0.1 Table of integrals

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Basic integrals

1. $\int u^{n} d u = \frac{u^{n + 1}}{n + 1} + C, n \neq − 1$

2. $\int \frac{d u}{u} = ln | u | + C$

3. $\int e^{u} d u = e^{u} + C$

4. $\int a^{u} d u = \frac{a^{u}}{ln a} + C$

5. $\int sin u d u = −cos u + C$

6. $\int cos u d u = sin u + C$

7. $\int {sec}^{2} u d u = tan u + C$

8. $\int {csc}^{2} u d u = −cot u + C$

9. $\int sec u tan u d u = sec u + C$

10. $\int csc u cot u d u = −csc u + C$

11. $\int tan u d u = ln | sec u | + C$

12. $\int cot u d u = ln | sin u | + C$

13. $\int sec u d u = ln | sec u + tan u | + C$

14. $\int csc u d u = ln | csc u - cot u | + C$

15. $\int \frac{d u}{\sqrt{a^{2} - u^{2}}} = {sin}^{−1} \frac{u}{a} + C$

16. $\int \frac{d u}{a^{2} + u^{2}} = \frac{1}{a} {tan}^{−1} \frac{u}{a} + C$

17. $\int \frac{d u}{u \sqrt{u^{2} - a^{2}}} = \frac{1}{a} {sec}^{−1} \frac{u}{a} + C$

Trigonometric integrals

18. $\int {sin}^{2} u d u = \frac{1}{2} u - \frac{1}{4} sin 2 u + C$

19. $\int {cos}^{2} u d u = \frac{1}{2} u + \frac{1}{4} sin 2 u + C$

20. $\int {tan}^{2} u d u = tan u - u + C$

21. $\int {cot}^{2} u d u = − cot u - u + C$

22. $\int {sin}^{3} u d u = - \frac{1}{3} (2 + {sin}^{2} u) cos u + C$

23. $\int {cos}^{3} u d u = \frac{1}{3} (2 + {cos}^{2} u) sin u + C$

24. $\int {tan}^{3} u d u = \frac{1}{2} {tan}^{2} u + ln | cos u | + C$

25. $\int {cot}^{3} u d u = - \frac{1}{2} {cot}^{2} u - ln | sin u | + C$

26. $\int {sec}^{3} u d u = \frac{1}{2} sec u tan u + \frac{1}{2} ln | sec u + tan u | + C$

27. $\int {csc}^{3} u d u = - \frac{1}{2} csc u cot u + \frac{1}{2} ln | csc u - cot u | + C$

28. $\int {sin}^{n} u d u = - \frac{1}{n} {sin}^{n - 1} u cos u + \frac{n - 1}{n} \int {sin}^{n - 2} u d u$

29. $\int {cos}^{n} u d u = \frac{1}{n} {cos}^{n - 1} u sin u + \frac{n - 1}{n} \int {cos}^{n - 2} u d u$

30. $\int {tan}^{n} u d u = \frac{1}{n - 1} {tan}^{n - 1} u - \int {tan}^{n - 2} u d u$

31. $\int {cot}^{n} u d u = \frac{−1}{n - 1} {cot}^{n - 1} u - \int {cot}^{n - 2} u d u$

32. $\int {sec}^{n} u d u = \frac{1}{n - 1} tan u {sec}^{n - 2} u + \frac{n - 2}{n - 1} \int {sec}^{n - 2} u d u$

33. $\int {csc}^{n} u d u = \frac{−1}{n - 1} cot u {csc}^{n - 2} u + \frac{n - 2}{n - 1} \int {csc}^{n - 2} u d u$

34. $\int sin a u sin b u d u = \frac{sin (a - b) u}{2 (a - b)} - \frac{sin (a + b) u}{2 (a + b)} + C$

35. $\int cos a u cos b u d u = \frac{sin (a - b) u}{2 (a - b)} + \frac{sin (a + b) u}{2 (a + b)} + C$

36. $\int sin a u cos b u d u = - \frac{cos (a - b) u}{2 (a - b)} - \frac{cos (a + b) u}{2 (a + b)} + C$

37. $\int u sin u d u = sin u - u cos u + C$

38. $\int u cos u d u = cos u + u sin u + C$

39. $\int u^{n} sin u d u = − u^{n} cos u + n \int u^{n - 1} cos u d u$

40. $\int u^{n} cos u d u = u^{n} sin u - n \int u^{n - 1} sin u d u$

41. $\begin{matrix} \int {sin}^{n} u {cos}^{m} u d u & = - \frac{{sin}^{n - 1} u {cos}^{m + 1} u}{n + m} + \frac{n - 1}{n + m} \int {sin}^{n - 2} u {cos}^{m} u d u \\ = \frac{{sin}^{n + 1} u {cos}^{m - 1} u}{n + m} + \frac{m - 1}{n + m} \int {sin}^{n} u {cos}^{m - 2} u d u \end{matrix}$

Exponential and logarithmic integrals

42. $\int u e^{a u} d u = \frac{1}{a^{2}} (a u - 1) e^{a u} + C$

43. $\int u^{n} e^{a u} d u = \frac{1}{a} u^{n} e^{a u} - \frac{n}{a} \int u^{n - 1} e^{a u} d u$

44. $\int e^{a u} sin b u d u = \frac{e^{a u}}{a^{2} + b^{2}} (a sin b u - b cos b u) + C$

45. $\int e^{a u} cos b u d u = \frac{e^{a u}}{a^{2} + b^{2}} (a cos b u + b sin b u) + C$

46. $\int ln u d u = u ln u - u + C$

47. $\int u^{n} ln u d u = \frac{u^{n + 1}}{{(n + 1)}^{2}} [(n + 1) ln u - 1] + C$

48. $\int \frac{1}{u ln u} d u = ln | ln u | + C$

Hyperbolic integrals

49. $\int sinh u d u = cosh u + C$

50. $\int cosh u d u = sinh u + C$

51. $\int tanh u d u = ln cosh u + C$

52. $\int coth u d u = ln | sinh u | + C$

53. $\int sech u d u = {tan}^{−1} | sinh u | + C$

54. $\int csch u d u = ln | tanh \frac{1}{2} u | + C$

55. $\int {sech}^{2} u d u = tanh u + C$

56. $\int {csch}^{2} u d u = − coth u + C$

57. $\int sech u tanh u d u = − sech u + C$

58. $\int csch u coth u d u = − csch u + C$

Inverse trigonometric integrals

59. $\int {sin}^{−1} u d u = u {sin}^{−1} u + \sqrt{1 - u^{2}} + C$

60. $\int {cos}^{−1} u d u = u {cos}^{−1} u - \sqrt{1 - u^{2}} + C$

61. $\int {tan}^{−1} u d u = u {tan}^{−1} u - \frac{1}{2} ln (1 + u^{2}) + C$

62. $\int u {sin}^{−1} u d u = \frac{2 u^{2} - 1}{4} {sin}^{−1} u + \frac{u \sqrt{1 - u^{2}}}{4} + C$

63. $\int u {cos}^{−1} u d u = \frac{2 u^{2} - 1}{4} {cos}^{−1} u - \frac{u \sqrt{1 - u^{2}}}{4} + C$

64. $\int u {tan}^{−1} u d u = \frac{u^{2} + 1}{2} {tan}^{−1} u - \frac{u}{2} + C$

65. $\int u^{n} {sin}^{−1} u d u = \frac{1}{n + 1} [u^{n + 1} {sin}^{−1} u - \int \frac{u^{n + 1} d u}{\sqrt{1 - u^{2}}}], n \neq − 1$

66. $\int u^{n} {cos}^{−1} u d u = \frac{1}{n + 1} [u^{n + 1} {cos}^{−1} u + \int \frac{u^{n + 1} d u}{\sqrt{1 - u^{2}}}], n \neq − 1$

67. $\int u^{n} {tan}^{−1} u d u = \frac{1}{n + 1} [u^{n + 1} {tan}^{−1} u - \int \frac{u^{n + 1} d u}{1 + u^{2}}], n \neq − 1$

Integrals involving a ² + u ² , a >0

68. $\int \sqrt{a^{2} + u^{2}} d u = \frac{u}{2} \sqrt{a^{2} + u^{2}} + \frac{a^{2}}{2} ln (u + \sqrt{a^{2} + u^{2}}) + C$

69. $\int u^{2} \sqrt{a^{2} + u^{2}} d u = \frac{u}{8} (a^{2} + 2 u^{2}) \sqrt{a^{2} + u^{2}} - \frac{a^{4}}{8} ln (u + \sqrt{a^{2} + u^{2}}) + C$

70. $\int \frac{\sqrt{a^{2} + u^{2}}}{u} d u = \sqrt{a^{2} + u^{2}} - a ln | \frac{a + \sqrt{a^{2} + u^{2}}}{u} | + C$

71. $\int \frac{\sqrt{a^{2} + u^{2}}}{u^{2}} d u = - \frac{\sqrt{a^{2} + u^{2}}}{u} + ln (u + \sqrt{a^{2} + u^{2}}) + C$

72. $\int \frac{d u}{\sqrt{a^{2} + u^{2}}} = ln (u + \sqrt{a^{2} + u^{2}}) + C$

73. $\int \frac{u^{2} d u}{\sqrt{a^{2} + u^{2}}} = \frac{u}{2} (\sqrt{a^{2} + u^{2}}) - \frac{a^{2}}{2} ln (u + \sqrt{a^{2} + u^{2}}) + C$

74. $\int \frac{d u}{u \sqrt{a^{2} + u^{2}}} = - \frac{1}{a} ln | \frac{\sqrt{a^{2} + u^{2}} + a}{u} | + C$

75. $\int \frac{d u}{u^{2} \sqrt{a^{2} + u^{2}}} = - \frac{\sqrt{a^{2} + u^{2}}}{a^{2} u} + C$

76. $\int \frac{d u}{{(a^{2} + u^{2})}^{3 / 2}} = \frac{u}{a^{2} \sqrt{a^{2} + u^{2}}} + C$

Integrals involving u ² − a ² , a >0

77. $\int \sqrt{u^{2} - a^{2}} d u = \frac{u}{2} \sqrt{u^{2} - a^{2}} - \frac{a^{2}}{2} ln | u + \sqrt{u^{2} - a^{2}} | + C$

78. $\int u^{2} \sqrt{u^{2} - a^{2}} d u = \frac{u}{8} (2 u^{2} - a^{2}) \sqrt{u^{2} - a^{2}} - \frac{a^{4}}{8} ln | u + \sqrt{u^{2} - a^{2}} | + C$

79. $\int \frac{\sqrt{u^{2} - a^{2}}}{u} d u = \sqrt{u^{2} - a^{2}} - a {cos}^{−1} \frac{a}{| u |} + C$

80. $\int \frac{\sqrt{u^{2} - a^{2}}}{u^{2}} d u = - \frac{\sqrt{u^{2} - a^{2}}}{u} + ln | u + \sqrt{u^{2} - a^{2}} | + C$

81. $\int \frac{d u}{\sqrt{u^{2} - a^{2}}} = ln | u + \sqrt{u^{2} - a^{2}} | + C$

82. $\int \frac{u^{2} d u}{\sqrt{u^{2} - a^{2}}} = \frac{u}{2} \sqrt{u^{2} - a^{2}} + \frac{a^{2}}{2} ln | u + \sqrt{u^{2} - a^{2}} | + C$

83. $\int \frac{d u}{u^{2} \sqrt{u^{2} - a^{2}}} = \frac{\sqrt{u^{2} - a^{2}}}{a^{2} u} + C$

84. $\int \frac{d u}{{(u^{2} - a^{2})}^{3 / 2}} = − \frac{u}{a^{2} \sqrt{u^{2} - a^{2}}} + C$

Integrals involving a ² − u ² , a >0

85. $\int \sqrt{a^{2} - u^{2}} d u = \frac{u}{2} \sqrt{a^{2} - u^{2}} + \frac{a^{2}}{2} {sin}^{−1} \frac{u}{a} + C$

86. $\int u^{2} \sqrt{a^{2} - u^{2}} d u = \frac{u}{8} (2 u^{2} - a^{2}) \sqrt{a^{2} - u^{2}} + \frac{a^{4}}{8} {sin}^{−1} \frac{u}{a} + C$

87. $\int \frac{\sqrt{a^{2} - u^{2}}}{u} d u = \sqrt{a^{2} - u^{2}} - a ln | \frac{a + \sqrt{a^{2} - u^{2}}}{u} | + C$

88. $\int \frac{\sqrt{a^{2} - u^{2}}}{u^{2}} d u = - \frac{1}{u} \sqrt{a^{2} - u^{2}} - {sin}^{−1} \frac{u}{a} + C$

89. $\int \frac{u^{2} d u}{\sqrt{a^{2} - u^{2}}} = - \frac{u}{u} \sqrt{a^{2} - u^{2}} + \frac{a^{2}}{2} {sin}^{−1} \frac{u}{a} + C$

90. $\int \frac{d u}{u \sqrt{a^{2} - u^{2}}} = - \frac{1}{a} ln | \frac{a + \sqrt{a^{2} - u^{2}}}{u} | + C$

91. $\int \frac{d u}{u^{2} \sqrt{a^{2} - u^{2}}} = - \frac{1}{a^{2} u} \sqrt{a^{2} - u^{2}} + C$

92. $\int {(a^{2} - u^{2})}^{3 / 2} d u = - \frac{u}{8} (2 u^{2} - 5 a^{2}) \sqrt{a^{2} - u^{2}} + \frac{3 a^{4}}{8} {sin}^{−1} \frac{u}{a} + C$

93. $\int \frac{d u}{{(a^{2} - u^{2})}^{3 / 2}} = - \frac{u}{a^{2} \sqrt{a^{2} - u^{2}}} + C$

Integrals involving 2 au − u ² , a >0

94. $\int \sqrt{2 a u - u^{2}} d u = \frac{u - a}{2} \sqrt{2 a u - u^{2}} + \frac{a^{2}}{2} {cos}^{−1} (\frac{a - u}{a}) + C$

95. $\int \frac{d u}{\sqrt{2 a u - u^{2}}} = {cos}^{−1} (\frac{a - u}{a}) + C$

96. $\int u \sqrt{2 a u - u^{2}} d u = \frac{2 u^{2} - a u - 3 a^{2}}{6} \sqrt{2 a u - u^{2}} + \frac{a^{3}}{2} {cos}^{−1} (\frac{a - u}{a}) + C$

97. $\int \frac{d u}{u \sqrt{2 a u - u^{2}}} = - \frac{\sqrt{2 a u - u^{2}}}{a u} + C$

Integrals involving a + bu , a ≠ 0

98. $\int \frac{u d u}{a + b u} = \frac{1}{b^{2}} (a + b u - a ln | a + b u |) + C$

99. $\int \frac{u^{2} d u}{a + b u} = \frac{1}{2 b^{3}} [{(a + b u)}^{2} - 4 a (a + b u) + 2 a^{2} ln | a + b u |] + C$

100. $\int \frac{d u}{u (a + b u)} = \frac{1}{a} ln | \frac{u}{a + b u} | + C$

101. $\int \frac{d u}{u^{2} (a + b u)} = - \frac{1}{a u} + \frac{b}{a^{2}} ln | \frac{a + b u}{u} | + C$

102. $\int \frac{u d u}{{(a + b u)}^{2}} = \frac{a}{b^{2} (a + b u)} + \frac{1}{b^{2}} ln | a + b u | + C$

103. $\int \frac{u d u}{u {(a + b u)}^{2}} = \frac{1}{a (a + b u)} - \frac{1}{a^{2}} ln | \frac{a + b u}{u} | + C$

104. $\int \frac{u^{2} d u}{{(a + b u)}^{2}} = \frac{1}{b^{3}} (a + b u - \frac{a^{2}}{a + b u} - 2 a ln | a + b u |) + C$

105. $\int u \sqrt{a + b u} d u = \frac{2}{15 b^{2}} (3 b u - 2 a) {(a + b u)}^{3 / 2} + C$

106. $\int \frac{u d u}{\sqrt{a + b u}} = \frac{2}{3 b^{2}} (b u - 2 a) \sqrt{a + b u} + C$

107. $\int \frac{u^{2} d u}{\sqrt{a + b u}} = \frac{2}{15 b^{3}} (8 a^{2} + 3 b^{2} u^{2} - 4 a b u) \sqrt{a + b u} + C$

108. $\begin{matrix} \int \frac{d u}{u \sqrt{a + b u}} & = \frac{1}{\sqrt{a}} ln | \frac{\sqrt{a + b u} - \sqrt{a}}{\sqrt{a + b u} + \sqrt{a}} | + C, & if a > 0 \\ = \frac{2}{\sqrt{− a}} tan - 1 \sqrt{\frac{a + b u}{− a}} + C, & if a < 0 \end{matrix}$

109. $\int \frac{\sqrt{a + b u}}{u} d u = 2 \sqrt{a + b u} + a \int \frac{d u}{u \sqrt{a + b u}}$

110. $\int \frac{\sqrt{a + b u}}{u^{2}} d u = - \frac{\sqrt{a + b u}}{u} + \frac{b}{2} \int \frac{d u}{u \sqrt{a + b u}}$

111. $\int u^{n} \sqrt{a + b u} d u = \frac{2}{b (2 n + 3)} [u^{n} {(a + b u)}^{3 / 2} - n a \int u^{n - 1} \sqrt{a + b u} d u]$

112. $\int \frac{u^{n} d u}{\sqrt{a + b u}} = \frac{2 u^{n} \sqrt{a + b u}}{b (2 n + 1)} - \frac{2 n a}{b (2 n + 1)} \int \frac{u^{n - 1} d u}{\sqrt{a + b u}}$

113. $\int \frac{d u}{u^{n} \sqrt{a + b u}} = - \frac{\sqrt{a + b u}}{a (n - 1) u^{n - 1}} - \frac{b (2 n - 3)}{2 a (n - 1)} \int \frac{d u}{u^{n - 1} \sqrt{a + b u}}$

Questions & Answers

find the equation of the tangent to the curve y=2x³-x²+3x+1 at the points x=1 and x=3

Esther Reply

derivative of logarithms function

Iqra Reply

how to solve this question

sidra

ex 2.1 question no 11

khansa

anyone can help me

khansa

question please

Rasul

ex 2.1 question no. 11

khansa

i cant type here

khansa

Find the derivative of g(x)=−3.

Abdullah Reply

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How can I help you?

Tlou

evaluate the following computation (x³-8/x-2)

Murtala Reply

teach me how to solve the first law of calculus.

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teach me also how to solve the first law of calculus

Bilson

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abdulkadir

f(x) = x-2 g(x) = 3x + 5 fog(x)? f(x)/g(x)

Naufal Reply

fog(x)= f(g(x)) = x-2 = 3x+5-2 = 3x+3 f(x)/g(x)= x-2/3x+5

diron

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how to use fundamental theorem to solve exponential

JULIA Reply

find the bounded area of the parabola y^2=4x and y=16x

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what is absolute value means?

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Chicken nuggets

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🐔🦃 nuggets

(mathematics) For a complex number a+bi, the principal square root of the sum of the squares of its real and imaginary parts, √a2+b2 . Denoted by | |. The absolute value |x| of a real number x is √x2 , which is equal to x if x is non-negative, and −x if x is negative.

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find integration of loge x

Game Reply

find the volume of a solid about the y-axis, x=0, x=1, y=0, y=7+x^3

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Brad Reply

Can calculus give the answers as same as other methods give in basic classes while solving the numericals?

Cosmos Reply

log tan (x/4+x/2)

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Rohan

y=(x^2 + 3x).(eipix)

Claudia

is this a answer

Ismael

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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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0.1 Table of integrals

Basic integrals

Trigonometric integrals

Exponential and logarithmic integrals

Hyperbolic integrals

Inverse trigonometric integrals

Integrals involving a 2 + u 2 , a >0

Integrals involving u 2 − a 2 , a >0

Integrals involving a 2 − u 2 , a >0

Integrals involving 2 au − u 2 , a >0

Integrals involving a + bu , a ≠ 0

Questions & Answers

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Integrals involving a ² + u ² , a >0

Integrals involving u ² − a ² , a >0

Integrals involving a ² − u ² , a >0

Integrals involving 2 au − u ² , a >0