プリンキピア

付録

Table of Integrals積分公式表

Basic Integrals基本的な積分

1. ∫undu=un+1n+1+C,n≠−1
1. ∫undu=un+1n+1+C,n≠−1
2. ∫duu=ln|u|+C
2. ∫duu=ln|u|+C
3. ∫eudu=eu+C
3. ∫eudu=eu+C
4. ∫audu=aulna+C
4. ∫audu=aulna+C
5. ∫sinudu=−cosu+C
5. ∫sinudu=−cosu+C
6. ∫cosudu=sinu+C
6. ∫cosudu=sinu+C
7. ∫sec2udu=tanu+C
7. ∫sec2udu=tanu+C
8. ∫csc2udu=−cotu+C
8. ∫csc2udu=−cotu+C
9. ∫secutanudu=secu+C
9. ∫secutanudu=secu+C
10. ∫cscucotudu=−cscu+C
10. ∫cscucotudu=−cscu+C
11. ∫tanudu=ln|secu|+C
11. ∫tanudu=ln|secu|+C
12. ∫cotudu=ln|sinu|+C
12. ∫cotudu=ln|sinu|+C
13. ∫secudu=ln|secu+tanu|+C
13. ∫secudu=ln|secu+tanu|+C
14. ∫cscudu=ln|cscu−cotu|+C
14. ∫cscudu=ln|cscu−cotu|+C
15. ∫dua2−u2=sin−1ua+C
15. ∫dua2−u2=sin−1ua+C
16. ∫dua2+u2=1atan−1ua+C
16. ∫dua2+u2=1atan−1ua+C
17. ∫duuu2−a2=1asec−1ua+C
17. ∫duuu2−a2=1asec−1ua+C

Trigonometric Integrals三角関数の積分

18. ∫sin2udu=12u−14sin2u+C
18. ∫sin2udu=12u−14sin2u+C
19. ∫cos2udu=12u+14sin2u+C
19. ∫cos2udu=12u+14sin2u+C
20. ∫tan2udu=tanu−u+C
20. ∫tan2udu=tanu−u+C
21. ∫cot2udu=−cotu−u+C
21. ∫cot2udu=−cotu−u+C
22. ∫sin3udu=−13(2+sin2u)cosu+C
22. ∫sin3udu=−13(2+sin2u)cosu+C
23. ∫cos3udu=13(2+cos2u)sinu+C
23. ∫cos3udu=13(2+cos2u)sinu+C
24. ∫tan3udu=12tan2u+ln|cosu|+C
24. ∫tan3udu=12tan2u+ln|cosu|+C
25. ∫cot3udu=−12cot2u−ln|sinu|+C
25. ∫cot3udu=−12cot2u−ln|sinu|+C
26. ∫sec3udu=12secutanu+12ln|secu+tanu|+C
26. ∫sec3udu=12secutanu+12ln|secu+tanu|+C
27. ∫csc3udu=−12cscucotu+12ln|cscu−cotu|+C
27. ∫csc3udu=−12cscucotu+12ln|cscu−cotu|+C
28. ∫sinnudu=−1nsinn−1ucosu+n−1n∫sinn−2udu
28. ∫sinnudu=−1nsinn−1ucosu+n−1n∫sinn−2udu
29. ∫cosnudu=1ncosn−1usinu+n−1n∫cosn−2udu
29. ∫cosnudu=1ncosn−1usinu+n−1n∫cosn−2udu
30. ∫tannudu=1n−1tann−1u−∫tann−2udu
30. ∫tannudu=1n−1tann−1u−∫tann−2udu
31. ∫cotnudu=−1n−1cotn−1u−∫cotn−2udu
31. ∫cotnudu=−1n−1cotn−1u−∫cotn−2udu
32. ∫secnudu=1n−1tanusecn−2u+n−2n−1∫secn−2udu
32. ∫secnudu=1n−1tanusecn−2u+n−2n−1∫secn−2udu
33. ∫cscnudu=−1n−1cotucscn−2u+n−2n−1∫cscn−2udu
33. ∫cscnudu=−1n−1cotucscn−2u+n−2n−1∫cscn−2udu
34. ∫sinausinbudu=sin((a−b)u)2(a−b)−sin((a+b)u)2(a+b)+C
34. ∫sinausinbudu=sin((a−b)u)2(a−b)−sin((a+b)u)2(a+b)+C
35. ∫cosaucosbudu=sin((a−b)u)2(a−b)+sin((a+b)u)2(a+b)+C
35. ∫cosaucosbudu=sin((a−b)u)2(a−b)+sin((a+b)u)2(a+b)+C
36. ∫sinaucosbudu=−cos((a−b)u)2(a−b)−cos((a+b)u)2(a+b)+C
36. ∫sinaucosbudu=−cos((a−b)u)2(a−b)−cos((a+b)u)2(a+b)+C
37. ∫usinudu=sinu−ucosu+C
37. ∫usinudu=sinu−ucosu+C
38. ∫ucosudu=cosu+usinu+C
38. ∫ucosudu=cosu+usinu+C
39. ∫unsinudu=−uncosu+n∫un−1cosudu
39. ∫unsinudu=−uncosu+n∫un−1cosudu
40. ∫uncosudu=unsinu−n∫un−1sinudu
40. ∫uncosudu=unsinu−n∫un−1sinudu
41. ∫sinnucosmudu=−sinn−1ucosm+1un+m+n−1n+m∫sinn−2ucosmudu=sinn+1ucosm−1un+m+m−1n+m∫sinnucosm−2udu
41. ∫sinnucosmudu=−sinn−1ucosm+1un+m+n−1n+m∫sinn−2ucosmudu=sinn+1ucosm−1un+m+m−1n+m∫sinnucosm−2udu

Exponential and Logarithmic Integrals指数関数・対数関数の積分

42. ∫ueaudu=1a2(au−1)eau+C
42. ∫ueaudu=1a2(au−1)eau+C
43. ∫uneaudu=1auneau−na∫un−1eaudu
43. ∫uneaudu=1auneau−na∫un−1eaudu
44. ∫eausinbudu=eaua2+b2(asinbu−bcosbu)+C
44. ∫eausinbudu=eaua2+b2(asinbu−bcosbu)+C
45. ∫eaucosbudu=eaua2+b2(acosbu+bsinbu)+C
45. ∫eaucosbudu=eaua2+b2(acosbu+bsinbu)+C
46. ∫lnudu=ulnu−u+C
46. ∫lnudu=ulnu−u+C
47. ∫unlnudu=un+1(n+1)2[(n+1)lnu−1]+C
47. ∫unlnudu=un+1(n+1)2[(n+1)lnu−1]+C
48. ∫1ulnudu=ln|lnu|+C
48. ∫1ulnudu=ln|lnu|+C

Hyperbolic Integrals双曲線関数の積分

49. ∫sinhudu=coshu+C
49. ∫sinhudu=coshu+C
50. ∫coshudu=sinhu+C
50. ∫coshudu=sinhu+C
51. ∫tanhudu=lncoshu+C
51. ∫tanhudu=lncoshu+C
52. ∫cothudu=ln|sinhu|+C
52. ∫cothudu=ln|sinhu|+C
53. ∫sechudu=tan−1(sinhu)+C
53. ∫sechudu=tan−1(sinhu)+C
54. ∫cschudu=ln|tanh12u|+C
54. ∫cschudu=ln|tanh12u|+C
55. ∫sech2udu=tanhu+C
55. ∫sech2udu=tanhu+C
56. ∫csch2udu=−cothu+C
56. ∫csch2udu=−cothu+C
57. ∫sechutanhudu=−sechu+C
57. ∫sechutanhudu=−sechu+C
58. ∫cschucothudu=−cschu+C
58. ∫cschucothudu=−cschu+C

Inverse Trigonometric Integrals逆三角関数の積分

59. ∫sin−1udu=usin−1u+1−u2+C
59. ∫sin−1udu=usin−1u+1−u2+C
60. ∫cos−1udu=ucos−1u−1−u2+C
60. ∫cos−1udu=ucos−1u−1−u2+C
61. ∫tan−1udu=utan−1u−12ln(1+u2)+C
61. ∫tan−1udu=utan−1u−12ln(1+u2)+C
62. ∫usin−1udu=2u2−14sin−1u+u1−u24+C
62. ∫usin−1udu=2u2−14sin−1u+u1−u24+C
63. ∫ucos−1udu=2u2−14cos−1u−u1−u24+C
63. ∫ucos−1udu=2u2−14cos−1u−u1−u24+C
64. ∫utan−1udu=u2+12tan−1u−u2+C
64. ∫utan−1udu=u2+12tan−1u−u2+C
65. ∫unsin−1udu=1n+1[un+1sin−1u−∫un+1du1−u2],n≠−1
65. ∫unsin−1udu=1n+1[un+1sin−1u−∫un+1du1−u2],n≠−1
66. ∫uncos−1udu=1n+1[un+1cos−1u+∫un+1du1−u2],n≠−1
66. ∫uncos−1udu=1n+1[un+1cos−1u+∫un+1du1−u2],n≠−1
67. ∫untan−1udu=1n+1[un+1tan−1u−∫un+1du1+u2],n≠−1
67. ∫untan−1udu=1n+1[un+1tan−1u−∫un+1du1+u2],n≠−1

Integrals Involving a2 + u2, a > 0a2 + u2 を含む積分（a > 0）

68. ∫a2+u2du=u2a2+u2+a22ln(u+a2+u2)+C
68. ∫a2+u2du=u2a2+u2+a22ln(u+a2+u2)+C
69. ∫u2a2+u2du=u8(a2+2u2)a2+u2−a48ln(u+a2+u2)+C
69. ∫u2a2+u2du=u8(a2+2u2)a2+u2−a48ln(u+a2+u2)+C
70. ∫a2+u2udu=a2+u2−aln|a+a2+u2u|+C
70. ∫a2+u2udu=a2+u2−aln|a+a2+u2u|+C
71. ∫a2+u2u2du=−a2+u2u+ln(u+a2+u2)+C
71. ∫a2+u2u2du=−a2+u2u+ln(u+a2+u2)+C
72. ∫dua2+u2=ln(u+a2+u2)+C
72. ∫dua2+u2=ln(u+a2+u2)+C
73. ∫u2dua2+u2=u2(a2+u2)−a22ln(u+a2+u2)+C
73. ∫u2dua2+u2=u2(a2+u2)−a22ln(u+a2+u2)+C
74. ∫duua2+u2=−1aln|a2+u2+au|+C
74. ∫duua2+u2=−1aln|a2+u2+au|+C
75. ∫duu2a2+u2=−a2+u2a2u+C
75. ∫duu2a2+u2=−a2+u2a2u+C
76. ∫du(a2+u2)3/2=ua2a2+u2+C
76. ∫du(a2+u2)3/2=ua2a2+u2+C

Integrals Involving u2 − a2, a > 0u2 − a2 を含む積分（a > 0）

77. ∫u2−a2du=u2u2−a2−a22ln|u+u2−a2|+C
77. ∫u2−a2du=u2u2−a2−a22ln|u+u2−a2|+C
78. ∫u2u2−a2du=u8(2u2−a2)u2−a2−a48ln|u+u2−a2|+C
78. ∫u2u2−a2du=u8(2u2−a2)u2−a2−a48ln|u+u2−a2|+C
79. ∫u2−a2udu=u2−a2−acos−1a|u|+C
79. ∫u2−a2udu=u2−a2−acos−1a|u|+C
80. ∫u2−a2u2du=−u2−a2u+ln|u+u2−a2|+C
80. ∫u2−a2u2du=−u2−a2u+ln|u+u2−a2|+C
81. ∫duu2−a2=ln|u+u2−a2|+C
81. ∫duu2−a2=ln|u+u2−a2|+C
82. ∫u2duu2−a2=u2u2−a2+a22ln|u+u2−a2|+C
82. ∫u2duu2−a2=u2u2−a2+a22ln|u+u2−a2|+C
83. ∫duu2u2−a2=u2−a2a2u+C
83. ∫duu2u2−a2=u2−a2a2u+C
84a. ∫du(u2−a2)3/2=−ua2u2−a2+C
84a. ∫du(u2−a2)3/2=−ua2u2−a2+C
84b. ∫duu2-a2=12alnu-au+a+C
84b. ∫duu2-a2=12alnu-au+a+C

Integrals Involving a2 − u2, a > 0a2 − u2 を含む積分（a > 0）

85. ∫a2−u2du=u2a2−u2+a22sin−1ua+C
85. ∫a2−u2du=u2a2−u2+a22sin−1ua+C
86. ∫u2a2−u2du=u8(2u2−a2)a2−u2+a48sin−1ua+C
86. ∫u2a2−u2du=u8(2u2−a2)a2−u2+a48sin−1ua+C
87. ∫a2−u2udu=a2−u2−aln|a+a2−u2u|+C
87. ∫a2−u2udu=a2−u2−aln|a+a2−u2u|+C
88. ∫a2−u2u2du=−1ua2−u2−sin−1ua+C
88. ∫a2−u2u2du=−1ua2−u2−sin−1ua+C
89. ∫u2dua2−u2=−u2a2−u2+a22sin−1ua+C
89. ∫u2dua2−u2=−u2a2−u2+a22sin−1ua+C
90. ∫duua2−u2=−1aln|a+a2−u2u|+C
90. ∫duua2−u2=−1aln|a+a2−u2u|+C
91. ∫duu2a2−u2=−1a2ua2−u2+C
91. ∫duu2a2−u2=−1a2ua2−u2+C
92. ∫(a2−u2)3/2du=−u8(2u2−5a2)a2−u2+3a48sin−1ua+C
92. ∫(a2−u2)3/2du=−u8(2u2−5a2)a2−u2+3a48sin−1ua+C
93a. ∫du(a2−u2)3/2=ua2a2−u2+C
93a. ∫du(a2−u2)3/2=ua2a2−u2+C
93b. ∫dua2-u2=12alnu+au-a+C
93b. ∫dua2-u2=12alnu+au-a+C

Integrals Involving 2au − u2, a > 02au − u2 を含む積分（a > 0）

94. ∫2au−u2du=u−a22au−u2+a22cos−1(a−ua)+C
94. ∫2au−u2du=u−a22au−u2+a22cos−1(a−ua)+C
95. ∫du2au−u2=cos−1(a−ua)+C
95. ∫du2au−u2=cos−1(a−ua)+C
96. ∫u2au−u2du=2u2−au−3a262au−u2+a32cos−1(a−ua)+C
96. ∫u2au−u2du=2u2−au−3a262au−u2+a32cos−1(a−ua)+C
97. ∫duu2au−u2=−2au−u2au+C
97. ∫duu2au−u2=−2au−u2au+C

Integrals Involving a + bu, a ≠ 0a + bu を含む積分（a ≠ 0）

98. ∫udua+bu=1b2(a+bu−aln|a+bu|)+C
98. ∫udua+bu=1b2(a+bu−aln|a+bu|)+C
99. ∫u2dua+bu=12b3[(a+bu)2−4a(a+bu)+2a2ln|a+bu|]+C
99. ∫u2dua+bu=12b3[(a+bu)2−4a(a+bu)+2a2ln|a+bu|]+C
100. ∫duu(a+bu)=1aln|ua+bu|+C
100. ∫duu(a+bu)=1aln|ua+bu|+C
101. ∫duu2(a+bu)=−1au+ba2ln|a+buu|+C
101. ∫duu2(a+bu)=−1au+ba2ln|a+buu|+C
102. ∫udu(a+bu)2=ab2(a+bu)+1b2ln|a+bu|+C
102. ∫udu(a+bu)2=ab2(a+bu)+1b2ln|a+bu|+C
103. ∫duu(a+bu)2=1a(a+bu)−1a2ln|a+buu|+C
103. ∫duu(a+bu)2=1a(a+bu)−1a2ln|a+buu|+C
104. ∫u2du(a+bu)2=1b3(a+bu−a2a+bu−2aln|a+bu|)+C
104. ∫u2du(a+bu)2=1b3(a+bu−a2a+bu−2aln|a+bu|)+C
105. ∫ua+budu=215b2(3bu−2a)(a+bu)3/2+C
105. ∫ua+budu=215b2(3bu−2a)(a+bu)3/2+C
106. ∫udua+bu=23b2(bu−2a)a+bu+C
106. ∫udua+bu=23b2(bu−2a)a+bu+C
107. ∫u2dua+bu=215b3(8a2+3b2u2−4abu)a+bu+C
107. ∫u2dua+bu=215b3(8a2+3b2u2−4abu)a+bu+C
108. ∫duua+bu=1aln|a+bu−aa+bu+a|+C,ifa>0=2−atan−1a+bu−a+C,ifa<0
108. ∫duua+bu=1aln|a+bu−aa+bu+a|+C,条件：a>0=2−atan−1a+bu−a+C,条件：a<0
109. ∫a+buudu=2a+bu+a∫duua+bu
109. ∫a+buudu=2a+bu+a∫duua+bu
110. ∫a+buu2du=−a+buu+b2∫duua+bu
110. ∫a+buu2du=−a+buu+b2∫duua+bu
111. ∫una+budu=2b(2n+3)[un(a+bu)3/2−na∫un−1a+budu]
111. ∫una+budu=2b(2n+3)[un(a+bu)3/2−na∫un−1a+budu]
112. ∫undua+bu=2una+bub(2n+1)−2nab(2n+1)∫un−1dua+bu
112. ∫undua+bu=2una+bub(2n+1)−2nab(2n+1)∫un−1dua+bu
113. ∫duuna+bu=−a+bua(n−1)un−1−b(2n−3)2a(n−1)∫duun−1a+bu
113. ∫duuna+bu=−a+bua(n−1)un−1−b(2n−3)2a(n−1)∫duun−1a+bu