1. Multiple Choice If a function $f$ is continuous on the interval $[a,\,\infty)$, then $\int_{a}^{\infty }f(x)\,dx$ is called [(a) a definite,(b) an infinite, (c) an improper, (d) a proper ] integral.

2. Multiple Choice If the $\lim\limits_{b \rightarrow \infty }\int_{a}^{b} f(x)\,dx$ does not exist, the improper integral $\int_{a}^{\infty }f(x)\,dx$ [ (a) converges, (b) diverges, (c) equals $ab].$

3. True or False If a function $f$ is continuous for all $x$, then the improper integral $\int_{-\infty }^{\,\infty }f(x)\,dx$ always converges.

4. True or False If a function $f$ is continuous and nonnegative on the interval $[a,\,\infty)$, and $\int_{a}^{\infty } f( x) \,dx$ converges, then $\int_{a}^{\infty }f(x)\,dx$ represents the area under the graph of $y=f(x)$ for $x \ge a.$

5. True or False If a function $f$ is continuous for all $x,$ then the improper integral $\int_{-\infty }^{\infty }f(x)\,dx=\lim\limits_{a\,\rightarrow \,\infty }\int_{-a}^{a}f(x)\,dx$.

6. To determine whether the improper integral $\int_{a}^{b} f( x) \,dx$ converges or diverges, where $f$ is continuous on $[a,\,b),$ but is not defined at $b$, requires finding what limit?

7. $\int_{0}^{\infty }x^{2}\,dx$

8. $\int_{0}^{5}x^{3}dx$

9. $\int_{2}^{3}\dfrac{dx}{x-1}$

10. $\int_{1}^{2}\dfrac{dx}{x-1}$

11. $\int_{0}^{1}\dfrac{1}{x}\,dx$

12. $\int_{-1}^{1}\dfrac{x\, dx}{x^{2}+1}$

13. $\int_{0}^{1}\dfrac{x}{x^{2}-1}\,dx$

14. $\int_{0}^{\infty }e^{-2x}\,dx$

15. $\int_{1}^{\infty }\dfrac{dx}{x^{3}}$

16. $\int_{-\infty }^{-10}\dfrac{dx}{x^{2}}$

17. $\int_{0}^{\infty }e^{2x}\,dx$

18. $\int_{0}^{\infty }e^{-x}\,dx$

19. $\int_{-\infty }^{-1}\dfrac{4}{x}\,dx$

20. $\int_{1}^{\infty }\dfrac{4}{x}\,dx$

21. $\int_{3}^{\infty }\dfrac{dx}{(x-1)^{4}}$

22. $\int_{-\infty }^{0}\dfrac{dx}{(x-1) ^{4}}$

23. $\int_{-\infty }^{\infty }\dfrac{dx}{x^{2}+4}$

24. $\int_{-\infty }^{\infty }\dfrac{dx}{x^{2}+1}$

25. $\int_{0}^{1}\dfrac{dx}{x^{2}}$

26. $\int_{0}^{1}\dfrac{dx}{x^{3}}$

27. $\int_{0}^{1}\dfrac{dx}{x}$

28. $\int_{4}^{6}\dfrac{dx}{x-4}$

29. $\int_{0}^{4}\dfrac{dx}{\sqrt{4-x}}$

30. $\int_{1}^{5}\dfrac{x~dx}{\sqrt{5-x}}$

31. $\int_{-1}^{1}\dfrac{dx}{\sqrt[3]{x}}$

32. $\int_{0}^{3}\dfrac{dx}{(x-2)^{2}}$

33. $\int_{0}^{\infty }\cos x\,dx$

34. $\int_{0}^{\infty }\sin (\pi x) \,dx$

35. $\int_{-\infty }^{0}e^{x}\,dx$

36. $\int_{-\infty }^{0}e^{-x}\,dx$

37. $\int_{0}^{\pi /2}\dfrac{x\,dx}{\sin x^{2}}$

38. $\int_{0}^{1}\dfrac{\ln x\,dx}{x}$

39. $\int_{0}^{1}\dfrac{dx}{1-x^{2}}$

40. $\int_{1}^{2}\dfrac{dx}{\sqrt{x^{2}-1}}$

41. $\int_{0}^{1}\dfrac{x\,dx}{(1-x^{2})^{2}}$

42. $\int_{0}^{2}\dfrac{dx}{(x-1)^{2}}$

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

44. $\int_{0}^{\pi /2}\csc x\,dx$

45. $\int_{0}^{\infty }\dfrac{x\,dx}{\sqrt{x+1}}$

46. $\int_{2}^{\infty }\dfrac{dx}{x\sqrt{x^{2}-1}}$

47. $\int_{-\infty }^{\infty }\dfrac{dx}{x^{2}+4x+5}$

48. $\int_{-\infty }^{\infty }\dfrac{dx}{e^{x}+e^{-x}}$

49. $\int_{-\infty }^{2}\dfrac{dx}{\sqrt{4-x}}$

50. $\int_{-\infty }^{1}\dfrac{x\,dx}{\sqrt{2-x}}$

51. $\int_{2}^{4}\dfrac{2x\,dx}{\sqrt[3]{x^{2}-4}}$

52. $\int_{0}^{\pi }\dfrac{1}{1-\cos x}\,dx$

53. $\int_{-1}^{1}\dfrac{1}{x^{3}}\,dx$

54. $\int_{0}^{2}\dfrac{dx}{x-1}$

55. $\int_{0}^{2}\dfrac{dx}{(x-1)^{1/3}}$

56. $\int_{-1}^{1}\dfrac{dx}{x^{5/3}}$

57. $\int_{1}^{2}\dfrac{dx}{(2-x)^{3/4}}$

58. $\int_{0}^{4}\dfrac{dx}{\sqrt{8x-x^{2}}}$

59. $\int_{a}^{3a}\dfrac{2x\,dx}{(x^{2}-a^{2})^{3/2}}$, $a \gt 0$

60. $\int_{0}^{3}\dfrac{x\,dx}{(9-x^{2})^{3/2}}$

61. $\int_{0}^{\infty }xe^{-x^{2}}\,dx$

62. $\int_{0}^{\infty }e^{-x}\sin x\,dx$

63. $\int_{1}^{\infty }\dfrac{1}{\sqrt{x^{2}-1}}\,dx$

64. $\int_{2}^{\infty }\dfrac{2}{\sqrt{x^{2}-4}}\,dx$

65. $\int_{1}^{\infty }\dfrac{1+e^{-x}}{x}\,dx$

66. $\int_{1}^{\infty }\dfrac{3e^{-x}}{x}\,dx$

67. $\int_{1}^{\infty }\dfrac{\sin ^{2}x}{x^{2}}\,dx$

68. $\int_{1}^{\infty }\dfrac{\cos ^{2}x}{x^{2}}\,dx$

69. $\int_{1}^{\infty }\dfrac{dx}{(x+1) \sqrt{x}}$

70. $\int_{1}^{\infty }\dfrac{dx}{x\sqrt{1+x^{2}}}$

71. Area Between Graphs Find the area, if it is defined, of the region enclosed by the graphs of $y=\dfrac{1}{x+1}$ and $y=\dfrac{1}{x+2}$ on the interval $[0,\infty )$. See the figure.

    

72. Area Between Graphs Find the area, if it is defined, under the graph of $y=\dfrac{1}{1+x^{2}}$ to the right of $x=0$. See the figure below.

    

73. Volume of a Solid of Revolution Find the volume, if it is defined, of the solid of revolution generated by revolving the region bounded by the graph of $y=e^{-x}$ and the $x$-axis to the right of $x=0$ about the $x$-axis. See the figure below.

    

74. Volume of a Solid of Revolution Find the volume, if it is defined, of the solid of revolution generated by revolving the region bounded by the graph of $y=\dfrac{1}{\sqrt{x}}$ and the $x$-axis to the right of $x=1$ about the $x$-axis. See the figure below.

    

75. Area Between Graphs Find the area, if it is defined, between the graph of $y=\dfrac{8a^{3}}{x^{2}+4a^{2}}$, $a>0$, and its horizontal asymptote.

76. Drug Reaction The rate of reaction $r$ to a given dose of a drug at time $t$ hours after administration is given by $r(t)=t\,e^{-t^{2}}$ (measured in appropriate units).

    1. (a) Why is it reasonable to define the total reaction as the area under the graph of $y=r(t)$ on $[0,\infty)$?
    2. (b) Find the total reaction to the given dose of the drug.

77. Present Value of Money The present value $PV$ of a capital asset that provides a perpetual stream of revenue that flows continuously at a rate of $R(t)$ dollars per year is given by $$PV= \int_{0}^{\infty }R(t)e^{-rt}\,dt$$

    where $r,$ expressed as a decimal, is the annual rate of interest compounded continuously.

    1. (a) Find the present value of an asset if it provides a constant return of $\$100$ per year and $r=8\%$.
    2. (b) Find the present value of an asset if it provides a return of $R(t)=1000+80t$ dollars per year and $r=7\%$.

78. Electrical Engineering In a problem in electrical theory, the integral $\int_{0}^{\infty }Ri^{2}\,dt$ occurs, where the current $i=Ie^{-Rt/L}$, $t$ is time, and $R$, $I$, and $L$ are positive constants. Find the integral.

79. Magnetic Potential The magnetic potential $u$ at a point on the axis of a circular coil is given by $$u=\dfrac{2\pi NIr}{10}\! \int_{x}^{\infty }\dfrac{dy}{(r^{2}+y^{2})^{3/2}}$$

    where $N$, $I$, $r$, and $x$ are constants. Find the integral.

80. Electrical Engineering The field intensity $F$ around a long (“infinite”) straight wire carrying electric current is given by the integral $$F=\dfrac{rIm}{10}\int_{-\infty }^{\infty }\dfrac{dy}{(r^{2}+y^{2})^{3/2}}$$

    where $r,$ $I$, and $m$ are constants. Find the integral.

81. Work The force $F$ of gravitational attraction between two point masses $m$ and $M$ that are $r$ units apart is $F=\dfrac{GmM}{r^{2}}$ , where $G$ is the universal gravitational constant. Find the work done in moving the mass $m$ along a straight-line path from $r=1$ unit to $r=\infty$.

82. For what numbers $a$ does $\int_{0}^{1}x^{a}\,dx$ converge?

83. $\int_{0}^{\infty }xe^{-x}\,dx$

84. $\int_{0}^{1}x\ln x\,dx$

85. $\int_{0}^{\infty}e^{-x}\cos x\,dx$

86. $\int_{0}^{\infty }\tan^{-1}x\,dx$

87. Show that $\int_{0}^{\infty }\sin x\,dx$ and $\int_{-\infty }^{0}\sin x\,dx$ each diverge, yet $\lim\limits_{t\,\rightarrow \,\infty }\int_{-t}^{t}\sin x\, dx=0$.

532

88. Find a function $f$ for which $\int_{0}^{\infty }f(x)\,dx$ and $\int_{-\infty }^{0}f(x)\,dx$ each diverge, yet $\lim\limits_{t\,\rightarrow \,\infty }\int_{-t}^{t}f(x)\,dx=1$.

89. Use the Comparison Test for Improper Integrals to show that $\int_{0}^{\infty }\dfrac{1}{\sqrt{2+\sin x}}\,dx$ diverges.

90. Use the Comparison Test for Improper Integrals to show that $\int_{2}^{\infty }\dfrac{\ln x}{\sqrt{x^{2}-1}}\,dx$ diverges.

91. If $n$ is a positive integer, show that:

    1. (a) $\int_{0}^{\infty }x^{n}\,e^{-x}\,dx=n\int_{0}^{\infty }x^{n-1}\,e^{-x}\,dx$
    2. (b) $\int_{0}^{\infty }x^{n}\,e^{-x}\,dx=n!$

92. Show that $\int_{e}^{\infty }\dfrac{\,dx}{x ( \ln x) ^{p}}$ converges if $p \gt 1$ and diverges if $p \le 1$.

93. Show that $\int_{a}^{b}\dfrac{dx}{(x-a) ^{p}}$ converges if $0 \lt p \lt 1$ and diverges if $p \ge 1.$

94. Show that $\int_{a}^{b}\dfrac{dx}{(b-x)^{p}}$ converges if $0 \lt p \lt 1$ and diverges if $p \ge 1.$

95. Refer to Problems 93 and 94. Discuss the convergence or divergence of the integrals if $p \le 0.$ Support your explanation with an example.

96. Comparison Test for Improper Integrals Show that if two functions $f$ and $g$ are nonnegative and continuous on the interval $[ a,\infty )$, and if $f( x) \geq g( x)$ for all numbers $x > c,$ where $c \geq a,$ then

    • If $\int_{a}^{\infty }f( x)\, dx$ converges, then $\int_{a}^{\infty }g( x)\,dx$ also converges.
    • If $\int_{a}^{\infty }g( x)\, dx$ diverges, then $\int_{a}^{\infty }f( x)\, dx$ also diverges.

97. $f(x) =x$

98. $f(x) =\cos x$

99. $f(x) =\sin x$

100. $f(x) =e^{x}$

101. $f(x) =e^{ax}$

102. $f(x) =1$

103. Find the arc length of $y=\sqrt{x-x^{2}}-\sin ^{-1}\sqrt{x}$.

104. Find $\int_{-\infty }^{a}e^{(x-e^{x})}\,dx.$

105. Find $\int_{-\infty }^{\infty }e^{(x-e^{x})}\,dx.$

106. 1. (a) Show that the area of the region in the first quadrant bounded by the graph of $y=e^{-x}$ and the $x$-axis is divided into two equal parts by the line $x=\ln 2$.
     2. (b) If the two regions with equal areas described in (a) are rotated about the $x$-axis, are the resulting volumes equal? If they are unequal, which one is larger and by how much?

107. Uniform Density Function Show that the function $f$ below is a probability density function. $$f(x)=\left\{ \begin{array}{c@{\qquad}l@{\quad}rl} 0 & \hbox{if} & x& \lt a \\ \dfrac{1}{b-a} & \hbox{if} & a & \le x \le b, a \lt b \\ 0 & \hbox{if} & x& \gt b \end{array} \right.$$

108. Exponential Density Function Show that the function $f$ is a probability density function for $a>0$. $$f(x)=\left\{ \begin{array}{l@{\qquad}l@{\quad}ll} \dfrac{1}{a}e^{-x/a} & \hbox{if}&x & \ge 0 \\ 0&\hbox{if}&x & \lt 0 \end{array} \right.$$

109. Find the expected value $\mu$ of the uniform density function $f$ given in Problem 107.

110. Find the expected value $\mu$ of the exponential probability density function $f$ defined in Problem 108.

111. Find the variance $\sigma ^{2}$ and standard deviation $\sigma$ of the uniform density function defined in Problem 107.

112. Find the variance $\sigma ^{2}$ and standard deviation $\sigma$ of the exponential density function defined in Problem 108.