2-5x+7)^3\,dx\)

5. \(\int x(x^2+1)^8\,dx\)

6. \(\int (12x+14)(3x^2+7x+7)^3\,dx\)

7. \(\int \frac{1}{2x+7}\,dx\)

8. \(\int \frac{1}{\sqrt{2x+3}}\,dx\)

9. \(\int \frac{x}{\sqrt{x+3}}\,dx\)

10. \(\int \frac{x^3-x}{\sqrt{x}}\,dx\)

11. \(\int \frac{e^{\sqrt{x}}}{\sqrt{x}}\,dx\)

12. \(\int \frac{x^4}{\sqrt{x^5+1}}\,dx\)

13. \(\int \frac{\frac{1}{x}+1}{x^2}\,dx\)

14. \(\int \frac{\ln (x)}{x}\,dx\)

In Exercises 15-23, use Substitution to evaluate the indefinite integral involving trigonometric functions.

15. \(\int \sin^2 (x) \cos (x)\,dx\)

16. \(\int \cos (3-6x)\,dx\)

17. \(\int \sec^2 (4-x)\,dx\)

18. \(\int \sec (2x)\,dx\)

19. \(\int \tan^2 (x)\sec^2 (x)\,dx\)

20. \(\int x \cos (x^2)\,dx\)

21. \(\int \tan^2 (x)\,dx\)

22. \(\int \cot x\,dx\). Do not just refer to Theorem 45 for the answer; justify it through Substitution.

23. \(\int \csc x\,dx\). Do not just refer to Theorem 45 for the answer; justify it through Substitution.

In Exercises 24-30, use Substitution to evaluate the indefinite integral involving exponential functions.

24. \(\int e^{3x-1}\,dx\)

25. \(\int e^{x^3}x^2\,dx\)

26. \(\int e^{x^2-2x+1}(x-1)\,dx\)

27. \(\int \frac{e^x+1}{e^x}\,dx\)

28. \(\int \frac{e^x-e^{-x}}{e^{2x}}\,dx\)

29. \(\int 3^{3x}\,dx\)

30. \(\int 4^{2x}\,dx\)

In Exercises 31-34, use Substitution to evaluate the indefinite integral involving logarithmic functions.

31. \(\int \frac{\ln x}{x}\,dx\)

32. \(\int \frac{(\ln x)^2}{x}\,dx\)

33. \(\int \frac{(\ln x)^3}{x}\,dx\)

34. \(\int \frac{1}{x\ln (x^2)}\,dx\)

In Exercises 35-40, use Substitution to evaluate the indefinite integral involving rational functions.

35. \(\int \frac{x^2+3x+1}{x}\,dx\)

36. \(\int \frac{x^3+x^2+x+1}{x}\,dx\)

37. \(\int \frac{x^3-1}{x+1}\,dx\)

38. \(\int \frac{x^2+2x-5}{x-3}\,dx\)

39. \(\int \frac{3x^2-5x+7}{x+1}\,dx\)

40. \(\int \frac{x^2+2x+1}{x^3+3x^2+3x}\,dx\)

In Exercises 41-50, use Substitution to evaluate the indefinite integral inverse trigonometric functions.

41. \(\int \frac{7}{x^2+7}\,dx\)

42. \(\int \frac{3}{\sqrt{9-x^2}}\,dx\)

43. \(\int \frac{14}{\sqrt{5-x^2}}\,dx\)

44. \(\int \frac{2}{x\sqrt{x^2-9}}\,dx\)

45. \(\int \frac{5}{\sqrt{x^4-16x^2}}\,dx\)

46. \(\int \frac{x}{\sqrt{1-x^4}}\,dx\)

47. \(\int \frac{1}{x^2-2x+8}\,dx\)

48. \(\int \frac{2}{\sqrt{-x^2+6x+7}}\,dx\)

49. \(\int \frac{3}{\sqrt{-x^2+8x+9}}\,dx\)

50. \(\int \frac{5}{x^2+6x+34}\,dx\)

In Exercises 51-75, evaluate the indefinite integral.

51. \(\int \frac{x^2}{(x^3+3)^2}\,dx\)

52. \(\int (3x^2+2x)(5x^3+5x^2+2)^8\,dx\)

53. \(\int \frac{x}{\sqrt{1-x^2}}\,dx\)

54. \(\int x^2 \csc^2 (x^3+1)\,dx\)

55. \(\int \sin (x) \sqrt{\cos (x)}\,dx\)

56. \(\int \frac{1}{x-5}\,dx\)

57. \(\int \frac{7}{3x+2}\,dx\)

58. \(\int \frac{3x^3+4x^2+2x-22}{x^2+3x+5}\,dx\)

59. \(\int \frac{2x+7}{x^2+7x+3}\,dx\)

60. \(\int \frac{9(2x+3)}{3x^2+9x+7}\,dx\)

61. \(\int \frac{-x^3+14x^2-46x-7}{x^2-7x+1}\,dx\)

62. \(\int \frac{x}{x^2+81}\,dx\)

63. \(\int \frac{2}{4x^2+1}\,dx\)

64. \(\int \frac{1}{x\sqrt{4x^2-1}}\,dx\)

65. \(\int \frac{1}{\sqrt{16-9x^2}}\,dx\)

66. \(\int \frac{3x-2}{x^2-2x+10}\,dx\)

67. \(\int \frac{7-2x}{x^2+12x+61}\,dx\)

68. \(\int \frac{x^2+5x-2}{x^2-10x+32}\,dx\)

69. \(\int \frac{x^3}{x^2+9}\,dx\)

70. \(\int \frac{x^3-x}{x^2+4x+9}\,dx\)

71. \(\int \frac{\sin (x)}{\cos^2 (x)+1}\,dx\)

72. \(\int \frac{\cos (x)}{\sin^2 (x)+1}\,dx\)

73. \(\int \frac{\cos (x)}{1-\sin^2 (x)}\,dx\)

74. \(\int \frac{3x-3}{\sqrt{x^2-2x-6}}\,dx\)

75. \(\int \frac{x-3}{\sqrt{x^2-6x+8}}\,dx\)

In Exercises 76-83, evaluate the definite integral.

76. \(\int_1^3 \frac{1}{x-5}\,dx\)

77. \(\int_2^6 x\sqrt{x-2}\,dx\)

78. \(\int_{-\pi/2}^{\pi/2} \sin^2 (x)\cos (x)\,dx\)

79. \(\int_0^1 2x (1-x^2)^4\,dx\)

80. \(\int_{-2}^{-1} (x+1)e^{x^2+2x+1}\,dx\)

81. \(\int_{-1}^1 \frac{1}{x+x^2}\,dx\)

82. \(\int_2^4 \frac{1}{x^2-6x+10}\,dx\)

83. \(\int_1^{\sqrt{3}} \frac{1}{\sqrt{4-x^2}}\,dx\)

6.2: Integration by Parts

Terms and Concepts

1. T/F: Integration by Parts is useful in evaluating integrands that contain products of function.

2. T/F: Integration by Parts can be thought of as the "opposite of the Chain Rule."

3. For what is "LIATE" useful?

Problems

In Exercises 4-33, evaluate the given indefinite integral.

4. \(\int x\sin x\,dx\)

5. \(\int xe^{-x}\,dx\)

6. \(\int x^2\sin x\,dx\)

7. \(\int x^3\sin x\,dx\)

8. \(\int xe^{x^2}\,dx\)

9. \(\int x^3e^x\,dx\)

10. \(\int xe^{-2x}\,dx\)

11. \(\int e^x \sin x\,dx\)

12. \(\int e^{2x}\cos x\,dx\)

13. \(\int e^{2x}\sin (3x)\,dx\)

14. \(\int e^{5x}\cos (5x)\,dx\)

15. \(\int \sin x \cos x\,dx\)

16. \(\int \sin^{-1} x\,dx\)

17. \(\int \tan^{-1} (2x)\,dx\)

18. \(\int x\tan^{-1} x\,dx\)

19. \(\int \sin^{-1} x\,dx\)

20. \(\int x\ln x\,dx\)

21. \(\int (x-2)\ln x\,dx\)

22. \(\int x\ln (x-1)\,dx\)

23. \(\int x\ln (x^2)\,dx\)

24. \(\int x^2 \ln x\,dx\)

25. \(\int (\ln x)^2\,dx\)

26. \(\int (\ln (x+1))^2\,dx\)

27. \(\int x\sec^2 x\,dx\)

28. \(\int x\csc^2 x\,dx\)

29. \(\int x\sqrt{x-2}\,dx\)

30. \(\int x\sqrt{x^2-2}\,dx\)

31. \(\int \sec x \tan x\,dx\)

32. \(\int x\sec x \tan x\,dx\)

33. \(\int x\csc x \cot x\,dx\)

In Exercises 34-38, evaluate the indefinite integral after first making a substitution.

34. \(\int \sin (\ln x)\,dx\)

35. \(\int \sin (\sqrt{x})\,dx\)

36. \(\int \ln (\sqrt{x})\,dx\)

37. \(\int e^{\sqrt{x}}\,dx\)

38. \(\int e^{\ln x}\,dx\)

In Exercises 39-47, evaluate the definite integral. Note: the corresponding indefinite integrals appear in Exercises 4-12.

39. \(\int_0^{\pi} x\sin x\,dx\)

40. \(\int_{-1}^1 xe^{-x}\,dx\)

41. \(\int_{-\pi/4}{^\pi/4} x^2\sin x\,dx\)

42. \(\int_{-\pi/2}^{\pi/2} x^3\sin x\,dx\)

43. \(\int_0^{\sqrt{\ln 2}} xe^{x^2}\,dx\)

44. \(\int_0^1 x^3e^x\,dx\)

45. \(\int_1^2 xe^{-2x}\,dx\)

46. \(\int_0^{\pi} e^x \sin x\,dx\)

47. \(\int_{-\pi/2}^{\pi/2} e^{2x}\cos x\,dx\)

6.3: Trigonometric Integrals

Terms and Concepts

1. T/F: \(\int \sin^2 (x) \cos^2 x \,dx\) cannot be evaluate using the techniques described in this section since both powers of \(\sin x\) and \(\cos x\) are even.

2. T/F: \(\sin^3 x \cos^3 x \,dx\) cannot be evaluated using the techniques described in this section since both powers of \(\sin x\text{ and }\cos x\) are odd.

3. T/F: This section addresses how to evaluate indefinite integrals such as \(\int \sin^5 x \tan^3 x\,dx\).

Problems

In Exercises 4-26, evaluate the indefinite integral.

4. \(\int \sin x \cos^4 x\,dx\)

5. \(\int \sin^3 x \cos x\,dx\)

6. \(\int \sin^3 x \cos^2 x\,dx\)

7. \(\int \sin^3 x \cos^3 x\,dx\)

8. \(\int \sin^6 x \cos^5 x\,dx\)

9. \(\int \sin^2 x \cos^7 x\,dx\)

10. \(\int \sin^2 x \cos^2 x\,dx\)

11. \(\int \sin (5x) \cos (3x)\,dx\)

12. \(\int \sin (x) \cos (2x)\,dx\)

13. \(\int \sin (3x) \sin (7x)\,dx\)

14. \(\int \sin (\pi x) \sin (2\pi x)\,dx\)

15. \(\int \cos (x) \cos (2x)\,dx\)

16. \(\int \cos \left (\frac{\pi}{2}x\right ) \cos (\pi x)\,dx\)

17. \(\int \tan^4 x \sec^2 x\,dx\)

18. \(\int \tan^2 x \sec^4 x\,dx\)

19. \(\int \tan^3 x \sec^4 x\,dx\)

20. \(\int \tan^3 x \sec^2 x\,dx\)

21. \(\int \tan^3 x \sec^3 x\,dx\)

22. \(\int \tan^5 x \sec^5 x\,dx\)

23. \(\int \tan^4 (x)\,dx\)

24. \(\int \sec^5 x\,dx\)

25. \(\int \tan^2 x \sec x\,dx\)

26. \(\int \tan^2 x \sec^3 x\,dx\)

In Exercises 27-33, evaluate the definite integral. Note: the corresponding indefinite integrals appear in the previous set.

27. \(\int_{0}^{\pi}\sin x \cos^4 x \,dx\)

28. \(\int_{-\pi}^{\pi}\sin^3 x \cos x \,dx\)

29. \(\int_{-\pi/2}^{\pi/2} \sin^2 x \cos^7 x \,dx\)

30. \(\int_{0}^{\pi/2} \sin (5x) \cos (3x) \,dx\)

31. \(\int_{-\pi/2}^{\pi/2} \cos (x) \cos (2x) \,dx\)

32. \(\int_{0}^{\pi/4} \tan^4 x \sec^2 x \,dx\)

33. \(\int_{-\pi/4}^{\pi/4} \tan^2 x \sec^4 x \,dx\)

6.4: Trigonometric Substitution

Terms and Concepts

1. Trigonometric Substitution works on the same principles as Integration by Substitution, though it can feel "_____".

2. If one uses Trigonometric Substitution on an integrand containing \(\sqrt{25-x^2}\), then one should set x = ______.

3. Consider the Pythagorean Identity \(\sin^2 \theta +\cos^2 \theta =1\).
(a) What identity is obtained when both sides are divided by \(\cos^2 \theta\)?
(b) Use the new identity to simplify \(9\tan^2 \theta +9\).

4. Why does Key Idea 13(a) state that \(\sqrt{a^2-x^2} = a\cos \theta\), and not \(|a \cos \theta |\)?

Problems

In Exercises 5-16, apply Trigonometric Substitution to evaluate the indefinite integrals.

5. \(\int \sqrt{x^2+1}\,dx\)

6. \(\int \sqrt{x^2+4}\,dx\)

7. \(\int \sqrt{1-x^2}\,dx\)

8. \(\int \sqrt{9-x^2}\,dx\)

9. \(\int \sqrt{x^2-1}\,dx\)

10. \(\int \sqrt{x^2-16}\,dx\)

11. \(\int \sqrt{4x^2+1}\,dx\)

12. \(\int \sqrt{1-9x^2}\,dx\)

13. \(\int \sqrt{16x^2-1}\,dx\)

14. \(\int \frac{3}{\sqrt{x^2+2}}\,dx\)

15. \(\int \frac{3}{\sqrt{7-x^2}}\,dx\)

16. \(\int \frac{5}{\sqrt{x^2-8}}\,dx\)

In Exercises 17-26, evaluate the indefinite integrals. Some may be evaluated without Trigonometric Substitution.

17. \(\int \frac{\sqrt{x^2-11}}{x}\,dx\)

18. \(\int \frac{1}{(x^2+1)^2}\,dx\)

19. \(\int \frac{x}{\sqrt{x^2-3}}\,dx\)

20. \(\int x^2 \sqrt{1-x^2}\,dx\)

21. \(\int \frac{x}{(x^2+0)^{3/2}}\,dx\)

22. \(\int \frac{5x^2}{\sqrt{x^2-10}}\,dx\)

23. \(\int \frac{1}{(x^2+4x+13)^2}\,dx\)

24. \(\int x^2(1-x^2)^{-3/2}\,dx\)

25. \(\int \frac{\sqrt{5-x^2}}{7x^2}\,dx\)

26. \(\int \frac{x^2}{\sqrt{x^2+3}}\,dx\)

In Exercises 27-32, evaluate the definite integrals by making the proper trigonometric substitution and changing the bounds of integration. (Note: each of the corresponding indefinite integrals has appeared previously in the Exercise set.)

27. \(\int_{-1}^{1}\sqrt{1-x^2} \,dx\)

28. \(\int_{4}^{8}\sqrt{x^2-16} \,dx\)

29. \(\int_{0}^{2}\sqrt{x^2+4} \,dx\)

30. \(\int_{-1}^{1} \frac{1}{(x^2+1)^2} \,dx\)

31. \(\int_{-1}^{1} \sqrt{9x^2} \,dx\)

32. \(\int_{-1}^{1}x^2\sqrt{1-x^2} \,dx\)

6.5 Partial Fraction Decomposition

Terms and Concepts

1. Fill in the blank: Partial Fraction Decomposition is a method of rewriting _____ functions.

2. T/F: It is sometimes necessary to use polynomial division before using Partial Fraction Decomposition.

3. Decompose \(\frac{1}{x^2-3x}\) without solving for the coefficients, as done in Example 181.

4. Decompose \(\frac{7-x}{x^2-9}\) without solving for the coefficients, as done in Example 181.

5. Decompose \(\frac{x-3}{x^2-7}\) without solving for the coefficients, as done in Example 181.

6. Decompose \(\frac{2x+5}{x^3+7x}\) without solving for the coefficients, as done in Example 181.

Problems

In Exercises 7-25, evaluate the indefinite integral.

7. \(\int \frac{7x+7}{x^2+3x-10}\,dx\)

8. \(\int \frac{7x-2}{x^2+x}\,dx\)

9. \(\int \frac{-4}{3x^2-12}\,dx\)

10. \(\int \frac{x+7}{(x+5)^2}\,dx\)

11. \(\int \frac{-3x-20}{(x+8)^2}\,dx\)

12. \(\int \frac{9x^2+11x+7}{x(x+1)^2}\,dx\)

13. \(\int \frac{-12x^2-x+33}{(x-1)(x+3)(3-2x)}\,dx\)

14. \(\int \frac{94x^2-10x}{(7x+3)(5x-1)(3x-1)}\,dx\)

15. \(\int \frac{x^2+2+1}{x^2+x-2}\,dx\)

16. \(\int \frac{x^3}{x^2-2x-20}\,dx\)

17. \(\int \frac{2x^2-4x+6}{x^2-2x+3}\,dx\)

18. \(\int \frac{1}{x^2+3x^2+3x}\,dx\)

19. \(\int \frac{x^2+x+5}{x^2+4x+10}\,dx\)

20. \(\int \frac{12x^2+21x+3}{(x+1)(3x^2+5x-1)}\,dx\)

21. \(\int \frac{6x^2+8x-4}{(x-3)(x^2+6x+10)}\,dx\)

22. \(\int \frac{2x^2+x+1}{(x+1)(x^2+9)}\,dx\)

23. \(\int \frac{x^2-20x-69}{(x-7)(x^2+2x+17)}\,dx\)

24. \(\int \frac{9x^2-60x+33}{(x-9)(x^2-2x+11)}\,dx\)

25. \(\int \frac{6x^2+45x+121}{(x+2)(x^2+10x+27)}\,dx\)

In Exercises 26-29, evaluate the definite integral.

26. \(\int_{1}^{2} \frac{8x+21}{(x+2)(x+3)} \,dx\)

27. \(\int_{0}^{5} \frac{14x+6}{(3x+2)(x+4)} \,dx\)

28. \(\int_{-1}^{1} \frac{x^2+5x-5}{(x-10)(x^2+4x+5)} \,dx\)

29. \(\int_{0}^{1} \frac{x}{(x+1)(x^2+2x+1)} \,dx\)

6.6: Hyperbolic Functions

Terms and Concepts

1. In Key Idea 16, the equation \(\int \tanh x\,dx = \ln (\cosh x)+C\) is given. Why is "\(\ln |\cosh x|\)" not used -i.e., why are absolute values no necessary?

2. The hyperbolic functions are used to define points on the right hand portion of the hyperbola \(x^2-y^2=1\), as shown in Figure 6.13. How can we use the hyperbolic functions to define points on the left hand portion of the hyperbola?

Problems

In Exercises 3-10, verify the given identity using Definition 23, as done in Example 186.

3. \(\coth^2 x-\text{csch }^2 x=1\)

4. \(\cosh 2x = \cosh^2 x+\sinh^2 x\)

5. \(\cosh^2 x = \frac{\cosh 2x+1}{2}\)

6. \(\sinh^2 x = \frac{\cosh 2x-1}{2}\)

7. \(\frac{d}{dx} [\text{sech } x] = -\text{sech } x \tanh x\)

8. \(\frac{d}{dx} [\coth x] = -\text{sech } x \tanh x\)

9. \(\int \tanh x\,dx = \ln (\cosh x)+C\)

10. \(\int \coth x\,dx = \ln |\sinh x|+C\)

In Exercises 11-21, find the derivative of the given function.

11. \(f(x) = \cosh 2x\)

12. \(f(x) = \tanh (x^2)\)

13. \(f(x) = \ln (\sinh x)\)

14. \(f(x) = \sinh x\cosh x\)

15. \(f(x) = x\sinh x -\cosh x\)

16. \(f(x) = \text{sech }^{-1}(x^2)\)

17. \(f(x) = \sinh^{-1}(3x)\)

18. \(f(x) = \cosh^{-1}(2x^2)\)

19. \(f(x) = \tanh^{-1}(x+5)\)

20. \(f(x) = \tanh^{-1} (\cos x)\)

21. \(f(x) = \cosh^{-1} (\sec x)\)

In Exercises 22-26, find the equation of the line tangent to the function at the given x-value.

22. \(f(x) = \sinh x\text{ at }x=0\)

23. \(f(x) = \cosh x\text{ at }x=\ln 2\)

24. \(f(x) = \text{sech }^2 x\text{ at }x=\ln3\)

25. \(f(x) = \sinh^{-1} x\text{ at }x=0\)

26. \(f(x) = \cosh^{-1} x\text{ at }x=\sqrt{2}\)

In Exercises 27-40, evaluate the given indefinite integral.

27. \(\int \tanh (2x)\,dx\)

28. \(\int \cosh (3x-7) \,dx\)

29. \(\int \sinh x \cosh x\,dx\)

30. \(\int x\cosh x \,dx\)

31. \(\int x\sinh x\,dx\)

32. \(\int \frac{1}{9-x^2}\,dx\)

33. \(\int \frac{2x}{\sqrt{x^4-4}}\,dx\)

34. \(\int \frac{\sqrt{x}}{\sqrt{1+x^3}}\,dx\)

35. \(\int \frac{1}{x^2-16}\,dx\)

36. \(\int \frac{1}{x^2+x}\,dx\)

37. \(\int \frac{e^x}{x^{2x}+1}\,dx\)

38. \(\int \sinh^{-1} x\,dx\)

39. \(\int \tanh^{-1}x\,dx\)

40. \(\int \text{sech } x\,dx\) (Hint: multiply by \(\frac{\cosh x}{\cosh x}\); set \(u=\sinh x\).)

In Exercises 41-43, evaluate the given definite integral.

41. \(\int_{-1}^{1}\sinh x\,dx\)

42. \(\int_{-\ln 2}^{\ln 2}\cosh x\,dx\)

43. \(\int_{0}^1 \tanh^{-1}x\,dx\).

6.7: L'Hopital's Rule

Terms and Concepts

1. List the different indeterminate forms described in this section.

2. T/F: l'Hopital's Rule provides a faster method of computing derivatives.

3. T/F: l'Hopitals Rule states that \(\frac{d}{dx} \left ( \frac{f(x)}{g(x)}\right ) = \frac{f'(x)}{g'(x)}\).

4. Explain what the indeterminate form "\(1^{\infty}\)" means.

5. Fill in the blanks" The Quotient Rule is applied to \(\frac{f(x)}{g(x)}\) when taking _____; l'Hopital's Rule is applied when taking certain_______.

6. Create (but do not evaluate) a limit that returns "\(\infty^0\)".

7. Create a function \(f(x)\) such that \(\lim\limits_{x\to1}f(x)\) returns "\(0^0\)".

Problems

In Exercises 8-52, evaluate the given limit.

8. \(\lim\limits_{x\to 1}\frac{x^2+x-2}{x-1}\)

9. \(\lim\limits_{x\to 2}\frac{x^2+x-6}{x^2-7x+10}\)

10. \(\lim\limits_{x\to \pi} \frac{\sin x}{x-\pi}\)

11. \(\lim\limits_{x\to\pi/4}\frac{\sin x-\cos x}{\cos (2x)}\)

12. \(\lim\limits_{x\to 0}\frac{\sin (5x)}{x}\)

13. \(\lim\limits_{x\to 0} \frac{\sin (2x)}{x+2}\)

14. \(\lim\limits_{x\to 0} \frac{\sin (2x)}{\sin (3x)}\)

15. \(\lim\limits_{x\to 0} \frac{\sin (ax)}{\sin (bx)}\)

16. \(\lim\limits_{x\to 0^+}\frac{e^x-1}{x^2}\)

17. \(\lim\limits_{x\to 0^+}\frac{e^x-x-1}{x^2}\)

18. \(\lim\limits_{x\to 0^+} \frac{x-\sin x}{x^3-x^2}\)

19. \(\lim\limits_{x\to \infty} \frac{x^4}{e^x}\)

20. \(\lim\limits_{x\to \infty} \frac{\sqrt{x}}{e^x}\)

21. \(\lim\limits_{x\to \infty} \frac{e^x}{\sqrt{x}}\)

22. \(\lim\limits_{x\to \infty} \frac{e^x}{2^x}\)

23. \(\lim\limits_{x\to \infty}\frac{e^x}{3^x}\)

24. \(\lim\limits_{x\to 3} \frac{x^3-5x^2+3x+9}{x^3-7x^2+15x-9}\)

25. \(\lim\limits_{x\to -2}\frac{x^3+4x^2+4x}{x^3+7x^2+16x+12}\)

26. \(\lim\limits_{x\to \infty} \frac{\ln x}{x}\)

27. \(\lim\limits_{x\to \infty} \frac{\ln (x^2)}{x}\)

28. \(\lim\limits_{x\to \infty} \frac{\left ( \ln x\right )^2}{x}\)

29. \(\lim\limits_{x\to 0^+}x\cdot \ln x\)

30. \(\lim\limits_{x\to 0^+}\sqrt{x}\cdot \ln x\)

31. \(\lim\limits_{x\to 0^+} xe^{1/x}\)

32. \(\lim\limits_{x\to \infty} x^3-x^2\)

33. \(\lim\limits_{x\to\infty} \sqrt{x}-\ln x\)

34. \(\lim\limits_{x\to -\infty} xe^x\)

35. \(\lim\limits_{x\to 0^+}\frac{1}{x^2}e^{-1/x}\)

36. \(\lim\limits_{x\to 0^+} (1+x)^{1/x}\)

37. \(\lim\limits_{x\to 0+} (2x)^x\)

38. \(\lim\limits_{x\to 0^+} (2/x)^x\)

39. \(\lim\limits_{x\to 0^+} (\sin x)^x\) Hint: use the Squeeze Theorem.

40. \(\lim\limits_{x\to 1^+} (1-x)^{1-x}\)

41. \(\lim\limits_{x\to \infty} (x)^{1/x}\)

42. \(\lim\limits_{x\to \infty} (1/x)^x\)

43. \(\lim\limits_{x\to 1^1} (\ln x)^{1-x}\)

44. \(\lim\limits_{x\to \infty} (1+x)^{1/x}\)

45. \(\lim\limits_{x\to \infty}(1+x^2)^{1/x}\)

46. \(\lim\limits_{x\to \pi/2} \tan x \cos x\)

47. \(\lim\limits_{x\to \pi /2} \tan x \sin (2x)\)

48. \(\lim\limits_{x\to 1^+} \frac{1}{\ln x}-\frac{1}{1-x}\)

49. \(\lim\limits_{x\to 3^+} \frac{5}{x^2-9}-\frac{x}{x-3}\)

50. \(\lim\limits_{x\to \infty}x\tan (1/x)\)

51. \(\lim\limits_{x\to \infty} \frac{(\ln x)^3}{x}\)

52. \(\lim\limits_{x\to 1}\frac{x^2+x-2}{\ln x}\)

6.8: Improper Integration

Terms and Concepts

1. The definite integral was defined with what two stipulations?

2. If \(\lim\limits_{b\to \infty}\int_0^b f(x)\,dx\) exists, then the integral \(\int_0^{\infty}f(x)\,dx\) is said to __________.

3. If \(\int_1^{\infty} f(x)\,dx=10,\text{ and }0\le g(x)\le f(x)\) for all x, then we know that \(\int_1^{\infty}g(x)\,dx\) ______.

4. For what values of p will \(\int_1^{\infty} \frac{1}{x^p}\,dx\) converge?

5. For what values of p will \(\int_{10}^{\infty} \frac{1}{x^p}\,dx\) converge?

6. For what values of p will \(\int_{0}^{1} \frac{1}{x^p}\,dx\) converge?

Problems

In Exercises 7-33, evaluate the given improper integral.

7. \(\int_0^{\infty}e^{5-2x}\,dx\)

8. \(\int_{1}^{\infty} \frac{1}{x^3} \,dx\)

9. \(\int_{1}^{\infty}x^{-4} \,dx\)

10. \(\int_{-\infty}^{\infty}\frac{1}{x^2+9} \,dx\)

11. \(\int_{-\infty}^{0}2^x \,dx\)

12. \(\int_{-\infty}^{0}\left ( \frac{1}{2}\right )^x \,dx\)

13. \(\int_{-\infty}^{\infty} \frac{x}{x^2+1} \,dx\)

14. \(\int_{-\infty}^{\infty} \frac{x}{x^2+4} \,dx\)

15. \(\int_{2}^{\infty} \frac{1}{(x-1)^2} \,dx\)

16. \(\int_{1}^{2} \frac{1}{(x-1)^2} \,dx\)

17. \(\int_{2}^{\infty} \frac{1}{x-1} \,dx\)

18. \(\int_{1}^{2}\frac{1}{x-1} \,dx\)

19. \(\int_{-1}^{1}\frac{1}{x} \,dx\)

20. \(\int_{1}^{3}\frac{1}{x-2} \,dx\)

21. \(\int_{0}^{\pi} \sec^2 x \,dx\)

22. \(\int_{-2}^{1} \frac{1}{\sqrt{|x|}} \,dx\)

23. \(\int_{0}^{\infty}xe^{-x} \,dx\)

24. \(\int_{0}^{\infty}xe^{-x^2} \,dx\)

25. \(\int_{-\infty}^{\infty}xe^{-x^2} \,dx\)

26. \(\int_{-\infty}^{\infty} \frac{1}{e^x+e^{-x}} \,dx\)

27. \(\int_{0}^{1}x\ln x \,dx\)

28. \(\int_{1}^{\infty} \frac{\ln x}{x} \,dx\)

29. \(\int_{0}^{1}\ln x \,dx\)

30. \(\int_{1}^{\infty} \frac{\ln x}{x^2} \,dx\)

31. \(\int_{1}^{\infty}\frac{\ln x}{\sqrt{x}} \,dx\)

32. \(\int_{0}^{\infty}e^{-x}\sin x \,dx\)

33. \(\int_{0}^{\infty} e^{-x}\cos x \,dx\)

In Exercises 34-43, use the Direct Comparison Test or the Limit Comparison Test to determine whether the given definite integral converges or diverges. Clearly state what test is being used and what function the integrand is being compared to.

34. \(\int_{10}^{\infty}\frac{3}{\sqrt{3x^2+2x-5}} \,dx\)

35. \(\int_{2}^{\infty} \frac{4}{\sqrt{7x^3-x}} \,dx\)

36. \(\int_{0}^{\infty} \frac{\sqrt{x+3}}{\sqrt{x^3-x^2+x+1}} \,dx\)

37. \(\int_{1}^{\infty} e^{-x}\ln x \,dx\)

38. \(\int_{5}^{\infty} e^{-x^2+3x-1} \,dx\)

39. \(\int_{0}^{\infty} \frac{\sqrt{x}}{e^x} \,dx\)

40. \(\int_{2}^{\infty} \frac{1}{x^2+\sin x} \,dx\)

41. \(\int_{0}^{\infty}\frac{x}{x^2+\cos x} \,dx\)

42. \(\int_{0}^{\infty}\frac{1}{x+e^x} \,dx\)

43. \(\int_{0}^{\infty} \frac{1}{e^x-x} \,dx\)