Problem Set: Integrals Involving Exponential and Logarithmic Functions

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

57. [latex]\displaystyle\int \text{ln}xdx[/latex]

(Hint:[latex]\displaystyle\int \text{ln}xdx=\frac{1}{2}\displaystyle\int x\text{ln}({x}^{2})dx)[/latex])

58. [latex]\displaystyle\int {x}^{2}{\text{ln}}^{2}xdx[/latex]

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

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

61. Write an integral to express the area under the graph of [latex]y=\frac{1}{t}[/latex] from [latex]t=1[/latex] to ex and evaluate the integral.

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

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

63. [latex]\displaystyle\int \tan (2x)dx[/latex]

64. [latex]\displaystyle\int \frac{ \sin (3x)- \cos (3x)}{ \sin (3x)+ \cos (3x)}dx[/latex]

65. [latex]\displaystyle\int \frac{x \sin ({x}^{2})}{ \cos ({x}^{2})}dx[/latex]

66. [latex]\displaystyle\int x \csc ({x}^{2})dx[/latex]

67. [latex]\displaystyle\int \text{ln}( \cos x) \tan xdx[/latex]

68. [latex]\displaystyle\int \text{ln}( \csc x) \cot xdx[/latex]

69. [latex]\displaystyle\int \frac{{e}^{x}-{e}^{\text{−}x}}{{e}^{x}+{e}^{\text{−}x}}dx[/latex]

In the following exercises, evaluate the definite integral.

70. [latex]{\displaystyle\int }_{1}^{2}\frac{1+2x+{x}^{2}}{3x+3{x}^{2}+{x}^{3}}dx[/latex]

In the following exercises, integrate using the indicated substitution.

71. [latex]\displaystyle\int \frac{x}{x-100}dx;u=x-100[/latex]

72. [latex]\displaystyle\int \frac{y-1}{y+1}dy;u=y+1[/latex]

73. [latex]\displaystyle\int \frac{1-{x}^{2}}{3x-{x}^{3}}dx;u=3x-{x}^{3}[/latex]

74. [latex]\displaystyle\int \frac{ \sin x+ \cos x}{ \sin x- \cos x}dx;u= \sin x- \cos x[/latex]

75. [latex]\displaystyle\int {e}^{2x}\sqrt{1-{e}^{2x}}dx;u={e}^{2x}[/latex]

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

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] [latex]y={e}^{x}[/latex] over [latex]\left[0,1\right][/latex]

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

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

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

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

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