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Basic integrals

1. u n d u = u n + 1 n + 1 + C , n 1

2. d u u = ln | u | + C

3. e u d u = e u + C

4. a u d u = a u ln a + C

5. sin u d u = −cos u + C

6. cos u d u = sin u + C

7. sec 2 u d u = tan u + C

8. csc 2 u d u = −cot u + C

9. sec u tan u d u = sec u + C

10. csc u cot u d u = −csc u + C

11. tan u d u = ln | sec u | + C

12. cot u d u = ln | sin u | + C

13. sec u d u = ln | sec u + tan u | + C

14. csc u d u = ln | csc u cot u | + C

15. d u a 2 u 2 = sin −1 u a + C

16. d u a 2 + u 2 = 1 a tan −1 u a + C

17. d u u u 2 a 2 = 1 a sec −1 u a + C

Trigonometric integrals

18. sin 2 u d u = 1 2 u 1 4 sin 2 u + C

19. cos 2 u d u = 1 2 u + 1 4 sin 2 u + C

20. tan 2 u d u = tan u u + C

21. cot 2 u d u = cot u u + C

22. sin 3 u d u = 1 3 ( 2 + sin 2 u ) cos u + C

23. cos 3 u d u = 1 3 ( 2 + cos 2 u ) sin u + C

24. tan 3 u d u = 1 2 tan 2 u + ln | cos u | + C

25. cot 3 u d u = 1 2 cot 2 u ln | sin u | + C

26. sec 3 u d u = 1 2 sec u tan u + 1 2 ln | sec u + tan u | + C

27. csc 3 u d u = 1 2 csc u cot u + 1 2 ln | csc u cot u | + C

28. sin n u d u = 1 n sin n 1 u cos u + n 1 n sin n 2 u d u

29. cos n u d u = 1 n cos n 1 u sin u + n 1 n cos n 2 u d u

30. tan n u d u = 1 n 1 tan n 1 u tan n 2 u d u

31. cot n u d u = −1 n 1 cot n 1 u cot n 2 u d u

32. sec n u d u = 1 n 1 tan u sec n 2 u + n 2 n 1 sec n 2 u d u

33. csc n u d u = −1 n 1 cot u csc n 2 u + n 2 n 1 csc n 2 u d u

34. sin a u sin b u d u = sin ( a b ) u 2 ( a b ) sin ( a + b ) u 2 ( a + b ) + C

35. cos a u cos b u d u = sin ( a b ) u 2 ( a b ) + sin ( a + b ) u 2 ( a + b ) + C

36. sin a u cos b u d u = cos ( a b ) u 2 ( a b ) cos ( a + b ) u 2 ( a + b ) + C

37. u sin u d u = sin u u cos u + C

38. u cos u d u = cos u + u sin u + C

39. u n sin u d u = u n cos u + n u n 1 cos u d u

40. u n cos u d u = u n sin u n u n 1 sin u d u

41. sin n u cos m u d u = sin n 1 u cos m + 1 u n + m + n 1 n + m sin n 2 u cos m u d u = sin n + 1 u cos m 1 u n + m + m 1 n + m sin n u cos m 2 u d u

Exponential and logarithmic integrals

42. u e a u d u = 1 a 2 ( a u 1 ) e a u + C

43. u n e a u d u = 1 a u n e a u n a u n 1 e a u d u

44. e a u sin b u d u = e a u a 2 + b 2 ( a sin b u b cos b u ) + C

45. e a u cos b u d u = e a u a 2 + b 2 ( a cos b u + b sin b u ) + C

46. ln u d u = u ln u u + C

47. u n ln u d u = u n + 1 ( n + 1 ) 2 [ ( n + 1 ) ln u 1 ] + C

48. 1 u ln u d u = ln | ln u | + C

Hyperbolic integrals

49. sinh u d u = cosh u + C

50. cosh u d u = sinh u + C

51. tanh u d u = ln cosh u + C

52. coth u d u = ln | sinh u | + C

53. sech u d u = tan −1 | sinh u | + C

54. csch u d u = ln | tanh 1 2 u | + C

55. sech 2 u d u = tanh u + C

56. csch 2 u d u = coth u + C

57. sech u tanh u d u = sech u + C

58. csch u coth u d u = csch u + C

Inverse trigonometric integrals

59. sin −1 u d u = u sin −1 u + 1 u 2 + C

60. cos −1 u d u = u cos −1 u 1 u 2 + C

61. tan −1 u d u = u tan −1 u 1 2 ln ( 1 + u 2 ) + C

62. u sin −1 u d u = 2 u 2 1 4 sin −1 u + u 1 u 2 4 + C

63. u cos −1 u d u = 2 u 2 1 4 cos −1 u u 1 u 2 4 + C

64. u tan −1 u d u = u 2 + 1 2 tan −1 u u 2 + C

65. u n sin −1 u d u = 1 n + 1 [ u n + 1 sin −1 u u n + 1 d u 1 u 2 ] , n 1

66. u n cos −1 u d u = 1 n + 1 [ u n + 1 cos −1 u + u n + 1 d u 1 u 2 ] , n 1

67. u n tan −1 u d u = 1 n + 1 [ u n + 1 tan −1 u u n + 1 d u 1 + u 2 ] , n 1

Integrals involving a 2 + u 2 , a >0

68. a 2 + u 2 d u = u 2 a 2 + u 2 + a 2 2 ln ( u + a 2 + u 2 ) + C

69. u 2 a 2 + u 2 d u = u 8 ( a 2 + 2 u 2 ) a 2 + u 2 a 4 8 ln ( u + a 2 + u 2 ) + C

70. a 2 + u 2 u d u = a 2 + u 2 a ln | a + a 2 + u 2 u | + C

71. a 2 + u 2 u 2 d u = a 2 + u 2 u + ln ( u + a 2 + u 2 ) + C

72. d u a 2 + u 2 = ln ( u + a 2 + u 2 ) + C

73. u 2 d u a 2 + u 2 = u 2 ( a 2 + u 2 ) a 2 2 ln ( u + a 2 + u 2 ) + C

74. d u u a 2 + u 2 = 1 a ln | a 2 + u 2 + a u | + C

75. d u u 2 a 2 + u 2 = a 2 + u 2 a 2 u + C

76. d u ( a 2 + u 2 ) 3 / 2 = u a 2 a 2 + u 2 + C

Integrals involving u 2 a 2 , a >0

77. u 2 a 2 d u = u 2 u 2 a 2 a 2 2 ln | u + u 2 a 2 | + C

78. u 2 u 2 a 2 d u = u 8 ( 2 u 2 a 2 ) u 2 a 2 a 4 8 ln | u + u 2 a 2 | + C

79. u 2 a 2 u d u = u 2 a 2 a cos −1 a | u | + C

80. u 2 a 2 u 2 d u = u 2 a 2 u + ln | u + u 2 a 2 | + C

81. d u u 2 a 2 = ln | u + u 2 a 2 | + C

82. u 2 d u u 2 a 2 = u 2 u 2 a 2 + a 2 2 ln | u + u 2 a 2 | + C

83. d u u 2 u 2 a 2 = u 2 a 2 a 2 u + C

84. d u ( u 2 a 2 ) 3 / 2 = u a 2 u 2 a 2 + C

Integrals involving a 2 u 2 , a >0

85. a 2 u 2 d u = u 2 a 2 u 2 + a 2 2 sin −1 u a + C

86. u 2 a 2 u 2 d u = u 8 ( 2 u 2 a 2 ) a 2 u 2 + a 4 8 sin −1 u a + C

87. a 2 u 2 u d u = a 2 u 2 a ln | a + a 2 u 2 u | + C

88. a 2 u 2 u 2 d u = 1 u a 2 u 2 sin −1 u a + C

89. u 2 d u a 2 u 2 = u u a 2 u 2 + a 2 2 sin −1 u a + C

90. d u u a 2 u 2 = 1 a ln | a + a 2 u 2 u | + C

91. d u u 2 a 2 u 2 = 1 a 2 u a 2 u 2 + C

92. ( a 2 u 2 ) 3 / 2 d u = u 8 ( 2 u 2 5 a 2 ) a 2 u 2 + 3 a 4 8 sin −1 u a + C

93. d u ( a 2 u 2 ) 3 / 2 = u a 2 a 2 u 2 + C

Integrals involving 2 au u 2 , a >0

94. 2 a u u 2 d u = u a 2 2 a u u 2 + a 2 2 cos −1 ( a u a ) + C

95. d u 2 a u u 2 = cos −1 ( a u a ) + C

96. u 2 a u u 2 d u = 2 u 2 a u 3 a 2 6 2 a u u 2 + a 3 2 cos −1 ( a u a ) + C

97. d u u 2 a u u 2 = 2 a u u 2 a u + C

Integrals involving a + bu , a ≠ 0

98. u d u a + b u = 1 b 2 ( a + b u a ln | a + b u | ) + C

99. u 2 d u a + b u = 1 2 b 3 [ ( a + b u ) 2 4 a ( a + b u ) + 2 a 2 ln | a + b u | ] + C

100. d u u ( a + b u ) = 1 a ln | u a + b u | + C

101. d u u 2 ( a + b u ) = 1 a u + b a 2 ln | a + b u u | + C

102. u d u ( a + b u ) 2 = a b 2 ( a + b u ) + 1 b 2 ln | a + b u | + C

103. u d u u ( a + b u ) 2 = 1 a ( a + b u ) 1 a 2 ln | a + b u u | + C

104. u 2 d u ( a + b u ) 2 = 1 b 3 ( a + b u a 2 a + b u 2 a ln | a + b u | ) + C

105. u a + b u d u = 2 15 b 2 ( 3 b u 2 a ) ( a + b u ) 3 / 2 + C

106. u d u a + b u = 2 3 b 2 ( b u 2 a ) a + b u + C

107. u 2 d u a + b u = 2 15 b 3 ( 8 a 2 + 3 b 2 u 2 4 a b u ) a + b u + C

108. d u u a + b u = 1 a ln | a + b u a a + b u + a | + C , if a > 0 = 2 a tan 1 a + b u a + C , if a < 0

109. a + b u u d u = 2 a + b u + a d u u a + b u

110. a + b u u 2 d u = a + b u u + b 2 d u u a + b u

111. u n a + b u d u = 2 b ( 2 n + 3 ) [ u n ( a + b u ) 3 / 2 n a u n 1 a + b u d u ]

112. u n d u a + b u = 2 u n a + b u b ( 2 n + 1 ) 2 n a b ( 2 n + 1 ) u n 1 d u a + b u

113. d u u n a + b u = a + b u a ( n 1 ) u n 1 b ( 2 n 3 ) 2 a ( n 1 ) d u u n 1 a + b u

Questions & Answers

why n does not equal -1
K.kupar Reply
ask a complete question if you want a complete answer.
Andrew
I agree with Andrew
Bg
f (x) = a is a function. It's a constant function.
Darnell Reply
proof the formula integration of udv=uv-integration of vdu.?
Bg Reply
Find derivative (2x^3+6xy-4y^2)^2
Rasheed Reply
no x=2 is not a function, as there is nothing that's changing.
Vivek Reply
are you sure sir? please make it sure and reply please. thanks a lot sir I'm grateful.
The
i mean can we replace the roles of x and y and call x=2 as function
The
if x =y and x = 800 what is y
Joys Reply
y=800
Gift
800
Bg
how do u factor the numerator?
Drew Reply
Nonsense, you factor numbers
Antonio
You can factorize the numerator of an expression. What's the problem there? here's an example. f(x)=((x^2)-(y^2))/2 Then numerator is x squared minus y squared. It's factorized as (x+y)(x-y). so the overall function becomes : ((x+y)(x-y))/2
The
The problem is the question, is not a problem where it is, but what it is
Antonio
I think you should first know the basics man: PS
Vishal
Yes, what factorization is
Antonio
Antonio bro is x=2 a function?
The
Yes, and no.... Its a function if for every x, y=2.... If not is a single value constant
Antonio
you could define it as a constant function if you wanted where a function of "y" defines x f(y) = 2 no real use to doing that though
zach
Why y, if domain its usually defined as x, bro, so you creates confusion
Antonio
Its f(x) =y=2 for every x
Antonio
Yes but he said could you put x = 2 as a function you put y = 2 as a function
zach
F(y) in this case is not a function since for every value of y you have not a single point but many ones, so there is not f(y)
Antonio
x = 2 defined as a function of f(y) = 2 says for every y x will equal 2 this silly creates a vertical line and is equivalent to saying x = 2 just in a function notation as the user above asked. you put f(x) = 2 this means for every x y is 2 this creates a horizontal line and is not equivalent
zach
The said x=2 and that 2 is y
Antonio
that 2 is not y, y is a variable 2 is a constant
zach
So 2 is defined as f(x) =2
Antonio
No y its constant =2
Antonio
what variable does that function define
zach
the function f(x) =2 takes every input of x within it's domain and gives 2 if for instance f:x -> y then for every x, y =2 giving a horizontal line this is NOT equivalent to the expression x = 2
zach
Yes true, y=2 its a constant, so a line parallel to y axix as function of y
Antonio
Sorry x=2
Antonio
And you are right, but os not a function of x, its a function of y
Antonio
As function of x is meaningless, is not a finction
Antonio
yeah you mean what I said in my first post, smh
zach
I mean (0xY) +x = 2 so y can be as you want, the result its 2 every time
Antonio
OK you can call this "function" on a set {2}, but its a single value function, a constant
Antonio
well as long as you got there eventually
zach
volume between cone z=√(x^2+y^2) and plane z=2
Kranthi Reply
answer please?
Fatima
It's an integral easy
Antonio
V=1/3 h π (R^2+r2+ r*R(
Antonio
How do we find the horizontal asymptote of a function using limits?
Lerato Reply
Easy lim f(x) x-->~ =c
Antonio
solutions for combining functions
Amna Reply
what is a function? f(x)
Jeremy Reply
one that is one to one, one that passes the vertical line test
Andrew
It's a law f() that to every point (x) on the Domain gives a single point in the codomain f(x)=y
Antonio
is x=2 a function?
The
restate the problem. and I will look. ty
jon Reply
is x=2 a function?
The
What is limit
MaHeSh Reply
it's the value a function will take while approaching a particular value
Dan
don ger it
Jeremy
what is a limit?
Dlamini
it is the value the function approaches as the input approaches that value.
Andrew
Thanx
Dlamini
Its' complex a limit It's a metrical and topological natural question... approaching means nothing in math
Antonio
is x=2 a function?
The
3y^2*y' + 2xy^3 + 3y^2y'x^2 = 0 sub in x = 2, and y = 1, isolate y'
Andrew Reply
what is implicit of y³+x²y³=5 at (2,1)
Estelita Reply
tel mi about a function. what is it?
Jeremy
A function it's a law, that for each value in the domaon associate a single one in the codomain
Antonio
function is a something which another thing depends upon to take place. Example A son depends on his father. meaning here is the father is function of the son. let the father be y and the son be x. the we say F(X)=Y.
Bg
yes the son on his father
pascal
a function is equivalent to a machine. this machine makes x to create y. thus, y is dependent upon x to be produced. note x is an independent variable
moe
x or y those not matter is just to represent.
Bg

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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