Problem Set: Integrals Involving Exponential and Logarithmic Functions

In the following exercises, verify by differentiation that $\displaystyle\int \text{ln}xdx=x(\text{ln}x-1)+C,$ then use appropriate changes of variables to compute the integral.

57. $\displaystyle\int \text{ln}xdx$

(Hint: $\displaystyle\int \text{ln}xdx=\frac{1}{2}\displaystyle\int x\text{ln}({x}^{2})dx)$ )

58. $\displaystyle\int {x}^{2}{\text{ln}}^{2}xdx$

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$\frac{1}{9}{x}^{3}(\text{ln}({x}^{3})-1)+C$

59. $\displaystyle\int \frac{\text{ln}x}{{x}^{2}}dx$ $(Hint\text{:}\text{Set}u=\frac{1}{x}\text{.})$

60. $\displaystyle\int \frac{\text{ln}x}{\sqrt{x}}dx$ $(Hint\text{:}\text{Set}u=\sqrt{x}\text{.})$

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$2\sqrt{x}(\text{ln}x-2)+C$

61. Write an integral to express the area under the graph of $y=\frac{1}{t}$ from $t=1$ to e^x and evaluate the integral.

62. Write an integral to express the area under the graph of $y={e}^{t}$ between $t=0$ and $t=\text{ln}x,$ and evaluate the integral.

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${\displaystyle\int }_{0}^{\text{ln}x}{e}^{t}dt={e}^{t}{|}_{0}^{\text{ln}x}={e}^{\text{ln}x}-{e}^{0}=x-1$

In the following exercises, use appropriate substitutions to express the trigonometric integrals in terms of compositions with logarithms.

63. $\displaystyle\int \tan (2x)dx$

64. $\displaystyle\int \frac{ \sin (3x)- \cos (3x)}{ \sin (3x)+ \cos (3x)}dx$

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$-\frac{1}{3}\text{ln}( \sin (3x)+ \cos (3x))$

65. $\displaystyle\int \frac{x \sin ({x}^{2})}{ \cos ({x}^{2})}dx$

66. $\displaystyle\int x \csc ({x}^{2})dx$

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$-\frac{1}{2}\text{ln}| \csc ({x}^{2})+ \cot ({x}^{2})|+C$

67. $\displaystyle\int \text{ln}( \cos x) \tan xdx$

68. $\displaystyle\int \text{ln}( \csc x) \cot xdx$

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$-\frac{1}{2}{(\text{ln}( \csc x))}^{2}+C$

69. $\displaystyle\int \frac{{e}^{x}-{e}^{\text{−}x}}{{e}^{x}+{e}^{\text{−}x}}dx$

In the following exercises, evaluate the definite integral.

70. ${\displaystyle\int }_{1}^{2}\frac{1+2x+{x}^{2}}{3x+3{x}^{2}+{x}^{3}}dx$

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\frac{1}{3} ln (\frac{26}{7})

$\frac{1}{3}\text{ln}(\frac{26}{7})$

In the following exercises, integrate using the indicated substitution.

71. $\displaystyle\int \frac{x}{x-100}dx;u=x-100$

72. $\displaystyle\int \frac{y-1}{y+1}dy;u=y+1$

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$y-2\text{ln}|y+1|+C$

73. $\displaystyle\int \frac{1-{x}^{2}}{3x-{x}^{3}}dx;u=3x-{x}^{3}$

74. $\displaystyle\int \frac{ \sin x+ \cos x}{ \sin x- \cos x}dx;u= \sin x- \cos x$

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$\text{ln}| \sin x- \cos x|+C$

75. $\displaystyle\int {e}^{2x}\sqrt{1-{e}^{2x}}dx;u={e}^{2x}$

76. $\displaystyle\int \text{ln}(x)\frac{\sqrt{1-{(\text{ln}x)}^{2}}}{x}dx;u=\text{ln}x$

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$-\frac{1}{3}{(1-(\text{ln}{x}^{2}))}^{3\text{/}2}+C$

In the following exercises, does the right-endpoint approximation overestimate or underestimate the exact area? Calculate the right endpoint estimate R₅₀ and solve for the exact area.

37. [T] $y={e}^{x}$ over $\left[0,1\right]$

38. [T] $y={e}^{\text{−}x}$ over $\left[0,1\right]$

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Exact solution: $\frac{e-1}{e},{R}_{50}=0.6258.$ Since $f$ is decreasing, the right endpoint estimate underestimates the area.

39. [T] $y=\text{ln}(x)$ over $\left[1,2\right]$

40. [T] $y=\frac{x+1}{{x}^{2}+2x+6}$ over $\left[0,1\right]$

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Exact solution: $\frac{2\text{ln}(3)-\text{ln}(6)}{2},{R}_{50}=0.2033.$ Since $f$ is increasing, the right endpoint estimate overestimates the area.

41. [T] $y={2}^{x}$ over $\left[-1,0\right]$

42. [T] $y=\text{−}{2}^{\text{−}x}$ over $\left[0,1\right]$

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Exact solution: $-\frac{1}{\text{ln}(4)},{R}_{50}=-0.7164.$ Since $f$ is increasing, the right endpoint estimate overestimates the area (the actual area is a larger negative number).

Integration: Supplemental Content

Problem Set: Integrals Involving Exponential and Logarithmic Functions

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