Monthly Archives: May 2020

IITJEE Foundation Maths: Tutorial Problems II

  1. A, B, C start from the same place at the rates a, a+b, a+2b kilometres per hour respectively. B starts n hours after A, how long after B must C start in order that they may overtake A at the same instant, and how far will they then have walked?
  2. Find the distance between two towns when by increasing the speed 7 kilometres per hour a train can perform the journey in 1 hour less, and by reducing the speed 5 kilometres per hour can perform the journey in 1 hour more.
  3. A person buys a certain quantity of land. If he had bought 7 hectares more each hectare would have cost Rs 80 less; and if each hectare had cost Rs. 360 more, he would have obtained 15 hectares less, how much did he pay for the land?
  4. A can walk half a kilometre per hour faster than B; and three quarters of a kilometre per hour faster than C. To walk a certain distance C takes three-quarters of an hour more than B, and two hours more than A; find their rates of walking per hour.
  5. A person spends Rs. 15 in buying goods; if each kg had cost 25 paise more he would have got 5 kg less, but if each kg had cost 15 paise less, he would have received 5 kg more; what weight did he buy?
  6. Five silver coins weight 125 gm and are worth Rs. 6. Ten bronze coins weigh 500 gm and are worth 80 paise. A number of silver and bronze coins which are worth Rs. 134 weigh 11 kg and 250 gm. How many coins of each kind are there?
  7. A and B are playing for money; in the first game, A loses one half of his money, but in the second he wins one-quarter of what B then has. When they cease playing, A has won Rs. 6 and B has still Rs. 14.50 more than A; with what amounts did they begin?
  8. A, B, C each spend the same amount in buying different qualities of the same commodity. B pays 36 paise per kg less than A and obtains 750 gm more; C pays 60 paise per kg more than A and obtains one kg less; how much does each spend?

IITJEE Foundation practice or training problem sheet: I

  1. 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.
  2. 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.
  3. 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.
  4. 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.
  5. 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.
  6. At what time between 9 and 10 o’clock are the hands of a watch together?
  7. When are the hands of a clock 8 minutes apart between the hours at 5 and 6 ?
  8. At what time between 10 and 11 o’clock is the hour hand six minutes ahead of the minute hand?
  9. At what time between 1 and 2 o’clock are the hands of a watch in the same straight line?
  10. At what times between 12 and 1 o’clock are the hands of a watch at right angles?
  11. 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.
  12. 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.
  13. 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.
  14. 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?
  15. 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

Two cute problems in HP : IITJEE Foundations\Mains, pre RMO

Problem 1: 

If a^{2}, b^{2}, c^{2} are in AP, show that b+c, c+a, a+b are in HP.

Proof 1:

Note that a straight forward proof is not so easy.

Below is a nice clever solution:

By adding ab+bc+ca to each term, we see that:

a^{2}+ab+ac+bc, b^{2}+ab+ac+bc, c^{2}+ab+ac+bc are in AP.

that is, (a+b)(a+c), (b+c)(b+a), (c+a)(c+b) are in AP.

Dividing each term by (a+b)(b+c)(c+a).

\frac{1}{b+c}, \frac{1}{c+a}, \frac{1}{a+b} are in AP.

that is, b+c, c+a, a+b are in HP.

QED.

Problem 2:

If the p^{th}, q^{th}, r^{th}, s^{th} terms of an AP are in GP, show that p-q, q-r, r-s are in GP.

Proof 2:

Once again a straight forward proof is not at all easy.

Below is a “bingo” sort of proof 🙂

With the usual notation, we have

\frac{a+(p-1)d}{a+(q-1)d} = \frac{a+(q-1)d}{a+(r-1)d} = \frac{a+(r-1)d}{a+(s-1)d}

Hence, each of the ratios is equal to

\frac{(a+(p-1)d)-(a+(q-1)d)}{(a+(q-1)d)-(a+(r-1)d)} = \frac{(a+(q-1)d)-(a+(r-1)d)}{(a+(r-1)d)-(a+(s-1)d)}

which in turn is equal  to \frac{p-q}{q-r} = \frac{q-r}{r-s}

Hence, p-q, q-r, r-s are in GP.

Cheers,

Nalin Pithwa

born under an unlucky star

via born under an unlucky star

Fourier Transformation in Data Science

via Fourier Transformation in Data Science

Fourier Transform in AI

via Fourier Transform in AI

IMO (1988) 6th Problem

via IMO (1988) 6th Problem

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

an outlier

via an outlier