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1、1 1.() + g() =f() +g() 2.() - g() =f() -g() 3.f()dg(x) = f(x)g(x) -g(x)df(x) 4.ax = ax ln a + C，a 1，a 0 5.xn = xn + 1 n + 1 + C， - 1 6.1 x = ln |x| + C 7.ex = ex+ C 8.sinx =- cosx + C 9.cosx = sinx + C 10. sec2x = tanx + C 11. csc2x =- cot x + C 12. sec xtanx = sec x + C 13. cscxcot x =- cscx + C 14
2、. (ax + b)n = (ax + b)n + 1 a(n + 1) + C，a 0，n - 1 15. dx ax + b = 1 aln |ax + b| + C，a 0 16. x(ax + b) n = (ax + b)n + 1 a2 ( ax + b n + 2 - b n + 1) + C，a 0，n - 1, - 2 17. x ax + b = x a - b a2ln |ax + b| + C，a 0 18. x (ax + b)2 = 1 a2(ln |ax + b| + b ax + b) + C，a 0 19. x2 ax + bdx = 1 2a3(ax + b
3、) 2 - 4b(ax + b) + 2b2ln |ax + b|+ C 20. x2 (ax + b)2dx = 1 a3(ax + b - 2bln |ax + b| - b2 ax + b) + C 21. x2 (ax + b)3dx = 1 a3(ln |ax + b| + 2b ax + b - b2 2(ax + b)2) + C 22. 2 (ax + b)ndx = 1 a3( - 1 (n - 3)(ax + b)n - 3 + 2b (n - 2)(ax + b)n - 2 - b2 (n - 1)(ax + b)n - 1) 2 + C，n 1,2,3 23. x(ax
4、 + b) = 1 bln| x ax + b| + C，b 0 24. x2(ax + b) =- 1 bx + a b2ln| ax + b x |+ C 25. dx x2(ax + b)2 =- a( 1 b2(ax + b) + 1 ab2x - 2 b3ln| ax + b x |)+ C 26. x ax + bdx = 2 15a2(3ax - 2b)(ax + b) 3 2 + C 27. x2ax + bdx = 2 105a3(15a 2x2 - 12abx + 8b2)(ax + b) 3 2 + C 28. (ax + b)n = 2(ax + b)n + 2 a(n
5、 + 2) + C，a 0，n - 2 29. 循环xnax + bdx = 2 a(2n + 3)x n(ax + b) 3 2 - 2nb a(2n + 3)x n - 1 ax + bdx 计算 30. ax + b x dx = 2 ax + b + b dx x ax + b = 2 ax + b - 2 barctanh ax + b b + C 31. dx x ax + b = 2 - barctan ax + b - b + C，b 0 33. ax + b x2 =- ax + b x + a 2 dx x ax + b + C 34. 循 ax + b xn dx =-
6、(ax + b) 3 2 b(n - 1)xn - 1 - (2n - 5)a 2b(n - 1) ax + b xn - 1 dx，n 1 环计算 35. 循环 xn ax + bdx = 2 a(2n + 1)(x n ax + b - bn xn - 1 ax + b) + C 计算 36. dx x2ax + b =- ax + b bx - a 2b dx x ax + b + C，b 0 37. 循 dx xnax + b =- ax + b b(n - 1)xn - 1 - (2n - 3)a 2b(n - 1) ax + b xn - 1 dx，n 1 环计算 3 38. xn
7、ax + bdx = 2 2n + 1 循环计算(xn + 1ax + b + bxnax + b - nbxn - 1ax + bdx)+ C 39. dx a2+ x2 = 1 aarctan x a + C，a 0 40. dx (a2+ x2)2 = x 2a2(a2+ x2) + 1 2a3arctan x a + C，a 0 41. dx a2- x2 = 1 2aln| a + x a - x| + C = 1 aarctanh x a + C，a 0，|a| |x| 42. dx (a2- x2)2 = x 2a2(a2- x2) + 1 4a3ln| x + a x - a|
8、 + C 43. 1 x2- a2dx = 1 2aln| x - a x + a| + C =- 1 aarccoth x a + C，a 0，|x| |a| 44. dx a2+ x2 = ln(x + a2+ x2)+ C 45. a2+ x2dx = x 2 a2+ x2+ a2 2ln (x +a2+ x2)+ C 46. (a2+ x2)3dx = x(a2+ x2) 3 4 + 3 8a 2x a2 + x2+ 3 8a 4ln(x + a2+ x2)+ C 47. (a2+ x2)5dx = x(a2+ x2) 5 6 + 5 24a 2x( a2+ x2) 3 + 5 16a
9、 4x a2 + x2+ 5 16a 6 ln(x +a2+ x2)+ C 48. x(a2+ x2) 2n + 1dx = (a2+ x2)2n + 3 2n + 3 + C 49. x2a2+ x2dx = x 8(a 2 + 2x2)a2+ x2- a4 8ln (x +a2+ x2)+ C 50. x2(a2+ x2) 3dx = x(a2+ x2) 5 6 - a2x a2+ x2 24 - a4x a2+ x2 16 - a6 16 ln(x +a2+ x2)+ C 51. x3a2+ x2dx = (a2+ x2)5 5 - a2(a2+ x2) 3 3 + C 52. x3(a
10、2+ x2) 3dx = (a2+ x2)7 7 - a2(a2+ x2) 5 5 + C 53. x3(a2+ x2) 2n + 1dx = (a2+ x2)2n + 5 2n + 5 - a2(a2+ x2) 2n + 3 2n + 3 + C 54. x4a2+ x2dx = x3(a2+ x2) 3 6 - a2x(a2+ x2) 3 8 + a4x a2+ x2 16 + a6 16 ln(x +a2+ x2)+ C 55. x4(a2+ x2) 3dx = x3(a2+ x2) 5 8 - a2x(a2+ x2) 5 16 + a4x(a2+ x2) 3 64 + 3a6x a2+
11、 x2 128 4 + 3a8 128ln (x +a2+ x2)+ C 56. x5a2+ x2dx = (a2+ x2)7 7 - 2a2(a2+ x2) 5 5 + a4(a2+ x2) 3 3 + C 57. x5(a2+ x2) 3dx = (a2+ x2)9 9 - 2a2(a2+ x2) 7 7 + a4(a2+ x2) 5 5 + C 58. x5(a2+ x2) 2n + 1dx = (a2+ x2)2n + 7 2n + 7 - 2a2(a2+ x2) 2n + 5 2n + 5 + a4(a2+ x2) 2n + 3 2n + 3 + C 59. a2+ x2 x dx
12、=a2+ x2- aln| a +a2+ x2 x | + C =a2+ x2- aarcsinh a x + C 60. (a2+ x2)3 x dx = (a2+ x2)3 3 + a2a2+ x2- a3ln| a +a2+ x2 x | + C 61. (a2+ x2)5 x dx = (a2+ x2)5 5 + a2(a2+ x2) 3 3 + a4a2+ x2- a5ln| a +a2+ x2 x | + C 62. (a2+ x2)7 7 dx = (a2+ x2)7 7 + a2(a2+ x2) 5 5 + a4(a2+ x2) 3 3 + a6a2+ x2- a7 ln| a
13、 +a2+ x2 x | + C 63. a2+ x2 x2 dx = ln(x + a2+ x2)- a2+ x2 x + C 64. x2 a2+ x2dx = - a2 2ln (x +a2+ x2)+ x a2+ x2 2 + C =- a2 2arcsinh x a + x a2+ x2 2 + C 65. dx x a2+ x2 =- 1 aln| a +a2+ x2 x | + C =- 1 aarcsinh a x + C 66. dx x2a2+ x2 =- a2+ x2 a2x + C，a 0 67. dx a2- x2 = arcsin x a + C，a 0，|x| |
14、a| 68. a2- x2dx = x 2 a2- x2+ a2 2arcsin x a + C，a 0，|x| |a| 69. a2- x2dx = 1 2(x a 2 - x2- sgnxarccosh| x a|) + C，|x| |a| 70. x a2- x2dx = - (a2- x2)3 3 + C，|x| |a| 5 71. x2a2- x2dx = a4 8arcsin x a - 1 8x a 2 - x2(a2- 2x2)+ C，a 0 72. a2- x2 x dx =a2- x2- aln| a +a2- x2 x | + C，|x| |a| 73. a2- x2 x
15、2 dx =- arcsin x a - a2- x2 x + C，a 0 74. x2 a2- x2dx = a2 2arcsin x a - x a2- x2 2 + C，a 0，x a2- x2 75. dx x a2- x2 =- 1 aln| a +a2- x2 x | + C，a 0 76. dx x2a2- x2 =- a2- x2 a2x + C，a 0 77. dx x2- a2 = ln|x + x2- a2|+ C 78. x2- a2dx = x 2 x2- a2- a2 2ln |x +x2- a2|+ C 79. 循环(x2- a2)ndx = x(x2- a2)
16、n n + 1 - na2 n + 1( x2- a2) n - 2dx，n - 1 计算 80. 循环 dx (x2- a2)n = x(x2- a2) 2 - n (2 - n)a2 + n - 3 (2 - n)a2 dx (x2- a2)n - 2，n 2 计算 81. x(x2- a2) ndx = (x2- a2)n + 2 n + 2 + C，n - 2 82. x2x2- a2dx = x 8(2x 2 - a2)x2- a2- a4 8ln |x +x2- a2|+ C 83. x2- a2 x dx =x2- a2- aarcsec| x a| + C =x2- a2- aarccos| a x| + C，a 0 84. x dx x2- a2 =x2- a2+ C 85.
