Mathematics Hothouse

If the numerator of a fraction is increased by 5, it reduces to $\frac{2}{3}$ , and if the denominator is increased by 9, it reduces to $\frac{1}{3}$ . Find the fraction.
Find a fraction such that it reduces to $\frac{3}{5}$ if 7 is subtracted from its denominator, and reduces to $\frac{3}{8}$ on subtracting 3 from its numerator.
If unity is taken from the denominator of a fraction, it reduces to $\frac{1}{2}$ ; if 3 is added to the numerator it reduces to $\frac{4}{7}$ , find the required fraction.
Find a fraction which becomes $\frac{3}{4}$ on adding 5 to the numerator and subtracting 1 from its denominator; and, reduces to $\frac{1}{3}$ on subtracting 4 from the numerator and adding 7 to the denominator.
If 9 is added to the numerator a certain fraction will be increased by $\frac{1}{3}$ ; if 6 is taken from the denominator the fraction reduces to $\frac{2}{3}$ ; find the required fraction.
At what time between 9 and 10 o’clock are the hands of a watch together?
When are the hands of a clock 8 minutes apart between the hours at 5 and 6 ?
At what time between 10 and 11 o’clock is the hour hand six minutes ahead of the minute hand?
At what time between 1 and 2 o’clock are the hands of a watch in the same straight line?
At what times between 12 and 1 o’clock are the hands of a watch at right angles?
A person buys 20 m of cloth and 25 m of canvas for Rs. 22.50. By selling the cloth at a gain of 15 per cent, and the canvas at a gain of 20 per cent, he clears Rs. 3.75. Find the price of each per metre.
A dealer spends Rs. 6950 in buying horses at Rs, 250/- each and cows at Rs. 200/- each; through disease, he loses 20 percent of the horses and 25 % of the cows. By selling the animals at the price he gave for them, he receives Rs. 5400/-. Find how many of each kind he bought.
The population of a certain district is 53000, of whom 835 can neither read nor write. These consists of 2 %, of all the males and 3 % of all the females; find the number of males and females.
Two persons C and D start simultaneously from two places a kilometre apart, and walk to meet each other; if C walks p kilometres per hour, and D one kilometre per hour faster than C, how far will D have walked when they meet?
A can walk a kilometres per hour faster than B; supposing that he gives B a start of c kilometres, and that B walks a kilometres per hour, how far will A have walked when he overtakes B?

Cheers,

Nalin Pithwa

Reference: Thomas’s Calculus: 12th edition.

Basic Forms:

$\int kdx = kx+C$ where k is any number
$\int {x^{n}}dx = \frac{x^{n+1}}{n+1} +C$ where $n \neq -1$
$\int \frac{dx}{x} = \ln {|x|}+C$
$\int {e^{x}}dx= e^{x}+C$
$\int a^{x}dx= \frac{a^{x}}{\ln {a}}$ where $a>0, a \neq 1$
$\int \sin{x} = -\cos{x}+C$
$\int \cos{x}dx = \sin{x}+C$
$\int {\sec^{2}x}dx= \tan{x}+C$
$\int \csc^{2}{x}dx = -\cot {x}+ C$
$\int{sec{x}}{\tan{x}}dx = \sec{x}+C$
$\int \csc{x}\cot{x}dx = -\csc {x}+C$
$\int{\tan{x}}dx = \ln{\sec{x}}+C$
$\int \cot{x}dx=\ln{|\sin{x}|}+C$
$\int \sinh{x}dx = \cosh{x}+C$
$\int \cosh{x}dx = \sinh{x}+C$
$\int \frac{dx}{\sqrt{a^{2}-x^{2}}} = \arcsin{\frac{x}{a}}+C$
$\int \frac{dx}{a^{2}+x^{2}} = \frac{1}{a}\arctan{\frac{x}{a}}+C$
$\int \frac{dx}{x\sqrt{x^{2}-a^{2}}}=\frac{1}{a}\sec^{-1}{\frac{|x|}{|a|}}+C$
$\int \frac{dx}{\sqrt{a^{2}+x^{2}}}=\sinh^{-1}{\frac{x}{a}}+C$ where $a>0$
$\int \frac{dx}{\sqrt{x^{2}-a^{2}}}=\cosh^{-1}{\frac{x}{a}}+C$ 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