Gilbert Strang; Edwin “Jed” Herman

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 a² + u², 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 u² − a², 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. $\int \frac{d u}{u^{2} \sqrt{u^{2} - a^{2}}} = \frac{\sqrt{u^{2} - a^{2}}}{a^{2} u} + C$

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

84b. $\int \frac{d u}{u^{2} - a^{2}} = \frac{1}{2 a} \ln |\frac{u - a}{u + a}| + C$

Integrals Involving a² − u², 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. $\int \frac{u^{2} d u}{\sqrt{a^{2} - u^{2}}} = - \frac{u}{2} \sqrt{a^{2} - u^{2}} + \frac{a^{2}}{2} {sin}^{−1} \frac{u}{a} + C$

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

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

93b. $\int \frac{d u}{a^{2} - u^{2}} = \frac{1}{2 a} \ln |\frac{u + a}{u - a}| + C$

Integrals Involving 2au − u², 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. $\int \frac{d u}{u^{n} \sqrt{a + b u}} = - \frac{\sqrt{a + b u}}{a (n - 1) u^{n - 1}} - \frac{b (2 n - 3)}{2 a (n - 1)} \int \frac{d u}{u^{n - 1} \sqrt{a + b u}}$

A | Table of Integrals