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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

A Function F(X)=Sinx+cosx is odd or even?
WIZARD Reply
neither
David
Neither
Lovuyiso
f(x)=1/1+x^2 |=[-3,1]
Yuliana Reply
apa itu?
fauzi
determine the area of the region enclosed by x²+y=1,2x-y+4=0
Gerald Reply
Hi
MP
Hi too
Vic
hello please anyone with calculus PDF should share
Adegoke
Which kind of pdf do you want bro?
Aftab
hi
Abdul
can I get calculus in pdf
Abdul
How to use it to slove fraction
Tricia Reply
Hello please can someone tell me the meaning of this group all about, yes I know is calculus group but yet nothing is showing up
Shodipo
You have downloaded the aplication Calculus Volume 1, tackling about lessons for (mostly) college freshmen, Calculus 1: Differential, and this group I think aims to let concerns and questions from students who want to clarify something about the subject. Well, this is what I guess so.
Jean
Im not in college but this will still help
nothing
how can we scatch a parabola graph
Dever Reply
Ok
Endalkachew
how can I solve differentiation?
Sir Reply
with the help of different formulas and Rules. we use formulas according to given condition or according to questions
CALCULUS
For example any questions...
CALCULUS
what is the procedures in solving number 1?
Vier Reply
review of funtion role?
Md Reply
for the function f(x)={x^2-7x+104 x<=7 7x+55 x>7' does limx7 f(x) exist?
find dy÷dx (y^2+2 sec)^2=4(x+1)^2
Rana Reply
Integral of e^x/(1+e^2x)tan^-1 (e^x)
naveen Reply
why might we use the shell method instead of slicing
Madni Reply
fg[[(45)]]²+45⅓x²=100
albert Reply
find the values of c such that the graph of f(x)=x^4+2x^3+cx^2+2x+2
Ramya Reply
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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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