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

John Conway, Simons Foundation, Science Lives, Mathematics, Mathematicians

via John Conway, Simons Foundation, Science Lives, Mathematics, Mathematicians

Binomial Theorem : Tutorial Problems II: IITJEE Mains practice

I. Find the (r+1)6{th} term in each of the following expansions:

  1. (1+x)^{-\frac{1}{2}}
  2. (1-x)^{-2}
  3. (1+3x)^{\frac{1}{3}}
  4. (1+x)^{-\frac{2}{3}}
  5. $latex(1+x^{2))^{-3}$
  6. (1-2x)^{-\frac{3}{2}}
  7. (a+bx)^{-1}
  8. (2-x)^{-2}
  9. \sqrt[3]{a^{2}-x^{2}}
  10. \frac{1}{\sqrt{1+2x}}
  11. \frac{1}{\sqrt[3]{(1-3x)^{2}}}
  12. \frac{1}{\sqrt[n]{(a^{n}-nx)}}

Find the greatest term in each of the following expressions:

  1. (1+x)^{-r} when x=\frac{4}{15}
  2. (1+x)^{\frac{11}{2}} when x=\frac{2}{3}
  3. (1-7x)^{-\frac{11}{4}} when x=\frac{1}{8}
  4. (2x+5y)^{12}​ when x=8, y=3
  5. (b-4x)^{-7} when x=\frac{1}{2}
  6. (3x^{2}=4y^{3})^{-n} when x=9, y=2, n=15

Find to five places of decimals the value of:

  1. \sqrt{98}
  2. \sqrt[3]{998}
  3. \sqrt[3]{1003}
  4. \sqrt[4]{2400}
  5. \frac{1}{\sqrt[3]{128}}
  6. (\frac{601}{50})^{\frac{1}{3}}
  7. (630)^{-\frac{2}{3}}
  8. (3128)^{\frac{1}{4}}

Regards,

Nalin Pithwa.