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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

show that the set of all natural number form semi group under the composition of addition
Nikhil Reply
explain and give four Example hyperbolic function
Lukman Reply
_3_2_1
felecia
⅗ ⅔½
felecia
_½+⅔-¾
felecia
The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
SABAL Reply
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
Pawel
2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
Pawel
ok
Ifeanyi
on number 2 question How did you got 2x +2
Ifeanyi
combine like terms. x + x + 2 is same as 2x + 2
Pawel
x*x=2
felecia
2+2x=
felecia
Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
mariel Reply
Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
Pawel
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Harshika Reply
what is a solution set?
Harshika
find the subring of gaussian integers?
Rofiqul
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Shirley Reply
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Abdullahi
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Mark
I need quadratic equation link to Alpa Beta
Abdullahi Reply
find the value of 2x=32
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divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
Beyan
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use the y -intercept and slope to sketch the graph of the equation y=6x
Only Reply
how do we prove the quadratic formular
Seidu Reply
please help me prove quadratic formula
Darius
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
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thank you help me with how to prove the quadratic equation
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what is math number
Tric Reply
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
can you teacch how to solve that🙏
Mark
Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
Brenna
(61/11,41/11,−4/11)
Brenna
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Brenna
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Good
Leaves accumulate on the forest floor at a rate of 2 g/cm2/yr and also decompose at a rate of 90% per year. Write a differential equation governing the number of grams of leaf litter per square centimeter of forest floor, assuming at time 0 there is no leaf litter on the ground. Does this amount approach a steady value? What is that value?
Abdul Reply
You have a cup of coffee at temperature 70°C, which you let cool 10 minutes before you pour in the same amount of milk at 1°C as in the preceding problem. How does the temperature compare to the previous cup after 10 minutes?
Abdul

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Source:  OpenStax, Calculus volume 2. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11965/1.2
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