日期: 2024-04-06 14:12 查看: 28 次编辑导出本文

附录积分表

1.$\int{{{x}^{n}}dx=\frac{{{x}^{n+1}}}{n+1}}+C,(n\ne 1)$

2.$\int{kdx=kx+C}$

3.$\int{\frac{dx}{x}=\ln }\left| x \right|+C$

4.$\int{{{e}^{x}}dx={{e}^{x}}+C}$

5.$\int{{{a}^{x}}dx=\frac{{{a}^{x}}}{\ln a}+C},(a>0,a\ne 1)$

6.$\int{\sin xdx=-\cos x+C}$

7.$\int{\cos xdx=\sin x+C}$

8.$\int{{{\sec }^{2}}xdx=\tan x+C}$

9.$\int{{{\csc }^{2}}xdx=-\cot x+C}$

10.$\int{\sec x\tan xdx=\sec x+C}$

11.$\int{\csc x\cot xdx=-\csc x+C}$

12.$\int{\tan xdx=\ln \left| \sec x \right|}+C$

13.$\int{\cot xdx=\ln \left| \sin x \right|}+C$

14.$\int{\sinh xdx=\cosh x}+C$

15.$\int{\cosh xdx=\sinh x}+C$

16.$\int{\frac{dx}{\sqrt{{{a}^{2}}-{{x}^{2}}}}}={{\sin }^{-1}}\frac{x}{a}+C$

17.$\int{\frac{dx}{{{a}^{2}}+{{x}^{2}}}}=\frac{1}{a}{{\tan }^{-1}}\frac{x}{a}+C$

18.$\int{\frac{dx}{x\sqrt{{{x}^{2}}-{{a}^{2}}}}}=\frac{1}{a}{{\sec }^{-1}}\left| \frac{x}{a} \right|+C$

19.$\int{\frac{dx}{\sqrt{{{a}^{2}}+{{x}^{2}}}}}={{\sinh }^{-1}}\frac{x}{a}+C$

20.$\int{\frac{dx}{\sqrt{{{x}^{2}}+{{a}^{2}}}}}={{\cosh }^{-1}}\frac{x}{a}+C$

21.$\int{{{(ax+b)}^{n}}dx=\frac{{{(ax+b)}^{n+1}}}{a(n+1)}}+C,n\ne -1$

22.$\int{x{{(ax+b)}^{n}}dx=\frac{{{(ax+b)}^{n+1}}}{{{a}^{2}}}}\left[ \frac{ax+b}{n+2}-\frac{b}{n+1} 23.\right]+C,n\ne -1,-2$

24.$\int{{{(ax+b)}^{-1}}dx=\frac{1}{a}}\ln \left| ax+b \right|+C$

25.$\int{x{{(ax+b)}^{-1}}dx=\frac{x}{a}}-\frac{b}{{{a}^{2}}}\ln \left| ax+b \right|+C$

26.$\int{x{{(ax+b)}^{-2}}dx=\frac{1}{{{a}^{2}}}}\left[ \ln \left| ax+b \right|+\frac{b}{ax+b} \right]+C$

27.$\int{\frac{dx}{x(ax+b)}=\frac{1}{b}}\left[ \ln \frac{x}{ax+b} \right]+C$

28.$\int{{{(\sqrt{ax+b})}^{n}}dx=\frac{2}{a}}\frac{({{\sqrt{ax+b)}}^{n+2}}}{n+2}+C,n\ne -2$

$\int{\frac{\sqrt{ax+b}}{x}dx=2}\sqrt{ax+b}+b\int{-\frac{dx}{x\sqrt{ax+b}}}$

29.(a)$\int{\frac{dx}{x\sqrt{ax+b}}=\frac{1}{\sqrt{b}}\ln }\left| \frac{\sqrt{ax+b}-\sqrt{b}}{\sqrt{ax+b}+\sqrt{b}} \right|+C$

(b)$\int{\frac{dx}{x\sqrt{ax-b}}=\frac{2}{\sqrt{b}}{{\tan }^{-1}}}\frac{\sqrt{ax-b}}{b}+C$

30.$\int{\frac{\sqrt{ax+b}}{{{x}^{2}}}}dx=-\frac{\sqrt{ax+b}}{x}+\frac{a}{2}\int{\frac{dx}{x\sqrt{ax+b}}+C}$

31.

$\int{\frac{dx}{{{x}^{2}}\sqrt{ax+b}}}=-\frac{\sqrt{ax+b}}{bx}-\frac{a}{2b}\int{\frac{dx}{x\sqrt{ax+b}}+C}$

32.$\int{\frac{dx}{{{a}^{2}}+{{x}^{2}}}=\frac{1}{a}}{{\tan }^{-1}}\frac{x}{a}+C$

33.$\int{\frac{dx}{{{({{a}^{2}}+{{x}^{2}})}^{2}}}=\frac{x}{2{{a}^{2}}({{a}^{2}}+{{x}^{2}})}}+\frac{1}{2{{a}^{3}}}{{\tan }^{-1}}\frac{x}{a}+C$

34.$\int{\frac{dx}{\sqrt{{{a}^{2}}+{{x}^{2}}}}={{\sinh }^{-1}}\frac{x}{a}}+C=\ln \left( x+\sqrt{{{a}^{2}}+{{x}^{2}}} \right)+C$

35.$\int{\sqrt{{{a}^{2}}+{{x}^{2}}}}dx=\frac{x}{2}\sqrt{{{a}^{2}}+{{x}^{2}}}+\frac{{{a}^{2}}}{2}\ln x\left( +\sqrt{{{a}^{2}}+{{x}^{2}}} \right)+C$

36.$\int{{{x}^{2}}\sqrt{{{a}^{2}}+{{x}^{2}}}}dx=\frac{x}{8}\left( {{a}^{2}}+2{{x}^{2}} \right)\sqrt{{{a}^{2}}+{{x}^{2}}}-\frac{{{a}^{4}}}{8}\ln \left( x+\sqrt{{{a}^{2}}+{{x}^{2}}} \right)+C$

37.$\int{\frac{\sqrt{{{a}^{2}}+{{x}^{2}}}}{x}}dx=\sqrt{{{a}^{2}}+{{x}^{2}}}-a\ln \left| \frac{a+\sqrt{{{a}^{2}}+{{x}^{2}}}}{x} \right|+C$

38.$\int{\frac{\sqrt{{{a}^{2}}+{{x}^{2}}}}{{{x}^{2}}}}dx=\ln \left( x+\sqrt{{{a}^{2}}+{{x}^{2}}} \right)-\frac{\sqrt{{{a}^{2}}+{{x}^{2}}}}{x}+C$

39.$\int{\frac{{{x}^{2}}}{\sqrt{{{a}^{2}}+{{x}^{2}}}}}dx=-\frac{{{a}^{2}}}{2}\ln \left( x+\sqrt{{{a}^{2}}+{{x}^{2}}} \right)+\frac{x\sqrt{{{a}^{2}}+{{x}^{2}}}}{2}+C$

40.$\int{\frac{dx}{x\sqrt{{{a}^{2}}+{{x}^{2}}}}}=-\frac{1}{a}\ln \left| \frac{a+\sqrt{{{a}^{2}}+{{x}^{2}}}}{x} \right|+C$

41.$\int{\frac{dx}{{{x}^{2}}\sqrt{{{a}^{2}}+{{x}^{2}}}}}=-\frac{\sqrt{{{a}^{2}}+{{x}^{2}}}}{{{a}^{2}}x}+C$

42. $\int{\frac{dx}{{{a}^{2}}-{{x}^{2}}}}=\frac{1}{2a}\ln \left| \frac{x+a}{x-a} \right|+C$

43. $\int{\frac{dx}{{{\left( {{a}^{2}}-{{x}^{2}} \right)}^{2}}}}=\frac{x}{2{{a}^{2}}({{a}^{2}}-{{x}^{2}})}+\frac{1}{4{{a}^{3}}}\ln \left| \frac{x+a}{x-a} \right|+C$

44. $\int{\frac{dx}{\sqrt{{{a}^{2}}-{{x}^{2}}}}}={{\sin }^{-1}}\frac{x}{a}+C$

45. $\int{\sqrt{{{a}^{2}}-{{x}^{2}}}}dx=\frac{x}{2}\sqrt{{{a}^{2}}-{{x}^{2}}}+\frac{{{a}^{2}}}{2}{{\sin }^{-1}}\frac{x}{a}+C$

46. $\int{{{x}^{2}}\sqrt{{{a}^{2}}-{{x}^{2}}}}dx=\frac{{{a}^{4}}}{8}{{\sin }^{-1}}\frac{x}{a}-\frac{1}{8}x\sqrt{{{a}^{2}}-{{x}^{2}}}\left( {{a}^{2}}-2{{x}^{2}} \right)+C$

47. $\int{\frac{\sqrt{{{a}^{2}}-{{x}^{2}}}}{x}}dx=\sqrt{{{a}^{2}}-{{x}^{2}}}-a\ln \left| \frac{a+\sqrt{{{a}^{2}}-{{x}^{2}}}}{x} \right|+C$

48. $\int{\frac{\sqrt{{{a}^{2}}-{{x}^{2}}}}{{{x}^{2}}}}dx=-{{\sin }^{-1}}\frac{x}{a}-\frac{\sqrt{{{a}^{2}}-{{x}^{2}}}}{x}+C$

49. $\int{\frac{{{x}^{2}}}{\sqrt{{{a}^{2}}-{{x}^{2}}}}}dx=\frac{{{a}^{2}}}{2}{{\sin }^{-1}}\frac{x}{a}-\frac{1}{2}x\sqrt{{{a}^{2}}-{{x}^{2}}}+C$

50. $\int{\frac{dx}{x\sqrt{{{a}^{2}}-{{x}^{2}}}}}=-\frac{1}{a}\ln \left| \frac{a+\sqrt{{{a}^{2}}-{{x}^{2}}}}{x} \right|+C$

51. $\int{\frac{dx}{{{x}^{2}}\sqrt{{{a}^{2}}-{{x}^{2}}}}}=-\frac{\sqrt{{{a}^{2}}-{{x}^{2}}}}{{{a}^{2}}x}+C$

52. $\int{\frac{dx}{\sqrt{{{x}^{2}}-{{a}^{2}}}}}=\ln \left| x+\sqrt{{{x}^{2}}-{{a}^{2}}} \right|+C$

53. $\int{\sqrt{{{x}^{2}}-{{a}^{2}}}dx}=\frac{x}{2}\sqrt{{{x}^{2}}-{{a}^{2}}}-\frac{{{a}^{2}}}{2}\ln \left| x+\sqrt{{{x}^{2}}-{{a}^{2}}} \right|+C$

54. $\int{{{\left( \sqrt{{{x}^{2}}-{{a}^{2}}} \right)}^{n}}dx}=\frac{x{{\left( \sqrt{{{x}^{2}}-{{a}^{2}}} \right)}^{n}}}{n+1}-\frac{n{{a}^{2}}}{n+1}\int{{{\left( \sqrt{{{x}^{2}}-{{a}^{2}}} \right)}^{n-2}}dx,(n\ne -1)}$

55. $\int{\frac{dx}{{{\left( \sqrt{{{x}^{2}}-{{a}^{2}}} \right)}^{n}}}}=\frac{x{{\left( \sqrt{{{x}^{2}}-{{a}^{2}}} \right)}^{2-n}}}{\left( 2-n \right){{a}^{2}}}-\frac{n-3}{\left( n-2 \right){{a}^{2}}}\int{\frac{dx}{{{\left( \sqrt{{{x}^{2}}-{{a}^{2}}} \right)}^{n-2}}},(n\ne 2)}$

56. $\int{x{{\left( \sqrt{{{x}^{2}}-{{a}^{2}}} \right)}^{n}}}dx=\frac{{{\left( \sqrt{{{x}^{2}}-{{a}^{2}}} \right)}^{n+2}}}{n+2}+C,n\ne -2$

57. $\int{{{x}^{2}}\sqrt{{{x}^{2}}-{{a}^{2}}}}dx=\frac{x}{8}\left( 2{{x}^{2}}-{{a}^{2}} \right)\sqrt{{{x}^{2}}-{{a}^{2}}}-\frac{{{a}^{4}}}{8}\ln \left| x+\sqrt{{{x}^{2}}-{{a}^{2}}} \right|+C$

58. $\int{\frac{\sqrt{{{x}^{2}}-{{a}^{2}}}}{x}}dx=\left( {{x}^{2}}-{{a}^{2}} \right)-a{{\sec }^{-1}}\left| \frac{x}{a} \right|+C$

59. $\int{\frac{\sqrt{{{x}^{2}}-{{a}^{2}}}}{{{x}^{2}}}}dx=\ln \left| x+\sqrt{{{x}^{2}}-{{a}^{2}}} \right|-\frac{\sqrt{{{x}^{2}}-{{a}^{2}}}}{x}+C$

60. $\int{\frac{{{x}^{2}}}{\sqrt{{{x}^{2}}-{{a}^{2}}}}}dx=\frac{{{a}^{2}}}{2}\ln \left| x+\sqrt{{{x}^{2}}-{{a}^{2}}} \right|+\frac{x}{2}\sqrt{{{x}^{2}}-{{a}^{2}}}+C$

61. $\int{\frac{dx}{x\sqrt{{{x}^{2}}-{{a}^{2}}}}}=\frac{1}{a}{{\sec }^{-1}}\left| \frac{x}{a} \right|+C=\frac{1}{a}{{\cos }^{-1}}\left| \frac{x}{a} \right|+C$

62. $\int{\frac{dx}{{{x}^{2}}\sqrt{{{x}^{2}}-{{a}^{2}}}}}=\frac{\sqrt{{{x}^{2}}-{{a}^{2}}}}{{{a}^{2}}x}+C$

63. $\int{\sin ax}dx=-\frac{1}{a}\cos ax+C$

64. $\int{\cos ax}dx=\frac{1}{a}\sin ax+C$

65. $\int{{{\sin }^{2}}ax}dx=\frac{x}{2}-\frac{\sin 2ax}{4a}+C$

66. $\int{{{\cos }^{2}}ax}dx=\frac{x}{2}+\frac{\sin 2ax}{4a}+C$

67. $\int{{{\sin }^{n}}ax}dx=-\frac{{{\sin }^{n-1}}ax\cos ax}{na}+\frac{n-1}{n}\int{{{\sin }^{n-2}}ax}dx$

68. $\int{{{\cos }^{n}}ax}dx=\frac{{{\cos }^{n-1}}ax\sin ax}{na}+\frac{n-1}{n}\int{{{\cos }^{n-2}}ax}dx$

69. (a)$\int{\sin ax\cos bx}dx=-\frac{\cos (a+b)x}{2(a+b)}-\frac{\cos (a-b)x}{2(a-b)}+C,{{a}^{2}}\ne {{b}^{2}}$

(b) $\int{\sin ax\sin bx}dx=-\frac{\sin (a-b)x}{2(a-b)}-\frac{\sin (a+b)x}{2(a+b)}+C,{{a}^{2}}\ne {{b}^{2}}$

70. $\int{\sin ax\cos ax}dx=-\frac{\cos 2ax}{4a}+C$

71. $\int{{{\sin }^{n}}ax\cos ax}dx=\frac{{{\sin }^{n+1}}ax}{(n+1)a}+C,n\ne 1$

72. $\int{\frac{\cos ax}{\sin ax}}dx=\frac{1}{a}\ln \left| \sin ax \right|+C$

73. $\int{{{\cos }^{n}}ax\sin ax}dx=-\frac{{{\cos }^{n+1}}ax}{(n+1)a}+C,n\ne 1$

74. $\int{\frac{\sin ax}{\cos ax}}dx=-\frac{1}{a}\ln \left| \cos ax \right|+C$

75. $\int{{{\sin }^{n}}ax{{\cos }^{m}}ax}dx=-\frac{{{\sin }^{n-1}}ax{{\cos }^{m+1}}ax}{a(m+n)}+\frac{n-1}{m+n}\int{{{\sin }^{n-2}}ax{{\cos }^{m}}ax}dx,n\ne m$,(reduces ${{\sin }^{n}}ax$)

76. $\int{{{\sin }^{n}}ax{{\cos }^{m}}ax}dx=-\frac{{{\sin }^{n+1}}ax{{\cos }^{m-1}}ax}{a(m+n)}+\frac{m-1}{m+n}\int{{{\sin }^{n}}ax{{\cos }^{m-2}}ax}dx,n\ne m$,(reduces ${{\cos }^{m}}ax$)

77. $\int{\frac{dx}{b+c\sin ax}=\frac{-2}{a\sqrt{{{b}^{2}}-{{c}^{2}}}}}{{\tan }^{-1}}\left[ \sqrt{\frac{b-c}{b+c}}\tan \left( \frac{\pi }{4}-\frac{ax}{2} \right) \right]+C,{{b}^{2}}>{{c}^{2}}$

78. $\int{\frac{dx}{b+c\sin ax}=\frac{-1}{a\sqrt{{{c}^{2}}-{{b}^{2}}}}}\ln \left| \frac{c+b\sin ax+\sqrt{{{c}^{2}}-{{b}^{2}}}\cos ax}{b+c\sin ax} \right|+C,{{b}^{2}}<{{c}^{2}}$

79. $\int{\frac{dx}{1+\sin ax}=-\frac{1}{a}\tan }\left( \frac{\pi }{4}-\frac{ax}{2} \right)+C$

80. $\int{\frac{dx}{1-\sin ax}=\frac{1}{a}\tan }\left( \frac{\pi }{4}+\frac{ax}{2} \right)+C$

81. $\int{\frac{dx}{b+c\cos ax}=\frac{2}{a\sqrt{{{b}^{2}}-{{c}^{2}}}}{{\tan }^{-1}}}\left( \sqrt{\frac{b-c}{b+c}}\tan \frac{ax}{2} \right)+C,{{b}^{2}}>{{c}^{2}}$

82. $\int{\frac{dx}{b+c\cos ax}=\frac{1}{a\sqrt{{{c}^{2}}-{{b}^{2}}}}}\ln \left| \frac{c+b\cos ax+\sqrt{{{c}^{2}}-{{b}^{2}}}\sin ax}{b+c\cos ax} \right|+C,{{c}^{2}}>{{b}^{2}}$

83. $\int{\frac{dx}{1+\cos ax}=\frac{1}{a}}\tan \frac{ax}{2}+C$

84. $\int{\frac{dx}{1-\cos ax}=-\frac{1}{a}}\cot \frac{ax}{2}+C$

85. $\int{x\sin ax}dx=\frac{1}{{{a}^{2}}}\sin ax-\frac{x}{a}\cos ax+C$

86. $\int{x\cos ax}dx=\frac{1}{{{a}^{2}}}\cos ax+\frac{x}{a}\sin ax+C$

87. $\int{{{x}^{n}}\sin ax}dx=-\frac{{{x}^{n}}}{a}\cos ax+\frac{n}{a}\int{{{x}^{n-1}}\cos ax}dx$

88. $\int{{{x}^{n}}\cos ax}dx=\frac{{{x}^{n}}}{a}\sin ax-\frac{n}{a}\int{{{x}^{n-1}}\sin ax}dx$

89. $\int{\tan ax}dx=\frac{1}{a}\ln \left| \sec ax \right|+C$

90. $\int{\cot ax}dx=\frac{1}{a}\ln \left| \sin ax \right|+C$

91. $\int{{{\tan }^{2}}ax}dx=\frac{1}{a}\tan ax-x+C$

92. $\int{{{\cot }^{2}}ax}dx=-\frac{1}{a}\cot ax-x+C$

93. $\int{{{\tan }^{n}}ax}dx=\frac{{{\tan }^{n-1}}ax}{a(n-1)}-\int{{{\tan }^{n-2}}ax}dx,n\ne 1$

94. $\int{{{\cot }^{n}}ax}dx=\frac{{{\cot }^{n-1}}ax}{a(n-1)}-\int{{{\cot }^{n-2}}ax}dx,n\ne 1$

95. $\int{\sec ax}dx=\frac{1}{a}\ln \left| \sec ax+\tan ax \right|+C$

96. $\int{\csc ax}dx=-\frac{1}{a}\ln \left| \csc ax+\cot ax \right|+C$

97. $\int{{{\sec }^{2}}ax}dx=\frac{1}{a}\tan ax+C$

98. $\int{{{\csc }^{2}}ax}dx=-\frac{1}{a}\cot ax+C$

99. $\int{{{\sec }^{n}}ax}dx=\frac{{{\sec }^{n-2}}ax\tan ax}{a(n-1)}+\frac{n-2}{n-1}\int{{{\sec }^{n-2}}ax}dx,n\ne 1$

100. $\int{{{\csc }^{n}}ax}dx=-\frac{{{\csc }^{n-2}}ax\cot ax}{a(n-1)}+\frac{n-2}{n-1}\int{{{\csc }^{n-2}}ax}dx,n\ne 1$

101. $\int{{{\sec }^{n}}ax\tan ax}dx=\frac{{{\sec }^{n}}ax}{na}+C,n\ne 0$

102. $\int{{{\csc }^{n}}ax\cot ax}dx=-\frac{{{\csc }^{n}}ax}{na}+C,n\ne 0$

103. $\int{{{\sin }^{-1}}ax}dx=x{{\sin }^{-1}}ax+\frac{1}{a}\sqrt{1-{{a}^{2}}{{x}^{2}}}+C$

104. $\int{{{\cos }^{-1}}ax}dx=x{{\cos }^{-1}}ax-\frac{1}{a}\sqrt{1-{{a}^{2}}{{x}^{2}}}+C$

105. $\int{{{\tan }^{-1}}ax}dx=x{{\tan }^{-1}}ax-\frac{1}{2a}\ln \left( 1+{{a}^{2}}{{x}^{2}} \right)+C$

106. $\int{{{x}^{n}}{{\sin }^{-1}}ax}dx=\frac{{{x}^{n+1}}}{n+1}{{\sin }^{-1}}ax-\frac{a}{n+1}\int{\frac{{{x}^{n+1}}dx}{\sqrt{1-{{a}^{2}}{{x}^{2}}}},n\ne -1}$

107. $\int{{{x}^{n}}{{\cos }^{-1}}ax}dx=\frac{{{x}^{n+1}}}{n+1}{{\cos }^{-1}}ax+\frac{a}{n+1}\int{\frac{{{x}^{n+1}}dx}{\sqrt{1-{{a}^{2}}{{x}^{2}}}},n\ne -1}$

108. $\int{{{x}^{n}}{{\tan }^{-1}}ax}dx=\frac{{{x}^{n+1}}}{n+1}{{\tan }^{-1}}ax-\frac{a}{n+1}\int{\frac{{{x}^{n+1}}dx}{1+{{a}^{2}}{{x}^{2}}},n\ne -1}$

109. $\int{{{e}^{ax}}dx}=\frac{1}{a}{{e}^{ax}}+C$

110. $\int{{{b}^{ax}}dx}=\frac{1}{a}\frac{{{b}^{ax}}}{\ln b}+C,(b>0,b\ne 1)$

111. $\int{x{{e}^{ax}}dx}=\frac{{{e}^{ax}}}{{{a}^{2}}}(ax-1)+C$

112. $\int{{{x}^{n}}{{e}^{ax}}dx}=\frac{1}{a}{{x}^{n}}{{e}^{ax}}-\frac{n}{a}\int{{{x}^{n-1}}{{e}^{ax}}}dx$

113. $\int{{{x}^{n}}{{b}^{ax}}dx}=\frac{{{x}^{n}}{{b}^{ax}}}{a\ln b}-\frac{n}{a\ln b}\int{{{x}^{n-1}}{{b}^{ax}}}dx,(b>0,b\ne 1)$

114. $\int{{{e}^{ax}}}\sin bxdx=\frac{{{e}^{ax}}}{{{a}^{2}}+{{b}^{2}}}(a\sin bx-b\cos bx)+C$

115. $\int{{{e}^{ax}}}\cos bxdx=\frac{{{e}^{ax}}}{{{a}^{2}}+{{b}^{2}}}(a\cos bx+b\sin bx)+C$

116. $\int{\ln ax}dx=x\ln ax-x+C$

117. $\int{{{x}^{n}}{{\left( \ln ax \right)}^{m}}}dx=\frac{{{x}^{n+1}}{{\left( \ln ax \right)}^{m}}}{n+1}-\frac{m}{n+1}\int{{{x}^{n}}{{\left( \ln ax \right)}^{m-1}}}dx,n\ne -1$

118. $\int{{{x}^{-1}}{{\left( \ln ax \right)}^{m}}}dx=\frac{{{\left( \ln ax \right)}^{m+1}}}{m+1}+C,m\ne -1$

119. $\int{\frac{dx}{x\ln ax}=\ln \left| \ln ax \right|}+C$

120. $\int{\frac{dx}{\sqrt{2ax-{{x}^{2}}}}}={{\sin }^{-1}}\left( \frac{x-a}{a} \right)+C$

121. $\int{\sqrt{2ax-{{x}^{2}}}}dx=\frac{x-a}{2}\sqrt{2ax-{{x}^{2}}}+\frac{{{a}^{2}}}{2}{{\sin }^{-1}}\left( \frac{x-a}{a} \right)+C$

122. $\int{{{\left( \sqrt{2ax-{{x}^{2}}} \right)}^{n}}}dx=\frac{\left( x-a \right){{\left( \sqrt{2ax-{{x}^{2}}} \right)}^{n}}}{n+1}+\frac{n{{a}^{2}}}{n+1}\int{{{\left( \sqrt{2ax-{{x}^{2}}} \right)}^{n-2}}}dx$

123. $\int{\frac{dx}{{{\left( \sqrt{2ax-{{x}^{2}}} \right)}^{n}}}}=\frac{\left( x-a \right){{\left( \sqrt{2ax-{{x}^{2}}} \right)}^{2-n}}}{\left( n-2 \right){{a}^{2}}}+\frac{n-3}{\left( n-2 \right){{a}^{2}}}\int{\frac{dx}{{{\left( \sqrt{2ax-{{x}^{2}}} \right)}^{n-2}}}}$

124. $\int{x\sqrt{2ax-{{x}^{2}}}}dx=\frac{\left( x+a \right)\left( 2x-3a \right)\sqrt{2ax-{{x}^{2}}}}{6}+\frac{{{a}^{3}}}{2}{{\sin }^{-1}}\left( \frac{x-a}{a} \right)+C$

125. $\int{\frac{\sqrt{2ax-{{x}^{2}}}}{x}}dx=\sqrt{2ax-{{x}^{2}}}+a{{\sin }^{-1}}\left( \frac{x-a}{a} \right)+C$

126. $\int{\frac{\sqrt{2ax-{{x}^{2}}}}{{{x}^{2}}}}dx=-2\sqrt{\frac{2a-x}{x}}-{{\sin }^{-1}}\left( \frac{x-a}{a} \right)+C$

127．$\int{\frac{xdx}{\sqrt{2ax-{{x}^{2}}}}}=a{{\sin }^{-1}}\left( \frac{x-a}{a} \right)-\sqrt{2ax-{{x}^{2}}}+C$

128． $\int{\frac{dx}{x\sqrt{2ax-{{x}^{2}}}}}=-\frac{1}{a}\sqrt{\frac{2a-x}{x}}+C$

129. $\int{\sinh }axdx=\frac{1}{a}\cosh ax+C$

130. $\int{\cosh }axdx=\frac{1}{a}\sinh ax+C$

131. $\int{{{\sinh }^{2}}}axdx=\frac{\sinh 2ax}{4a}-\frac{x}{2}+C$

132. $\int{{{\cosh }^{2}}}axdx=\frac{\sinh 2ax}{4a}+\frac{x}{2}+C$

133. $\int{{{\sinh }^{n}}}axdx=\frac{{{\sinh }^{n-1}}ax\cosh ax}{na}-\frac{n-1}{n}\int{{{\sinh }^{n-2}}axdx,n\ne 0}$

134. $\int{{{\cosh }^{n}}}axdx=\frac{{{\cosh }^{n-1}}ax\sinh ax}{na}+\frac{n-1}{n}\int{{{\cosh }^{n-2}}axdx,n\ne 0}$

135. $\int{x\sinh }axdx=\frac{x}{a}\cosh ax-\frac{1}{{{a}^{2}}}\sinh ax+C$

136. $\int{x\cosh }axdx=\frac{x}{a}\sinh ax-\frac{1}{{{a}^{2}}}\cosh ax+C$

137. $\int{{{x}^{n}}\sinh }axdx=\frac{{{x}^{n}}}{a}\cosh ax-\frac{n}{a}\int{{{x}^{n-1}}\cosh axdx}$

138. $\int{{{x}^{n}}\cosh }axdx=\frac{{{x}^{n}}}{a}\sinh ax-\frac{n}{a}\int{{{x}^{n-1}}\sinh axdx}$

139. $\int{\tanh axdx}=\frac{1}{a}\ln \left( \cosh ax \right)+C$

140. $\int{\coth axdx}=\frac{1}{a}\ln \left| \sinh ax \right|+C$

141. $\int{{{\tanh }^{2}}axdx}=x-\frac{1}{a}\tanh ax+C$

142. $\int{{{\coth }^{2}}axdx}=x-\frac{1}{a}\coth ax+C$

143. $\int{{{\tanh }^{n}}axdx}=-\frac{{{\tanh }^{n-1}}ax}{(n-1)a}+\int{{{\tanh }^{n-2}}axdx},n\ne 1$

144. $\int{{{\coth }^{n}}axdx}=-\frac{{{\coth }^{n-1}}ax}{(n-1)a}+\int{{{\coth }^{n-2}}axdx},n\ne 1$

145. $\int{\operatorname{sech}axdx}=-\frac{1}{a}{{\sin }^{-1}}\left( \tanh ax \right)+C$

146. $\int{\operatorname{csch}axdx}=\frac{1}{a}\ln \left| \tanh \frac{ax}{2} \right|+C$

147. $\int{{{\operatorname{sech}}^{2}}axdx}=\frac{1}{a}\tanh ax+C$

148. $\int{{{\operatorname{csch}}^{2}}axdx}=-\frac{1}{a}\coth ax+C$

149. $\int{{{\operatorname{sech}}^{n}}axdx}=\frac{{{\operatorname{sech}}^{n-2}}ax\tanh ax}{(n-1)a}+\frac{n-2}{n-1}\int{{{\operatorname{sech}}^{n-2}}axdx,n\ne 1}$

150. $\int{{{\operatorname{csch}}^{n}}axdx}=\frac{{{\operatorname{csch}}^{n-2}}ax\coth ax}{(n-1)a}+\frac{n-2}{n-1}\int{{{\operatorname{csch}}^{n-2}}axdx,n\ne 1}$

151. $\int{{{\operatorname{sech}}^{n}}ax\tanh axdx}=-\frac{{{\operatorname{sech}}^{n}}ax}{na}+C,n\ne 0$

152. $\int{{{\operatorname{csch}}^{n}}ax\coth axdx}=-\frac{{{\operatorname{csch}}^{n}}ax}{na}+C,n\ne 0$

153. $\int{{{e}^{ax}}\sinh bx}dx=\frac{{{e}^{ax}}}{2}\left( \frac{{{e}^{bx}}}{a+b}-\frac{{{e}^{-bx}}}{a-b} \right)+C,{{a}^{2}}\ne {{b}^{2}}$

154. $\int{{{e}^{ax}}\cosh bx}dx=\frac{{{e}^{ax}}}{2}\left( \frac{{{e}^{bx}}}{a+b}+\frac{{{e}^{-bx}}}{a-b} \right)+C,{{a}^{2}}\ne {{b}^{2}}$

155. $\int_{0}^{\infty }{{{x}^{n-1}}{{e}^{-x}}}dx=\Gamma (n)=(n-1)!,n>0$

156. $\int_{0}^{\infty }{{{e}^{-a{{x}^{2}}}}}dx=\frac{1}{2}\sqrt{\frac{\pi }{a}},a>0$

157. 
$$ \begin{aligned} \int_0^{\pi / 2} \sin ^n x d x= \\
& \int_0^{\pi / 2} \cos ^n x d x=\left\{\begin{array}{c}
\left(\frac{1 \cdot 3 \cdot 5 \cdots(n-1)}{2 \cdot 4 \cdot 6 \cdots n} \cdot \frac{\pi}{2}\right), \text { if } \mathrm{n} \text { is an even integer } \geq 2 \\
\left(\frac{2 \cdot 4 \cdot 6 \cdots(n-1)}{3 \cdot 5 \cdot 7 \cdot n}\right), \text { if } \mathrm{n} \text { is an odd integer } \geq 3
\end{array}\right.
\end{aligned}
$$

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