1. If the substitution $u=2x+3$ is used with $\int \sin (2x+3)\, dx,$ the result is $\int$ _________$du.$

2. True or False  If the substitution $u=x^{2}+3$ is used with $\int_{0}^{1}x(x^{2}+3)^{3}dx,$ the result is $\int_{0}^{1}x(x^{2}+3) ^{3}dx=\dfrac{1}{2}\int_{0}^{1}u^{3}du.$

3. Multiple Choice  $\int_{-4}^{4}x^{3}dx= [{\boldsymbol (a)}\, 128\enspace {\boldsymbol (b)}\, 4\enspace {\boldsymbol (c)}\,0\enspace {\boldsymbol (d)}\,64 ]$.

4. True or False $\int_{0}^{5}x^{2}dx=\dfrac{1}{2}\int_{-5}^{5}x^{2}dx$.

5. $\int e^{3x+1}dx$; let $u=3x+1$.

6. $\int \dfrac{dx}{x\ln x}$; let $u=\ln x$.

7. $\int {({1-t^{2}})^{6}t\,dt};$ let $u=1-t^{2}$.

8. $\int {\sin ^{5}x\cos x\,dx;}$ let $u=\sin x$.

9. $\int \dfrac{x^{2}\,dx}{\sqrt{1-x^{6}}}\ ;$ let $u=x^{3}$.

10. $\int \dfrac{e^{-x}}{6+e^{-x}} dx;$ let $u=6+e^{-x}$.

11. $\int \sin (3x) \,dx$

12. $\int {x}\sin x^{2}\,dx$

13. $\int {\sin x\cos^{2}x\,dx}$

14. $\int \tan ^{2}x\sec ^{2}x\,dx$

15. $\int \dfrac{e^{1/x}}{x^{2}} dx$

16. $\int \dfrac{e^{\sqrt[3]{x}}}{\sqrt[3]{x^{2}}} dx$

17. $\int \dfrac{x\;dx}{x^{2}-1}$

18. $\int \dfrac{5x\;dx}{1-x^{2}}$

19. $\int \dfrac{e^{x}}{\sqrt{1+e^{x}}} dx$

20. $\int \dfrac{dx}{x(\ln x)^{7}}$

21. $\int {{\dfrac{1}{{\sqrt{x}( {1+\sqrt{x}})^{4}}}}\,dx}$

22. $\int \dfrac{dx}{\sqrt{x}(1+\sqrt{x})}$

23. $\int \dfrac{3e^{x}}{\sqrt[4]{e^{x}-1}} dx$

24. $\int \dfrac{[ \ln ( 5x) ] ^{3}}{x} dx$

25. $\int \dfrac{\cos x\;dx}{2\sin x-1}$

26. $\int \dfrac{\cos (2x)\;dx}{\sin (2x)}$

27. $\int \sec (5x) \,dx$

28. $\int \tan (2x) \,dx$

29. $\int \sqrt{\tan x}\sec^{2}x\,dx$

30. $\int (2+3\cot x)^{3/2}\csc^{2}x\,dx$

31. $\int \dfrac{\sin x}{\cos ^{2}x}\,dx$

32. $\int \dfrac{\cos x}{\sin ^{2}x}\,dx$

33. $\int \sin x\cdot e^{\cos x}\,dx$

34. $\int \sec ^{2}x\cdot e^{\tan x}\,dx$

35. $\int {x\sqrt{x+3}\,dx}$

36. $\int {x\sqrt{4-x}\,dx}$

37. $\int\, [ \sin x+\cos (3x) ] dx$

38. $\int\, \Big[ x^{2}+\sqrt{3x+2}\Big] dx$

39. $\int \dfrac{dx}{x^{2}+25}$

40. $\int \dfrac{\cos x}{1+\sin ^{2}x}\,dx$

41. $\int \dfrac{dx}{\sqrt{9-x^{2}}}$

42. $\int \dfrac{dx}{\sqrt{16-9x^{2}}}$

43. $\int \sinh x\cosh x\,dx$

44. $\int \hbox{ sech}^{2}x\tanh x\,dx$

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

46. $\int_{-1}^{1}( s^{2}-1) ^{5}s ds$

47. $\int_{0}^{1}x^{2}e^{x^{3}+1} dx$

48. $\int_{0}^{1}xe^{x^{2}-2} dx$

49. $\int_{1}^{6}x\sqrt{x+3}\,dx$

50. $\int_{2}^{6}x^{2}\sqrt{x-2} dx$

51. $\int_{0}^{2}x\cdot 3^{2x^{2}} dx$

52. $\int_{0}^{1}x\cdot 10^{-x^{2}} dx$

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

54. $\int_{0}^{\pi /4}\dfrac{\sin (2x)}{\sqrt{5-2\cos (2x) }} dx$

55. $\int_{0}^{2}\dfrac{e^{2x}}{e^{2x}+1} dx$

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

57. $\int_{2}^{3}\dfrac{dx}{x\ln x}$

58. $\int_{2}^{3}\dfrac{dx}{x(\ln x)^{2}}$

59. $\int_{0}^{\pi }e^{x}\cos (e^{x}) \,dx$

60. $\int_{0}^{\pi }e^{-x}\cos (e^{-x}) \,dx$

61. $\int_{0}^{1}\dfrac{x\,dx}{1+x^{4}}$

62. $\int_{0}^{1}\dfrac{e^{x}}{1+e^{2x}} dx$

63. $\int_{-2}^{2}( x^{2}-4) dx$

64. $\int_{-1}^{1}( x^{3}-2x) dx$

397

65. $\int_{-\pi /2}^{\pi /2}\dfrac{1}{3}\sin \theta d\theta$

66. $\int_{-\pi /4}^{\pi /4}\sec^{2}x\,dx$

67. $\int_{-1}^{1}\dfrac{3}{1+x^{2}}\,dx$

68. ${{\int_{-5}^{5}}} \big(x^{1/3}+x\big){ dx}$

69. $\int_{-5}^{5}\vert 2x\vert \,dx$

70. $\int_{-1}^{1}[\vert x\vert -3 ] \,dx$

71. $\int \dfrac{x+1}{x^{2}+1}dx$

72. $\int \dfrac{2x-3}{1+x^{2}}dx$

73. $\int \left({2\sqrt{x^{2}+3}-{\dfrac{{4}}{{x}}}+9}\right) ^{6}\left({{\dfrac{{x}}{\sqrt{x^{2}+3}}}+{\dfrac{{2}}{{x^{2}}}}}\right) dx$

74. $\int {{\Big[\sqrt{({z^{2}+1})^{4}-3}\Big]}{\Big[z{({z^{2}+1})^{3}}\Big]}\,dz}$

75. $\int {{\dfrac{{x+4x^{3}}}{\sqrt{x}}}\,dx}$

76. $\int {{\dfrac{{z\,dz}}{{z+\sqrt{z^{2}+4}}}}}$

77. $\int {\sqrt{t}\sqrt{4+t\sqrt{t}}\,dt }$

78. $\int_{0}^{1}\dfrac{x+1}{x^{2}+3}dx$

79. $\int 3^{2x+1}dx$

80. $\int 2^{3x+5}dx$

81. $\int \dfrac{\sin x}{\sqrt{4-\cos ^{2}x}}dx$

82. $\int \dfrac{\sec^{2}x\,dx}{\sqrt{1-\tan^{2}x}}$

83. $\int_{0}^{1}\dfrac{(z^{2}+5)(z^{3}+15z-3)\,}{{196-(z^{3}+15z-3)^{2}}}dz$

84. $\int_{2}^{17}\dfrac{dx}{\sqrt{\sqrt{{x-1}}+(x-1)^{5/4}}}$

85. $\int \dfrac{dx}{x\log _{10}x}$

86. $\int \dfrac{dx}{x\log _{3}\sqrt[5]{x}}$

87. $\int_{10}^{100}\dfrac{dx}{x\log x}$

88. $\int_{3}^{32}\dfrac{dx}{x\log_2 x}$

89. $\int_{3}^{9}\dfrac{dx}{x\log _{3}x}$

90. $\int_{10}^{100}\dfrac{dx}{x\log _{5}x}$

91. If $\int_{1}^{b}t^{2}(5t^{3}-1)^{1/2}\,dt=\dfrac{38}{45},$ find $b.$

92. If $\int_{a}^{3}t\sqrt{9-t^{2}}\,dt=6,$ find $a.$

93. $\int (x+1) ^{2} dx$

94. $\int ( x^{2}+1) ^{2}x\, dx$

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

96. $\int_{{-}\pi /4}^{\pi /4}\sin ( 7\theta -\pi)\,d\theta$

97. Area  Find the area under the graph of $f(x) =\dfrac{x^{2}}{\sqrt{2x+1}}$ from $0$ to $4.$

98. Area  Find the area under the graph of $f(x) =\dfrac{x}{(x^{2}+1) ^{2}}$ from $0$ to $2.$

99. Area  Find the area under the graph of $y=\dfrac{1}{3x^{2}+1}$ from $x=0$ to $x=1$.

100. Area  Find the area under the graph of $y=\dfrac{1}{x\sqrt{x^{2}-4}}$ from $x=3$ to $x=4$.

101. Area  Find the area under the graph of the catenary, $$y=a\cosh \dfrac{x}{a}+b-a,$$

     from $x=0$ to $x=a$.

102. Area  Find $b$ so that the area under the graph of $$y=(x+1)\sqrt{x^{2}+2x+4}$$

     is $\dfrac{56}{3}$ for $0\leq x\leq b$.

103. Average Value  Find the average value of $y=\tan x$ on the interval $\left[ 0,\dfrac{\pi }{4}\right]$.

104. Average Value  Find the average value of $y=\sec x$ on the interval $\left[ 0,\dfrac{\pi}{4}\right]$.

105. If $\int_{0}^{2}f(x - 3)\,dx = 8$, find ${\int_{-3}^{-1}{f(x)\,dx. }}$

106. If $\int_{-2}^{1}f(x + 1)\,dx=\dfrac{5}{2}$, find ${\int_{-1}^{2}{f(x)\,dx.}}$

107. If $\int_{0}^{4}f\left( \dfrac{x}{2}\right) dx = 8$, find ${\int_{0}^{2}{f(x)\,dx.}}$

108. If $\int_{0}^{1}g(3x)\,dx = 6$, find ${\int_{0}^{3}}g{(x)\,dx.}$

109. Newton's Law of Cooling  Newton's Law of Cooling states that the rate of change of temperature with respect to time is proportional to the difference between the temperature of the object and the ambient temperature. A thermometer that reads $4{}^{\circ}{\rm C}$ is brought into a room that is $30{}^{\circ}{\rm C}$.

     1. (a) Write the differential equation that models the temperature $u=u(t)$ of the thermometer at time $t$ in minutes (min).
     2. (b) Find the general solution of the differential equation.
     3. (c) If the thermometer reads $10{}^{\circ}{\rm C}$ after 2 min, determine the temperature reading 5 min after the thermometer is first brought into the room.

110. Newton's Law of Cooling  A thermometer reading $70{}^{\circ}{\rm F}$ is taken outside where the ambient temperature is $22{}^{\circ}{\rm F}$. Four minutes later the reading is $32{}^{\circ}{\rm F}$.

     1. (a) Write the differential equation that models the temperature $u=u(t)$ of the thermometer at time $t.$
     2. (b) Find the general solution of the differential equation.
     3. (c) Find the particular solution to the differential equation, using the initial condition that when $t = 0$ min, then $u=70{}^{\circ}{\rm F}.$
     4. (d) Find the thermometer reading 7 min after the thermometer was brought outside.
     5. (e) Find the time it takes for the reading to change from $70{}^{\circ}{\rm F}$ to within $\dfrac{1}{2}{}^{\circ}{\rm F}$ of the air temperature.

111. Forensic Science  At $4$ p.m., a body was found floating in water whose temperature is $12{}^{\circ}{\rm C}.$ When the woman was alive, her body temperature was $37{}^{\circ}{\rm C}$ and now it is $20{}^{\circ}{\rm C}.$ Suppose the rate of change of the temperature $u =u(t)$ of the body with respect to the time $t$ in hours (h) is proportional to $u(t)-T$, where $T$ is the water temperature and the constant of proportionality is $-0.159.$

     1. (a) Write a differential equation that models the temperature $u=u(t)$ of the body at time $t$.
     2. (b) Find the general solution of the differential equation.
     3. (c) Find the particular solution to the differential equation, using the initial condition that at the time of death, when $t=0 h$, her body temperature was $u=37{}^{\circ}{\rm C}.$
     4. (d) At what time did the woman drown?
     5. (e) How long does it take for the woman's body to cool to 15$^{\circ}$C?

112. Newton's Law of Cooling  A pie is removed from a $350{}^{\circ}{\rm F}$ oven to cool in a room whose temperature is $72{}^{\circ}{\rm F}$.

     1. (a) Write the differential equation that models the temperature $u=u(t)$ of the pie at time $t.$
     2. (b) Find the general solution of the differential equation.
     3. (c) Find the particular solution to the differential equation, using the initial condition that when $t=0 min$, then $u=350{}^{\circ}{\rm F}.$
     4. (d) If $u(5) =$ $200{}^{\circ}{\rm F},$ what is the temperature of the pie after $15$ min?
     5. (e) How long will it take for the pie to be $100{}^{\circ}{\rm F}$ and ready to eat?

113. Electric Potential  The electric field strength a distance $z$ from the axis of a ring of radius $R$ carrying a charge $Q$ is given by the formula $$E(z)={\dfrac{{Qz}}{{(R^{2}+z^{2})^{3/2}}}}$$

     If the electric potential $V$ is related to $E$ by $E=-\dfrac{dV}{dz}$, what is $V(z)$?

114. Impulse During a Rocket Launch  The impulse $J$ due to a force $F$ is the product of the force times the amount of time $t$ for which the force acts. When the force varies over time, $$J=\int_{t_{1}}^{t_{2}}F(t)\,dt.$$

     

     We can model the force acting on a rocket during launch by an exponential function $F(t)=Ae^{bt}$, where $A$ and $b$ are constants that depend on the characteristics of the engine. At the instant lift-off occurs ($t=0$), the force must equal the weight of the rocket.

     1. (a) Suppose the rocket weighs 25,000 N (a mass of about 2500 kg or a weight of 5500lb), and 30 seconds after lift-off the force acting on the rocket equals twice the weight of the rocket. Find $A$ and $b.$
     2. (b) Find the impulse delivered to the rocket during the first 30 seconds after the launch.

115. Air Resistance on a Falling Object  If an object of mass $m$ is dropped, the air resistance on it when it has speed $v$ can be modeled as $F_{{\rm air}}=-kv$, where the constant $k$ depends on the shape of the object and the condition of the air. The minus sign is necessary because the direction of the force is opposite to the velocity. Using Newton's Second Law of Motion, this force leads to a downward acceleration $a(t)=ge^{-kt/m}$. See Problem 137. Using the equation for $a(t)$, find:

     1. (a) $v(t)$, if the object starts from rest $v_0=v(0)=0$, with the positive direction downward.
     2. (b) $s(t)$, if the object starts from the position $s_0=s(0)=0$, with the positive direction downward.
     3. (c) What limits do $a(t)$, $v(t)$, and $s(t)$ approach if the object falls for a very long time ($t\rightarrow \infty$)? Interpret each result and explain if it is physically reasonable.
     4. (d) Graph $a=a(t)$, $v=v(t)$, $s=s(t)$. Do the graphs support the conclusions obtained in part (c)? Use $g = 9.8$ m$/$s$^2$, $k = 5$, and $m = 10$ kg.

116. Area  Let $f(x)=k\sin ( kx)$, where $k$ is a positive constant.

     1. (a) Find the area of the region under one arch of the graph of $f.$
     2. (b) Find the area of the triangle formed by the $x$-axis and the tangent lines to one arch of $f$ at the points where the graph of $f$ crosses the $x$-axis.

117. Use an appropriate substitution to show that $$\int_{0}^{1}x^{m}(1 - x)^{n} dx = \int_{0}^{1}x^{n}(1 - x)^{m} dx,$$

     where $m,$ $n$ are positive integers.

118. Properties of Integrals  Find ${\int_{-1}^{1}{f(x)\,dx}}$ for the function given below: $$f(x)={\left\{ {{ \begin{array}{l@{ }l@{ }l} {x+1} & \hbox{if} & {x\lt0} \\ \cos (\pi x) & \hbox{if} & {x\geq 0} \end{array} }}\right. }$$

119. If $f$ is continuous on $[a,b]$, show that $$\int_{a}^{b}f(x)\,dx = \int_{a}^{b}f(a + b -\,x)\,dx$$

120. If $\int_{0}^{1}f(x) \,dx = 2$, find:

     1. (a) ${\int_{0}^{0.5}{f(2x)\,dx }}$
     2. (b) $\int_{0}^{3}f \left( \dfrac{1}{3}x\right) dx$
     3. (c) $\int_{0}^{1/5}f( 5x) dx$
     4. (d) Find the upper and lower limits of integration so that $$\int_{a}^{b}f\left( \dfrac{x}{4}\right) dx=8.$$
     5. (e) Generalize (d) so that $\int_{a}^{b}f( kx) dx=\dfrac{1}{k}\cdot 2$ for $k>0.$

121. If $\int_{0}^{2}f (s) \,ds = 5$, find:

     1. (a) ${\int_{-1}^{1}{f(s+1)\,ds }}$
     2. (b) $\int_{-3}^{-1}f( s+3) \,ds$
     3. (c) $\int_{4}^{6}f( s-4) \,ds$
     4. (d) Find the upper and lower limits of integration so that $$\int_{a}^{b}f(s - 2) ds=5.$$
     5. (e) Generalize (d) so that $\int_{a}^{b}f( s-k) \,ds=5$ for $k>0.$

122. Find $\int_{0}^{b}\vert 2x\vert \,dx$ for any real number $b.$

123. If $f$ is an odd function, show that $\int_{-a}^{a}f(x)\,dx = 0$.

124. Find the constant $k$, where $0\leq k\leq \,3$, for which $${\int_{0}^{3}{{\dfrac{{x}}{\sqrt{x^{2}+16}}}dx={\dfrac{{3k}}{\sqrt{k^{2}+16}}} }}$$

125. If $n$ is a positive integer, for what number is $$\int_{0}^{a}x^{n-1} dx= \dfrac{1}{n}$$

126. If $f$ is a continuous function defined on the interval $[0,1]$, show that $${\int_{0}^{\pi }{x f(\sin x)\,dx={\dfrac{{\pi }}{{2}}}{\int_{0}^{\pi }{f(\sin x)\,dx}}}}$$

127. Prove that $\int \csc x\,dx=\ln \left\vert \csc x-\cot x\right\vert +C$. [Hint: Multiply and divide the integrand by $(\csc x-\cot x)$.]

128. Describe a method for finding $\int_{a}^{b}\vert f(x)\vert \,dx$ in terms of $F(x)=\int f(x)\mathit{~}dx$ when $f(x)$ has finitely many zeros.

129. Find $\int \sqrt[n]{{a\,{+}\,bx}}\,dx$, where $a$ and $b$ are real numbers, $b\neq 0$, and $n\geq 2$ is an integer.

130. If $f$ is continuous for all $x$, which of the following integrals have the same value?

     1. (a) $\int_{a}^{b}{f(x)\,dx }$
     2. (b) $\int_{0}^{b-a}{f(x+a)\,dx }$
     3. (c) $\int_{a+c}^{b+c}{f(x+c)\,dx }$

131. Find $\int {{\dfrac{{x^{6}+3x^{4}+3x^{2}+x+1}}{{(x^{2}+1)^{2}}}}}\,dx$.

132. Find $\int \dfrac{\sqrt[4]{x}}{\sqrt{x}+\sqrt[3]{x}}dx$.

133. Find $\int \dfrac{3x+2}{x\sqrt{x+1}}\ dx$.

134. Find $\int \dfrac{dx}{(x\ln x) [\ln (\ln x)] }$.

135. Air Resistance on a Falling Object  (Refer to Problem 115.) If an object of mass $m$ is dropped, the air resistance on it when it has speed $v$ can be modeled as $$F_{{\rm air}}=-kv,$$

     where the constant $k$ depends on the shape of the object and the condition of the air. The minus sign is necessary because the direction of the force is opposite to the velocity. Using Newton's Second Law of Motion, show that the downward acceleration of the object is $$a(t) =ge^{-kt/m},$$

     where $g$ is the acceleration due to gravity. (Hint: The velocity of the object obeys the differential equation $$m\dfrac{dv}{dt}=mg-kv$$

     Solve the differential equation for $v$ and use the fact that $ma=mg-kv$.)

136. A separable differential equation can be written in the form $\dfrac{dy}{dx}=\dfrac{f(x)}{g (y)}$, where $f$ and $g$ are continuous. Then $$\int g(y)\,dy=\int f(x)\,dx$$

     and integrating (if possible) will give a solution to the differential equation. Use this technique to solve parts (a)–(c) below. (You may need to leave your answer in implicit form.)

     1. (a) $\dfrac{y^{2}}{x}\dfrac{dy}{dx}=1+x^{2}$
     2. (b) $\dfrac{dy}{dx}=y\dfrac{x^{2}-2x+1}{y+3}$
     3. (c) $y\dfrac{dy}{dx}=\dfrac{x^{2}}{y+4}$;  if $y=2$ when $x=8$