Tag Archives: integration formulae

A brief table of integrals

Reference: Thomas’s Calculus: 12th edition.

Basic Forms:

  1. \int kdx = kx+C where k is any number
  2. \int {x^{n}}dx = \frac{x^{n+1}}{n+1} +C where n \neq -1
  3. \int \frac{dx}{x} = \ln {|x|}+C
  4. \int {e^{x}}dx= e^{x}+C
  5. \int a^{x}dx= \frac{a^{x}}{\ln {a}} where a>0, a \neq 1
  6. \int \sin{x} = -\cos{x}+C
  7. \int \cos{x}dx = \sin{x}+C
  8. \int {\sec^{2}x}dx= \tan{x}+C
  9. \int \csc^{2}{x}dx = -\cot {x}+ C
  10. \int{sec{x}}{\tan{x}}dx = \sec{x}+C
  11. \int \csc{x}\cot{x}dx = -\csc {x}+C
  12. \int{\tan{x}}dx = \ln{\sec{x}}+C
  13. \int \cot{x}dx=\ln{|\sin{x}|}+C
  14. \int \sinh{x}dx = \cosh{x}+C
  15. \int \cosh{x}dx = \sinh{x}+C
  16. \int \frac{dx}{\sqrt{a^{2}-x^{2}}} = \arcsin{\frac{x}{a}}+C
  17. \int \frac{dx}{a^{2}+x^{2}} = \frac{1}{a}\arctan{\frac{x}{a}}+C
  18. \int \frac{dx}{x\sqrt{x^{2}-a^{2}}}=\frac{1}{a}\sec^{-1}{\frac{|x|}{|a|}}+C
  19. \int \frac{dx}{\sqrt{a^{2}+x^{2}}}=\sinh^{-1}{\frac{x}{a}}+C where a>0
  20. \int \frac{dx}{\sqrt{x^{2}-a^{2}}}=\cosh^{-1}{\frac{x}{a}}+C where x>a>0

Forms involving ax+b:

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

22. \int x(ax+b)^{n}dx = \frac{(ax+b)^{n+1}}{a^{2}}(\frac{ax+b}{n+2} - \frac{b}{n+1})+C, where n \neq -1, -2.

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

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

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

26. \int \frac{dx}{x(ax+b)}=\frac{1}{b}\ln{|\frac{x}{ax+b}|}+C

27. \int (\sqrt{ax+b})^{n}dx = \frac{2}{a}\frac{(\sqrt{ax+b})^{n+2}}{n+2}+C, where n \neq -2

28. \int \frac{\sqrt{ax+b}}{x}dx=2\sqrt{ax+b}+b\int \frac{dx}{x\sqrt{ax+b}}

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

$$$29b. \int \frac{dx}{x\sqrt{ax-b}} = \frac{2}{\sqrt{b}}\arctan{\sqrt{\frac{ax-b}{b}}} + C

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

Forms involving a^{2}+ x^{2}

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

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

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

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

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

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

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

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

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

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

Forms involving a^{2}-x^{2}

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

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

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

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

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

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

48. \int \frac{\sqrt{a^{2}-x^{2}}}{x^{2}}dx = - \arcsin{\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}\arcsin{\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 {|\frac{a+\sqrt{a^{2}-x^{2}}}{x}|} + C

51. \int \frac{dx}{x^{3}\sqrt{a^{2}-x^{2}}} = - \frac{\sqrt{a^{2}-x^{2}}}{a^{2}x} + C

Forms involving x^{2}-a^{2}

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

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

54. \int (\sqrt{x^{2}-a^{2}})^{n}dx = \frac{x(\sqrt{x^{2}-a^{2}})^{n+1}}{n+1} - \frac{na^{2}}{n+1} \int (\sqrt{x^{2}-a^{2}})^{n-2}dx + C, where n \neq -1

55.\frac{dx}{(\sqrt{x^{2}-a^{2}})^{n}} = \frac{x(x^{2}-a^{2})^{2-n}}{(2-n)a^{2}}  - \frac{n-3}{(n-2)a^{2}}\int \frac{dx}{(x^{2}-a^{2})^{n-2}}, where n \neq 2

56. \int x(\sqrt{x^{2}-a^{2}})^{n}dx = \frac{(\sqrt{x^{2}-a^{2}})^{n+2}}{n+2} + C, where n \neq -2

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

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

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

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

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

62. \int \frac{dx}{x^{3}\sqrt{x^{2}-a^{2}}} = \frac{\sqrt{x^{2}-a^{2}}}{a^{2}x} + C

Trigonometric Forms

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

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

69B. \int \sin{(ax)}\sin{(bx)}dx = \frac{\sin{(a-b)x}}{2(a-b)} - \frac{\sin{(a+b)x}}{2(a+b)} + C, where a^{2} \neq b^{2}

69C. \int \cos{(ax)}\cos{(bx)}dx = \frac{\sin{(a-b)x}}{2(a-b)}+ \frac{\sin{(a+b)x}}{2(a+b)} + C, where a^{2} \neq 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, where n \neq -1

72. \int \frac{\cos{(ax)}}{\sin{(ax)}}dx = \frac{1}{a}\ln {|\sin{(ax)}|} + C

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

74. \int \frac{\sin{(ax)}}{\cos{(ax)}}dx = - \frac{1}{a}\ln {|\cos{(ax)}|} + 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, where n \neq -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, where m \neq -n, (reduces \cos^{m}{(ax)})

77. \int \frac{dx}{b+c\sin{(ax)}} = \frac{-2}{a\sqrt{b^{2}-c^{2}}}\arctan{(\sqrt{\frac{b-c}{b+c}}\tan{(\frac{\pi}{4}-\frac{ax}{2})})} + C, where b^{2}>c^{2}

78/ \int \frac{dx}{b+c\sin{(ax)}} = \frac{-1}{a\sqrt{c^{2}-b^{2}}}\ln {|\frac{c+b\sin{(ax)}+\sqrt{c^{2}-b^{2}}\cos{(ax)}}{b+c\sin{(ax)}}|} +C, where b^{2}< c^{2}

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

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

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

82. \int \frac{dx}{b+c\cos{(ax)}} = \frac{1}{a\sqrt{c^{2}-b^{2}}}\ln {|\frac{c+b\cos{(ax)}+\sqrt{c^{2}-b^{2}}\sin{(ax)}}{b+c\cos{(ax)}}|} + C, where b^{2} < c^{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|\sec{(ax)}| + C

90. \int \cot {(ax)}dx = \frac{1}{a}\ln {\sin{(ax)}||} + 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 = \int \frac{\tan^{n-1}{(ax)}}{a(n-1)} - \int \tan^{n-2}{(ax)}dx, where n \neq 1

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

95. \int \sec {(ax)} dx = \frac{1}{a} \ln {|\sec{(ax)} + \tan{(ax)}|} +C

96. \int \csc{(ax)}dx = - \frac{1}{a} \ln{|\csc{(ax)} + \cot{(ax)}|} + 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, where n \neq 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, where n \neq 1

101. \int \sec^{n}{(ax)} \tan{(ax)}dx = \frac{\sec^{n}{(ax)}}{na} + C, where n \neq 0

102. \int \csc^{n}{(ax)}\cot{(ax)}dx = - \frac{\csc^{n}{(ax)}}{na} + C, where n \neq 0

Inverse Trigonometric Forms:

103. \int \arcsin{(ax)} dx = x \arcsin{(ax)} + \frac{1}{a}\sqrt{1-a^{2}x^{2}} + C

104. \int \arccos{(ax)} dx = x \arccos{(ax)} - \frac{1}{a}\sqrt{1-a^{2}x^{2}} + C

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

106. \int x^{n}\arcsin{(ax)} dx = \frac{x^{n+1}}{n+1}\arcsin{(ax)} - \frac{n}{n+1}\int \frac{x^{n-1}}{\sqrt{1-a^{2}x^{2}}} dx , where n \neq -1

107. \int x^{n} \arccos{(ax)}dx = \frac{x^{n+1}}{n+1}\arccos{(ax)} + \frac{a}{n+1}\int \frac{x^{n+1}}{\sqrt{1-a^{2}x^{2}}}dx, where n \neq -1

108. \int x^{n} \arctan{(ax)} dx = \frac{x^{n+1}}{n+1}\arctan{(ax)} - \frac{n}{n+1} \int \frac{x^{n+1}}{1+a^{2}x^{2}}dx, where n \neq -1

Exponential and Logarithmic Forms

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, where b >0, b \neq 1

111. \int  xe^{(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, where b>0, n \neq 1

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

115. \int e^{(ax)} \cos{(bx)} dx = \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} (\ln {(ax)})^{m}dx = \frac{x^{n+1}(\ln {(ax)})^{m}}{n+1} - \frac{m}{n+1} \int x^{n}(\ln {(ax)})^{m-1}dx, where n \neq -1

118. \int x^{-1}(\ln {(ax)})^{m+1} dx = \frac{(\ln {(ax)})^{m+1}}{m+1}, where m \neq -1

119. \int \frac{1}{x \ln {(ax)}} dx = \ln {|\ln {(ax)}|} + C

Forms involving \sqrt{2ax-x^{2}}, where a >0

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

121. \int \sqrt{2ax-x^{2}} dx = \frac{x-a}{2}\sqrt{2ax-x^{2}} + \frac{a^{2}}{2}\arcsin{(\frac{x-a}{a})} + C

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

123. \int \frac{1}{(\sqrt{2ax-x^{2}})^{n}} dx = \frac{(x-a)(\sqrt{2ax-x^{2}})^{2-n}}{(n-2)a^{2}} + \frac{n-3}{(n-2)a^{2}} \int \frac{1}{(\sqrt{2ax-x^{2}})^{n-2}} dx

124. \int x \sqrt{2ax-x^{2}} = \frac{(x+a)(2x-3a)\sqrt{2ax-x^{2}}}{6} + \frac{a^{2}}{2}\arcsin{(\frac{x-a}{a})} + C

125. \int \frac{\sqrt{2ax-x^{2}}}{x} dx = \sqrt{2ax-x^{2}} + a \arcsin{(\frac{x-a}{a})} + C

126. \int \frac{\sqrt{2ax-x^{2}}}{x^{2}} dx = -2\sqrt{\frac{2ax-x}{x}} - \arcsin{(\frac{x-a}{a})} + C

127. \int \frac{x}{\sqrt{2ax-x^{2}}} dx = a\arcsin{(\frac{x-a}{a})} - \sqrt{2ax-x^{2}} + C

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

Hyperbolic Forms

129. \int \sinh{(ax)} dx = \frac{1}{a}\cosh{(ax)} + C

130. \int \cosh{(ax)} dx = \frac{1}{a}\sinh{(ax)} + C

131. \int \sinh^{2}{(ax)} dx = \frac{\sinh{(2ax)}}{4a} -\frac{x}{2} + C

132. \int \cosh^{2}{ax} dx = \frac{\sinh{(ax)}}{4a} + \frac{x}{2} + C

133. \int \sinh^{n}{(ax)} dx = \frac{\sinh^{n-1}{(ax)}\cosh{(ax)}}{na} - \frac{n-1}{n}\int \sinh^{n-2}{(ax)} dx, where n \neq 0

134. \int \cosh^{n}{(ax0} dx = \frac{\cosh^{n-1}{(ax)}\sinh{(ax)}}{na} + \frac{n-1}{n}\int \cosh^{n-2}{(ax)}dx, where n \neq 0

135. \int x\sinh{(ax)} dx = \frac{1}{a}\cosh{(ax)} - \frac{1}{a^{2}}\sinh{(ax)} + C

136. \int x \cosh{(ax)} dx = \frac{x}{a}\sinh{(ax)} - \frac{1}{a^{2}}\cosh{(ax)} + C

137. \int x^{n}\sinh{(ax)} dx = \frac{x^{n}}{a}\cosh{(ax)} - \frac{n}{a}\int x^{n-1} \cosh{(ax)} dx

138. \int x^{n}\cosh{(ax)} dx = \frac{x^{n}}{a}\sinh{(ax)}- \frac{n}{a}\int x^{n-1}\cosh{(ax)} dx

139.\int \tanh{(ax)} dx = \frac{1}{a}\ln {\cosh{(ax)}} + C

140. \int \coth{(ax)} dx = \frac{1}{a}\ln {\sinh{(ax)}} +C

141. \int  \tanh^{2}{(ax)} dx = x - \frac{1}{a}\tanh{(ax)} +C

142. \int \coth^{2}{(ax)} dx = x - \frac{1}{a}\coth{(ax)} +C

143. \int \tanh^{n}{(ax)} dx = - \frac{\tanh^{n-1}{(ax)}}{(n-1)a} + \int \tanh^{n-2}{(ax)} dx, where n \neq 1

144. \int \coth^{n}{(ax)} dx = -\frac{\coth^{n-1}{(ax)}}{(n-1)a} + \int \coth^{n-2}{(ax)} dx, where n \neq 1

145. \int sech {(ax)} dx = \frac{1}{a}\arcsin{\tanh{(ax)}} + C

146. \int csch {(ax)} dx = \frac{1}{a}\ln{|\tanh{(\frac{ax}{2}}|} + C

147. \int sech^{2}{(ax)} dx = \frac{1}{a}\tanh{(ax)} +C

148. \int csch^{2} {(ax)} dx = -\frac{1}{a}\coth{(ax)} +C

149. \int sech^{n}{(ax)} dx = \frac{sech^{n-2}{(ax)}\tanh{(ax)}}{(n-1)a} + \frac{n-2}{n-1}\int sech^{n-2}{(ax)} dx, where n \neq 1

150. \int csch^{n}{(ax)} dx = \frac{csch^{(n-2)}{(ax)}\coth{(ax)}}{(n-1)a} - \frac{n-2}{n-1}\int csch^{(n-2)}{(ax)} dx, where n \neq 1

151. \int sech^{n}{(ax)}\tanh{(ax)} dx = -\frac{sech^{n}{(ax)}}{na} + C, where n \neq 0

152. \int csch^{n}{(ax)} \coth {(ax)} dx = - \frac{csch^{n}{(ax)}}{na} + C, where n \neq 0

153. \int e^{(ax)}\sinh{(bx)} dx = \frac{e^{ax}}{2}(\frac{e^{bx}}{a+b} - \frac{e^{-bx}}{a-b}) +C, where a^{2} \neq b^{2}

154. \int e^{(ax)}\cosh{(bx)} dx = \frac{e^{ax}}{2}(\frac{e^{bx}}{a+b} + \frac{e^{-bx}}{a-b}) + C, where a^{2} \neq b^{2}

Some definite integrals

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

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

157A. \int_{0}^{\frac{\pi}{2}}\sin^{n}{(x)} dx = \int_{0}^{\frac{\pi}{2}} \cos^{n}{(x)} dx = \frac{1.3.5.\ldots (n-1)}{2.4.6.\ldots n}.\frac{\pi}{2} when n is an even integer greater than or equal to 2

157B. \int_{0}^{\frac{\pi}{2}}\sin^{n}{(x)} dx = \int_{0}^{\frac{\pi}{2}} \cos^{n}{(x)} dx = \frac{2.4.6.\ldots (n-1)}{3.5.7.\ldots n}, if n is an odd integer greater than or equal to 3