# 0.1 Table of integrals

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## Basic integrals

1. $\int {u}^{n}\phantom{\rule{0.2em}{0ex}}du=\frac{{u}^{n+1}}{n+1}+C,n\ne \text{−}1$

2. $\int \frac{du}{u}=\text{ln}\phantom{\rule{0.1em}{0ex}}|u|+C$

3. $\int {e}^{u}\phantom{\rule{0.2em}{0ex}}du={e}^{u}+C$

4. $\int {a}^{u}\phantom{\rule{0.2em}{0ex}}du=\frac{{a}^{u}}{\text{ln}\phantom{\rule{0.1em}{0ex}}a}+C$

5. $\int \text{sin}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{−cos}\phantom{\rule{0.2em}{0ex}}u+C$

6. $\int \text{cos}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{sin}\phantom{\rule{0.2em}{0ex}}u+C$

7. $\int {\text{sec}}^{2}u\phantom{\rule{0.2em}{0ex}}du=\text{tan}\phantom{\rule{0.2em}{0ex}}u+C$

8. $\int {\text{csc}}^{2}u\phantom{\rule{0.2em}{0ex}}du=\text{−cot}\phantom{\rule{0.2em}{0ex}}u+C$

9. $\int \text{sec}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}\text{tan}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{sec}\phantom{\rule{0.2em}{0ex}}u+C$

10. $\int \text{csc}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}\text{cot}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{−csc}\phantom{\rule{0.2em}{0ex}}u+C$

11. $\int \text{tan}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{ln}\phantom{\rule{0.1em}{0ex}}|\text{sec}\phantom{\rule{0.2em}{0ex}}u|+C$

12. $\int \text{cot}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{ln}\phantom{\rule{0.1em}{0ex}}|\text{sin}\phantom{\rule{0.2em}{0ex}}u|+C$

13. $\int \text{sec}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{ln}\phantom{\rule{0.1em}{0ex}}|\text{sec}\phantom{\rule{0.2em}{0ex}}u+\text{tan}\phantom{\rule{0.2em}{0ex}}u|+C$

14. $\int \text{csc}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{ln}\phantom{\rule{0.1em}{0ex}}|\text{csc}\phantom{\rule{0.2em}{0ex}}u-\text{cot}\phantom{\rule{0.2em}{0ex}}u|+C$

15. $\int \frac{\phantom{\rule{0.2em}{0ex}}du}{\sqrt{{a}^{2}-{u}^{2}}}={\text{sin}}^{-1}\frac{u}{a}+C$

16. $\int \frac{\phantom{\rule{0.2em}{0ex}}du}{{a}^{2}+{u}^{2}}=\frac{1}{a}{\text{tan}}^{-1}\frac{u}{a}+C$

17. $\int \frac{\phantom{\rule{0.2em}{0ex}}du}{u\sqrt{{u}^{2}-{a}^{2}}}=\frac{1}{a}{\text{sec}}^{-1}\frac{u}{a}+C$

## Trigonometric integrals

18. $\int {\text{sin}}^{2}u\phantom{\rule{0.2em}{0ex}}du=\frac{1}{2}u-\frac{1}{4}\text{sin}\phantom{\rule{0.2em}{0ex}}2u+C$

19. $\int {\text{cos}}^{2}u\phantom{\rule{0.2em}{0ex}}du=\frac{1}{2}u+\frac{1}{4}\text{sin}\phantom{\rule{0.2em}{0ex}}2u+C$

20. $\int {\text{tan}}^{2}u\phantom{\rule{0.2em}{0ex}}du=\text{tan}\phantom{\rule{0.2em}{0ex}}u-u+C$

21. $\int {\text{cot}}^{2}u\phantom{\rule{0.2em}{0ex}}du=\text{−}\text{cot}\phantom{\rule{0.2em}{0ex}}u-u+C$

22. $\int {\text{sin}}^{3}u\phantom{\rule{0.2em}{0ex}}du=-\frac{1}{3}\left(2+{\text{sin}}^{2}u\right)\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}u+C$

23. $\int {\text{cos}}^{3}u\phantom{\rule{0.2em}{0ex}}du=\frac{1}{3}\left(2+{\text{cos}}^{2}u\right)\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}u+C$

24. $\int {\text{tan}}^{3}u\phantom{\rule{0.2em}{0ex}}du=\frac{1}{2}{\text{tan}}^{2}u+\text{ln}\phantom{\rule{0.1em}{0ex}}|\text{cos}\phantom{\rule{0.2em}{0ex}}u|+C$

25. $\int {\text{cot}}^{3}u\phantom{\rule{0.2em}{0ex}}du=-\frac{1}{2}{\text{cot}}^{2}u-\text{ln}\phantom{\rule{0.1em}{0ex}}|\text{sin}\phantom{\rule{0.2em}{0ex}}u|+C$

26. $\int {\text{sec}}^{3}u\phantom{\rule{0.2em}{0ex}}du=\frac{1}{2}\text{sec}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}\text{tan}\phantom{\rule{0.2em}{0ex}}u+\frac{1}{2}\text{ln}\phantom{\rule{0.1em}{0ex}}|\text{sec}\phantom{\rule{0.2em}{0ex}}u+\text{tan}\phantom{\rule{0.2em}{0ex}}u|+C$

27. $\int {\text{csc}}^{3}u\phantom{\rule{0.2em}{0ex}}du=-\frac{1}{2}\text{csc}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}\text{cot}\phantom{\rule{0.2em}{0ex}}u+\frac{1}{2}\text{ln}\phantom{\rule{0.1em}{0ex}}|\text{csc}\phantom{\rule{0.2em}{0ex}}u-\text{cot}\phantom{\rule{0.2em}{0ex}}u|+C$

28. $\int {\text{sin}}^{n}u\phantom{\rule{0.2em}{0ex}}du=-\frac{1}{n}{\text{sin}}^{n-1}u\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}u+\frac{n-1}{n}\int {\text{sin}}^{n-2}u\phantom{\rule{0.2em}{0ex}}du$

29. $\int {\text{cos}}^{n}u\phantom{\rule{0.2em}{0ex}}du=\frac{1}{n}{\text{cos}}^{n-1}u\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}u+\frac{n-1}{n}\int {\text{cos}}^{n-2}u\phantom{\rule{0.2em}{0ex}}du$

30. $\int {\text{tan}}^{n}u\phantom{\rule{0.2em}{0ex}}du=\frac{1}{n-1}{\text{tan}}^{n-1}u-\int {\text{tan}}^{n-2}u\phantom{\rule{0.2em}{0ex}}du$

31. $\int {\text{cot}}^{n}u\phantom{\rule{0.2em}{0ex}}du=\frac{-1}{n-1}{\text{cot}}^{n-1}u-\int {\text{cot}}^{n-2}u\phantom{\rule{0.2em}{0ex}}du$

32. $\int {\text{sec}}^{n}u\phantom{\rule{0.2em}{0ex}}du=\frac{1}{n-1}\text{tan}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}{\text{sec}}^{n-2}u+\frac{n-2}{n-1}\int {\text{sec}}^{n-2}u\phantom{\rule{0.2em}{0ex}}du$

33. $\int {\text{csc}}^{n}u\phantom{\rule{0.2em}{0ex}}du=\frac{-1}{n-1}\text{cot}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}{\text{csc}}^{n-2}u+\frac{n-2}{n-1}\int {\text{csc}}^{n-2}u\phantom{\rule{0.2em}{0ex}}du$

34. $\int \text{sin}\phantom{\rule{0.2em}{0ex}}au\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}bu\phantom{\rule{0.2em}{0ex}}du=\frac{\text{sin}\left(a-b\right)u}{2\left(a-b\right)}-\frac{\text{sin}\left(a+b\right)u}{2\left(a+b\right)}+C$

35. $\int \text{cos}\phantom{\rule{0.2em}{0ex}}au\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}bu\phantom{\rule{0.2em}{0ex}}du=\frac{\text{sin}\left(a-b\right)u}{2\left(a-b\right)}+\frac{\text{sin}\left(a+b\right)u}{2\left(a+b\right)}+C$

36. $\int \text{sin}\phantom{\rule{0.2em}{0ex}}au\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}bu\phantom{\rule{0.2em}{0ex}}du=-\frac{\text{cos}\left(a-b\right)u}{2\left(a-b\right)}-\frac{\text{cos}\left(a+b\right)u}{2\left(a+b\right)}+C$

37. $\int u\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{sin}\phantom{\rule{0.2em}{0ex}}u-u\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}u+C$

38. $\int u\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{cos}\phantom{\rule{0.2em}{0ex}}u+u\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}u+C$

39. $\int {u}^{n}\text{sin}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{−}{u}^{n}\text{cos}\phantom{\rule{0.2em}{0ex}}u+n\int {u}^{n-1}\text{cos}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du$

40. $\int {u}^{n}\text{cos}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du={u}^{n}\text{sin}\phantom{\rule{0.2em}{0ex}}u-n\int {u}^{n-1}\text{sin}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du$

41. $\begin{array}{cc}\hfill \int {\text{sin}}^{n}u\phantom{\rule{0.2em}{0ex}}{\text{cos}}^{m}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du& =-\frac{{\text{sin}}^{n-1}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}{\text{cos}}^{m+1}\phantom{\rule{0.2em}{0ex}}u}{n+m}+\frac{n-1}{n+m}\int {\text{sin}}^{n-2}u\phantom{\rule{0.2em}{0ex}}{\text{cos}}^{m}u\phantom{\rule{0.2em}{0ex}}du\hfill \\ & =\frac{{\text{sin}}^{n+1}u\phantom{\rule{0.2em}{0ex}}{\text{cos}}^{m-1}u}{n+m}+\frac{m-1}{n+m}\int {\text{sin}}^{n}u\phantom{\rule{0.2em}{0ex}}{\text{cos}}^{m-2}u\phantom{\rule{0.2em}{0ex}}du\hfill \end{array}$

## Exponential and logarithmic integrals

42. $\int u{e}^{au}\phantom{\rule{0.2em}{0ex}}du=\frac{1}{{a}^{2}}\left(au-1\right){e}^{au}+C$

43. $\int {u}^{n}{e}^{au}\phantom{\rule{0.2em}{0ex}}du=\frac{1}{a}{u}^{n}{e}^{au}-\frac{n}{a}\int {u}^{n-1}{e}^{au}\phantom{\rule{0.2em}{0ex}}du$

44. $\int {e}^{au}\text{sin}\phantom{\rule{0.2em}{0ex}}bu\phantom{\rule{0.2em}{0ex}}du=\frac{{e}^{au}}{{a}^{2}+{b}^{2}}\left(a\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}bu-b\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}bu\right)+C$

45. $\int {e}^{au}\text{cos}\phantom{\rule{0.2em}{0ex}}bu\phantom{\rule{0.2em}{0ex}}du=\frac{{e}^{au}}{{a}^{2}+{b}^{2}}\left(a\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}bu+b\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}bu\right)+C$

46. $\int \text{ln}\phantom{\rule{0.1em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=u\phantom{\rule{0.2em}{0ex}}\text{ln}\phantom{\rule{0.1em}{0ex}}u-u+C$

47. $\int {u}^{n}\text{ln}\phantom{\rule{0.1em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\frac{{u}^{n+1}}{{\left(n+1\right)}^{2}}\left[\left(n+1\right)\text{ln}\phantom{\rule{0.1em}{0ex}}u-1\right]+C$

48. $\int \frac{1}{u\phantom{\rule{0.2em}{0ex}}\text{ln}\phantom{\rule{0.1em}{0ex}}u}\phantom{\rule{0.2em}{0ex}}du=\text{ln}\phantom{\rule{0.1em}{0ex}}|\text{ln}\phantom{\rule{0.1em}{0ex}}u|+C$

## Hyperbolic integrals

49. $\int \text{sinh}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{cosh}\phantom{\rule{0.2em}{0ex}}u+C$

50. $\int \text{cosh}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{sinh}\phantom{\rule{0.2em}{0ex}}u+C$

51. $\int \text{tanh}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{ln}\phantom{\rule{0.1em}{0ex}}\text{cosh}\phantom{\rule{0.2em}{0ex}}u+C$

52. $\int \text{coth}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{ln}\phantom{\rule{0.1em}{0ex}}|\text{sinh}\phantom{\rule{0.2em}{0ex}}u|+C$

53. $\int \text{sech}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du={\text{tan}}^{-1}|\text{sinh}\phantom{\rule{0.2em}{0ex}}u|+C$

54. $\int \text{csch}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{ln}\phantom{\rule{0.1em}{0ex}}|\text{tanh}\phantom{\rule{0.1em}{0ex}}\frac{1}{2}u|+C$

55. $\int {\text{sech}}^{2}u\phantom{\rule{0.2em}{0ex}}du=\text{tanh}\phantom{\rule{0.2em}{0ex}}u+C$

56. $\int {\text{csch}}^{2}u\phantom{\rule{0.2em}{0ex}}du=\text{−}\text{coth}\phantom{\rule{0.2em}{0ex}}u+C$

57. $\int \text{sech}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}\text{tanh}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{−}\text{sech}\phantom{\rule{0.2em}{0ex}}u+C$

58. $\int \text{csch}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}\text{coth}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}du=\text{−}\text{csch}\phantom{\rule{0.2em}{0ex}}u+C$

## Inverse trigonometric integrals

59. $\int {\text{sin}}^{-1}u\phantom{\rule{0.2em}{0ex}}du=u\phantom{\rule{0.2em}{0ex}}{\text{sin}}^{-1}u+\sqrt{1-{u}^{2}}+C$

60. $\int {\text{cos}}^{-1}u\phantom{\rule{0.2em}{0ex}}du=u\phantom{\rule{0.2em}{0ex}}{\text{cos}}^{-1}u-\sqrt{1-{u}^{2}}+C$

61. $\int {\text{tan}}^{-1}u\phantom{\rule{0.2em}{0ex}}du=u\phantom{\rule{0.2em}{0ex}}{\text{tan}}^{-1}u-\frac{1}{2}\text{ln}\phantom{\rule{0.1em}{0ex}}\left(1+{u}^{2}\right)+C$

62. $\int u\phantom{\rule{0.2em}{0ex}}{\text{sin}}^{-1}u\phantom{\rule{0.2em}{0ex}}du=\frac{2{u}^{2}-1}{4}{\text{sin}}^{-1}u+\frac{u\sqrt{1-{u}^{2}}}{4}+C$

63. $\int u\phantom{\rule{0.2em}{0ex}}{\text{cos}}^{-1}u\phantom{\rule{0.2em}{0ex}}du=\frac{2{u}^{2}-1}{4}{\text{cos}}^{-1}u-\frac{u\sqrt{1-{u}^{2}}}{4}+C$

64. $\int u\phantom{\rule{0.2em}{0ex}}{\text{tan}}^{-1}u\phantom{\rule{0.2em}{0ex}}du=\frac{{u}^{2}+1}{2}{\text{tan}}^{-1}u-\frac{u}{2}+C$

65. $\int {u}^{n}{\text{sin}}^{-1}u\phantom{\rule{0.2em}{0ex}}du=\frac{1}{n+1}\left[{u}^{n+1}{\text{sin}}^{-1}u-\int \frac{{u}^{n+1}\phantom{\rule{0.2em}{0ex}}du}{\sqrt{1-{u}^{2}}}\right],n\ne \text{−}1$

66. $\int {u}^{n}{\text{cos}}^{-1}u\phantom{\rule{0.2em}{0ex}}du=\frac{1}{n+1}\left[{u}^{n+1}{\text{cos}}^{-1}u+\int \frac{{u}^{n+1}\phantom{\rule{0.2em}{0ex}}du}{\sqrt{1-{u}^{2}}}\right],n\ne \text{−}1$

67. $\int {u}^{n}{\text{tan}}^{-1}u\phantom{\rule{0.2em}{0ex}}du=\frac{1}{n+1}\left[{u}^{n+1}{\text{tan}}^{-1}u-\int \frac{{u}^{n+1}\phantom{\rule{0.2em}{0ex}}du}{1+{u}^{2}}\right],n\ne \text{−}1$

## Integrals involving a2 + u2 , a >0

68. $\int \sqrt{{a}^{2}+{u}^{2}}\phantom{\rule{0.2em}{0ex}}du=\frac{u}{2}\sqrt{{a}^{2}+{u}^{2}}+\frac{{a}^{2}}{2}\text{ln}\phantom{\rule{0.1em}{0ex}}\left(u+\sqrt{{a}^{2}+{u}^{2}}\right)+C$

69. $\int {u}^{2}\sqrt{{a}^{2}+{u}^{2}}\phantom{\rule{0.2em}{0ex}}du=\frac{u}{8}\left({a}^{2}+2{u}^{2}\right)\sqrt{{a}^{2}+{u}^{2}}-\frac{{a}^{4}}{8}\text{ln}\phantom{\rule{0.1em}{0ex}}\left(u+\sqrt{{a}^{2}+{u}^{2}}\right)+C$

70. $\int \frac{\sqrt{{a}^{2}+{u}^{2}}}{u}\phantom{\rule{0.2em}{0ex}}du=\sqrt{{a}^{2}+{u}^{2}}-a\phantom{\rule{0.2em}{0ex}}\text{ln}\phantom{\rule{0.1em}{0ex}}|\frac{a+\sqrt{{a}^{2}+{u}^{2}}}{u}|+C$

71. $\int \frac{\sqrt{{a}^{2}+{u}^{2}}}{{u}^{2}}\phantom{\rule{0.2em}{0ex}}du=-\frac{\sqrt{{a}^{2}+{u}^{2}}}{u}+\text{ln}\phantom{\rule{0.1em}{0ex}}\left(u+\sqrt{{a}^{2}+{u}^{2}}\right)+C$

72. $\int \frac{\phantom{\rule{0.2em}{0ex}}du}{\sqrt{{a}^{2}+{u}^{2}}}=\text{ln}\phantom{\rule{0.1em}{0ex}}\left(u+\sqrt{{a}^{2}+{u}^{2}}\right)+C$

73. $\int \frac{{u}^{2}\phantom{\rule{0.2em}{0ex}}du}{\sqrt{{a}^{2}+{u}^{2}}}=\frac{u}{2}\left(\sqrt{{a}^{2}+{u}^{2}}\right)-\frac{{a}^{2}}{2}\text{ln}\phantom{\rule{0.1em}{0ex}}\left(u+\sqrt{{a}^{2}+{u}^{2}}\right)+C$

74. $\int \frac{\phantom{\rule{0.2em}{0ex}}du}{u\sqrt{{a}^{2}+{u}^{2}}}=-\frac{1}{a}\text{ln}\phantom{\rule{0.1em}{0ex}}|\frac{\sqrt{{a}^{2}+{u}^{2}}+a}{u}|+C$

75. $\int \frac{\phantom{\rule{0.2em}{0ex}}du}{{u}^{2}\sqrt{{a}^{2}+{u}^{2}}}=-\frac{\sqrt{{a}^{2}+{u}^{2}}}{{a}^{2}u}+C$

76. $\int \frac{\phantom{\rule{0.2em}{0ex}}du}{{\left({a}^{2}+{u}^{2}\right)}^{3\text{/}2}}=\frac{u}{{a}^{2}\sqrt{{a}^{2}+{u}^{2}}}+C$

## Integrals involving u2 − a2 , a >0

77. $\int \sqrt{{u}^{2}-{a}^{2}}\phantom{\rule{0.2em}{0ex}}du=\frac{u}{2}\sqrt{{u}^{2}-{a}^{2}}-\frac{{a}^{2}}{2}\text{ln}\phantom{\rule{0.1em}{0ex}}|u+\sqrt{{u}^{2}-{a}^{2}}|+C$

78. $\int {u}^{2}\sqrt{{u}^{2}-{a}^{2}}\phantom{\rule{0.2em}{0ex}}du=\frac{u}{8}\left(2{u}^{2}-{a}^{2}\right)\sqrt{{u}^{2}-{a}^{2}}-\frac{{a}^{4}}{8}\text{ln}\phantom{\rule{0.1em}{0ex}}|u+\sqrt{{u}^{2}-{a}^{2}}|+C$

79. $\int \frac{\sqrt{{u}^{2}-{a}^{2}}}{u}\phantom{\rule{0.2em}{0ex}}du=\sqrt{{u}^{2}-{a}^{2}}-a{\text{cos}}^{-1}\frac{a}{|u|}+C$

80. $\int \frac{\sqrt{{u}^{2}-{a}^{2}}}{{u}^{2}}\phantom{\rule{0.2em}{0ex}}du=-\frac{\sqrt{{u}^{2}-{a}^{2}}}{u}+\text{ln}\phantom{\rule{0.1em}{0ex}}|u+\sqrt{{u}^{2}-{a}^{2}}|+C$

81. $\int \frac{\phantom{\rule{0.2em}{0ex}}du}{\sqrt{{u}^{2}-{a}^{2}}}=\text{ln}\phantom{\rule{0.1em}{0ex}}|u+\sqrt{{u}^{2}-{a}^{2}}|+C$

82. $\int \frac{{u}^{2}\phantom{\rule{0.2em}{0ex}}du}{\sqrt{{u}^{2}-{a}^{2}}}=\frac{u}{2}\sqrt{{u}^{2}-{a}^{2}}+\frac{{a}^{2}}{2}\text{ln}\phantom{\rule{0.1em}{0ex}}|u+\sqrt{{u}^{2}-{a}^{2}}|+C$

83. $\int \frac{\phantom{\rule{0.2em}{0ex}}du}{{u}^{2}\sqrt{{u}^{2}-{a}^{2}}}=\frac{\sqrt{{u}^{2}-{a}^{2}}}{{a}^{2}u}+C$

84. $\int \frac{\phantom{\rule{0.2em}{0ex}}du}{{\left({u}^{2}-{a}^{2}\right)}^{3\text{/}2}}=\text{−}\frac{u}{{a}^{2}\sqrt{{u}^{2}-{a}^{2}}}+C$

## Integrals involving a2 − u2 , a >0

85. $\int \sqrt{{a}^{2}-{u}^{2}}\phantom{\rule{0.2em}{0ex}}du=\frac{u}{2}\sqrt{{a}^{2}-{u}^{2}}+\frac{{a}^{2}}{2}{\text{sin}}^{-1}\frac{u}{a}+C$

86. $\int {u}^{2}\sqrt{{a}^{2}-{u}^{2}}\phantom{\rule{0.2em}{0ex}}du=\frac{u}{8}\left(2{u}^{2}-{a}^{2}\right)\sqrt{{a}^{2}-{u}^{2}}+\frac{{a}^{4}}{8}{\text{sin}}^{-1}\frac{u}{a}+C$

87. $\int \frac{\sqrt{{a}^{2}-{u}^{2}}}{u}\phantom{\rule{0.2em}{0ex}}du=\sqrt{{a}^{2}-{u}^{2}}-a\text{ln}\phantom{\rule{0.1em}{0ex}}|\frac{a+\sqrt{{a}^{2}-{u}^{2}}}{u}|+C$

88. $\int \frac{\sqrt{{a}^{2}-{u}^{2}}}{{u}^{2}}\phantom{\rule{0.2em}{0ex}}du=-\frac{1}{u}\sqrt{{a}^{2}-{u}^{2}}-{\text{sin}}^{-1}\frac{u}{a}+C$

89. $\int \frac{{u}^{2}\phantom{\rule{0.2em}{0ex}}du}{\sqrt{{a}^{2}-{u}^{2}}}=-\frac{u}{u}\sqrt{{a}^{2}-{u}^{2}}+\frac{{a}^{2}}{2}{\text{sin}}^{-1}\frac{u}{a}+C$

90. $\int \frac{\phantom{\rule{0.2em}{0ex}}du}{u\sqrt{{a}^{2}-{u}^{2}}}=-\frac{1}{a}\text{ln}\phantom{\rule{0.1em}{0ex}}|\frac{a+\sqrt{{a}^{2}-{u}^{2}}}{u}|+C$

91. $\int \frac{\phantom{\rule{0.2em}{0ex}}du}{{u}^{2}\sqrt{{a}^{2}-{u}^{2}}}=-\frac{1}{{a}^{2}u}\sqrt{{a}^{2}-{u}^{2}}+C$

92. $\int {\left({a}^{2}-{u}^{2}\right)}^{3\text{/}2}\phantom{\rule{0.2em}{0ex}}du=-\frac{u}{8}\left(2{u}^{2}-5{a}^{2}\right)\sqrt{{a}^{2}-{u}^{2}}+\frac{3{a}^{4}}{8}{\text{sin}}^{-1}\frac{u}{a}+C$

93. $\int \frac{\phantom{\rule{0.2em}{0ex}}du}{{\left({a}^{2}-{u}^{2}\right)}^{3\text{/}2}}=-\frac{u}{{a}^{2}\sqrt{{a}^{2}-{u}^{2}}}+C$

## Integrals involving 2 au − u2 , a >0

94. $\int \sqrt{2au-{u}^{2}}\phantom{\rule{0.2em}{0ex}}du=\frac{u-a}{2}\sqrt{2au-{u}^{2}}+\frac{{a}^{2}}{2}{\text{cos}}^{-1}\left(\frac{a-u}{a}\right)+C$

95. $\int \frac{\phantom{\rule{0.2em}{0ex}}du}{\sqrt{2au-{u}^{2}}}={\text{cos}}^{-1}\left(\frac{a-u}{a}\right)+C$

96. $\int u\sqrt{2au-{u}^{2}}\phantom{\rule{0.2em}{0ex}}du=\frac{2{u}^{2}-au-3{a}^{2}}{6}\sqrt{2au-{u}^{2}}+\frac{{a}^{3}}{2}{\text{cos}}^{-1}\left(\frac{a-u}{a}\right)+C$

97. $\int \frac{\phantom{\rule{0.2em}{0ex}}du}{u\sqrt{2au-{u}^{2}}}=-\frac{\sqrt{2au-{u}^{2}}}{au}+C$

## Integrals involving a + bu , a ≠ 0

98. $\int \frac{u\phantom{\rule{0.2em}{0ex}}du}{a+bu}=\frac{1}{{b}^{2}}\left(a+bu-a\text{ln}\phantom{\rule{0.1em}{0ex}}|a+bu|\right)+C$

99. $\int \frac{{u}^{2}\phantom{\rule{0.2em}{0ex}}du}{a+bu}=\frac{1}{2{b}^{3}}\left[{\left(a+bu\right)}^{2}-4a\left(a+bu\right)+2{a}^{2}\text{ln}\phantom{\rule{0.1em}{0ex}}|a+bu|\right]+C$

100. $\int \frac{\phantom{\rule{0.2em}{0ex}}du}{u\left(a+bu\right)}=\frac{1}{a}\text{ln}\phantom{\rule{0.1em}{0ex}}|\frac{u}{a+bu}|+C$

101. $\int \frac{\phantom{\rule{0.2em}{0ex}}du}{{u}^{2}\left(a+bu\right)}=-\frac{1}{au}+\frac{b}{{a}^{2}}\text{ln}\phantom{\rule{0.1em}{0ex}}|\frac{a+bu}{u}|+C$

102. $\int \frac{u\phantom{\rule{0.2em}{0ex}}du}{{\left(a+bu\right)}^{2}}=\frac{a}{{b}^{2}\left(a+bu\right)}+\frac{1}{{b}^{2}}\text{ln}\phantom{\rule{0.1em}{0ex}}|a+bu|+C$

103. $\int \frac{u\phantom{\rule{0.2em}{0ex}}du}{u\phantom{\rule{0.2em}{0ex}}{\left(a+bu\right)}^{2}}=\frac{1}{a\left(a+bu\right)}-\frac{1}{{a}^{2}}\text{ln}\phantom{\rule{0.1em}{0ex}}|\frac{a+bu}{u}|+C$

104. $\int \frac{{u}^{2}\phantom{\rule{0.2em}{0ex}}du}{{\left(a+bu\right)}^{2}}=\frac{1}{{b}^{3}}\left(a+bu-\frac{{a}^{2}}{a+bu}-2a\text{ln}\phantom{\rule{0.1em}{0ex}}|a+bu|\right)+C$

105. $\int u\sqrt{a+bu}\phantom{\rule{0.2em}{0ex}}du=\frac{2}{15{b}^{2}}\left(3bu-2a\right){\left(a+bu\right)}^{3\text{/}2}+C$

106. $\int \frac{u\phantom{\rule{0.2em}{0ex}}du}{\sqrt{a+bu}}=\frac{2}{3{b}^{2}}\left(bu-2a\right)\sqrt{a+bu}+C$

107. $\int \frac{{u}^{2}\phantom{\rule{0.2em}{0ex}}du}{\sqrt{a+bu}}=\frac{2}{15{b}^{3}}\left(8{a}^{2}+3{b}^{2}{u}^{2}-4abu\right)\sqrt{a+bu}+C$

108. $\begin{array}{ccc}\hfill \int \frac{\phantom{\rule{0.2em}{0ex}}du}{u\sqrt{a+bu}}& =\frac{1}{\sqrt{a}}\text{ln}\phantom{\rule{0.1em}{0ex}}|\frac{\sqrt{a+bu}-\sqrt{a}}{\sqrt{a+bu}+\sqrt{a}}|+C,\hfill & \text{if}\phantom{\rule{0.2em}{0ex}}a>0\hfill \\ & =\frac{2}{\sqrt{\text{−}a}}\text{tan}-1\sqrt{\frac{a+bu}{\text{−}a}}+C,\hfill & \text{if}\phantom{\rule{0.2em}{0ex}}a<0\hfill \end{array}$

109. $\int \frac{\sqrt{a+bu}}{u}\phantom{\rule{0.2em}{0ex}}du=2\sqrt{a+bu}+a\int \frac{\phantom{\rule{0.2em}{0ex}}du}{u\sqrt{a+bu}}$

110. $\int \frac{\sqrt{a+bu}}{{u}^{2}}\phantom{\rule{0.2em}{0ex}}du=-\frac{\sqrt{a+bu}}{u}+\frac{b}{2}\int \frac{\phantom{\rule{0.2em}{0ex}}du}{u\sqrt{a+bu}}$

111. $\int {u}^{n}\sqrt{a+bu}\phantom{\rule{0.2em}{0ex}}du=\frac{2}{b\left(2n+3\right)}\left[{u}^{n}{\left(a+bu\right)}^{3\text{/}2}-na\int {u}^{n-1}\sqrt{a+bu}\phantom{\rule{0.2em}{0ex}}du\right]$

112. $\int \frac{{u}^{n}\phantom{\rule{0.2em}{0ex}}du}{\sqrt{a+bu}}=\frac{2{u}^{n}\sqrt{a+bu}}{b\left(2n+1\right)}-\frac{2na}{b\left(2n+1\right)}\int \frac{{u}^{n-1}\phantom{\rule{0.2em}{0ex}}du}{\sqrt{a+bu}}$

113. $\int \frac{\phantom{\rule{0.2em}{0ex}}du}{{u}^{n}\sqrt{a+bu}}=-\frac{\sqrt{a+bu}}{a\left(n-1\right){u}^{n-1}}-\frac{b\left(2n-3\right)}{2a\left(n-1\right)}\int \frac{\phantom{\rule{0.2em}{0ex}}du}{{u}^{n-1}\sqrt{a+bu}}$

what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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Damian
yes that's correct
Professor
I think
Professor
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LITNING
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write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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what king of growth are you checking .?
Renato
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?
Kyle
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why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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research.net
kanaga
sciencedirect big data base
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Introduction about quantum dots in nanotechnology
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absolutely yes
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Privacy Information Security Software Version 1.1a
Good
Leaves accumulate on the forest floor at a rate of 2 g/cm2/yr and also decompose at a rate of 90% per year. Write a differential equation governing the number of grams of leaf litter per square centimeter of forest floor, assuming at time 0 there is no leaf litter on the ground. Does this amount approach a steady value? What is that value?
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Abdul