or INTEGRALS.

SECTION I. Definitions.—II. General Theorems.—III. Rational Fluxions.—IV. Irrational Fluxions.—V. Circular Fluxions.—VI. Logarithmic Fluxions.—VII. Exponential Fluxions.—VIII. Index of Fluxions.

SECT. I.—Definitions.

The fluents of such expressions, as are the most likely to occur in the solution of physical problems, may be very conveniently arranged in the form of a Table; the principal materials of which will be extracted from Meier Hirsch's Integraltafeln. 4. Berlin, 1810. It might have been somewhat enlarged by additional matter that may be found in the earlier publications of our countrymen Waring and Landen, which have been particularly consulted on the occasion; but Waring's improvements relate most commonly to cases so complicated, as seldom to be applicable to practical purposes; and Landen's theorems, though incomparably more distinct and better arranged than Waring's, tend rather to the investigation of some elegant analogies, than to the facilitation of actual computations. Some of these, however, will be briefly noticed, and an improvement in the mode of notation will be attempted, which, if universally adopted, would tend to save much unnecessary circumlocution in the enunciation of many general theorems.

1. The earlier letters of the alphabet, as far as q, and sometimes r, are commonly employed to denote constant quantities; the subsequent letters generally for quantities considered as variable. They are here employed as relating indifferently to quantities positive-or negative, and to numbers whole or fractional; except when they are used as indices or exponents.

2. The Italic character is employed, in preference to others, for denoting quantities in general, the Roman for characteristic marks, as d for a fluxion, or differential, sin, cos, or f, g, for sine and cosine; and hl for hyperbolic logarithm. The long Italic \( \int \), however, not being otherwise used, serves very conveniently as a characteristic, to denote a fluent.

3. When the Italic letters m, n, p, q, r, or any others, are employed as indices, they are to be here understood as denoting any numbers without limitation; the Roman small letters, m, n, will be applied to whole numbers only, excluding fractions, but either positive or negative, or 0; the small Italic Capitals M, N, to positive numbers, whether whole or fractional, excluding negative numbers only; and the small Roman Capitals M, N, to positive integers only, including however 0.

4. The characteristic \( \Sigma \) implies the sum of a finite number of terms, derived from all the possible variations of a quantity, which is here denoted by a small letter of the Greek alphabet.

5. A comma, in an index, denotes or.

6. The fluents, indicated by the table, are to be understood as corresponding equally to any particular values of the quantities concerned; so that, in order to obtain the expression of the definite quantity required by the conditions of any problem, we must always take the difference of the two values found by substituting two values of the elementary variable quantities; and this rule being general, it supersedes the necessity of introducing a constant correction of the fluent in each particular case.

7. Particular values of fluents, limited on both sides, are distinguished by accents, \( \int' \).

SECT. II.—General Theorems.

1. \( \int dx = x \). See Sect. I. Art. 6.

2. \( \int adx = ax \).

3. \( \int x^n dx = \frac{1}{n+1} x^{n+1} \). Cavallieri was acquainted with the fluent of \( x^N \); Wallis extended it to \( x^n \); but Newton first discovered, in 1672, the general expression, as comprehending the fluxion of an irrational quantity.

Exception. In the case \( n = -1 \), the theorem fails, and we must substitute

\[ \int \frac{dx}{x} = \text{hl} \ x \]

4. \( \int y dx = xy - \int x dy \)

5. \( \int y dz = \frac{dy}{dx} \int z dx - \frac{d^2 y}{dx^2} \int z dx^2 + \frac{d^3 y}{dx^3} \int z dx^3 - \ldots \); dx being any constant fluxion whatever. This very elegant theorem may be applied with great convenience to all the more complicated logarithmic functions. See n. 547, 556, 570, 572. Taylor, Meth. Incr.

6. \( \int' dy = \frac{dx}{dx} + \frac{x^6}{2} \frac{d^2 y}{dx^2} + \frac{x^3}{2.3} \frac{d^3 y}{dx^3} + \ldots \);

\( x \) being the initial value of \( y \), this theorem gives the increment of \( y \) corresponding to any increment of \( x \) beginning at the same time: it may be called the master key which opens a way to all the treasures of analysis. From Taylor, Meth. Incr.

SECT. III.—Rational Fluxions.

A. \( x^m (a + bx)^{-n} dx \)

a. \( \frac{x^m dx}{a + bx} \) 7. \( \int \frac{x^M dx}{a+bx} = \frac{x^M}{Mb} - \frac{ax^{M-1}}{(M-1)b^2} + \frac{a^2 x^{M-2}}{(M-2)b^3} \ldots \)

\( \frac{a^{M-1} x}{b^M} - \frac{a^M}{b^{M+1}} hl(a+bx) \)

Examples.

8. \( \int \frac{dx}{a+bx} = \frac{1}{b} hl(a+bx) \)

9. \( \int \frac{x dx}{a+bx} = \frac{x}{b} - \frac{a}{b^2} hl(a+bx) \)

10. \( \int \frac{x^2 dx}{a+bx} = \frac{x^2}{2b} - \frac{ax}{b^2} + \frac{a^2}{b^3} hl(a+bx) \)

11. \( \int \frac{x^3 dx}{a+bx} = \frac{x^3}{3b} - \frac{ax^2}{2b^2} + \frac{a^2 x}{b^3} - \frac{a^3}{b^4} hl(a+bx) \)

b. \( \frac{x^M dx}{(a+bx)^2} \)

12. \( \int \frac{dx}{(a+bx)^2} = -\frac{1}{b(a+bx)} \)

13. \( \int \frac{x dx}{(a+bx)^2} = \frac{a}{b^2(a+bx)} + \frac{1}{b^2} hl(a+bx) \)

14. \( \int \frac{x^2 dx}{(a+bx)^2} = \left( \frac{x^2}{b} - \frac{2a^2}{b^3} \right) \frac{1}{a+bx} - \frac{2a}{b^3} hl(a+bx) \)

15. \( \int \frac{x^3 dx}{(a+bx)^2} = \left( \frac{x^3}{2b} - \frac{3ax^2}{2b^2} + \frac{3a^2 x}{b^3} \right) \frac{1}{a+bx} + \frac{3a^2}{b^4} hl(a+bx) \)

c. \( \frac{x^M dx}{(a+bx)^3} \)

16. \( \int \frac{dx}{(a+bx)^3} = -\frac{1}{2b(a+bx)^2} \)

17. \( \int \frac{x dx}{(a+bx)^3} = -\left( \frac{x}{b} + \frac{a}{2b^2} \right) \frac{1}{(x+bx)^2} \)

18. \( \int \frac{x^2 dx}{(a+bx)^3} = \left( \frac{2ax}{b^3} + \frac{3a^2}{2b^3} \right) \frac{1}{(a+bx)^2} + \frac{1}{b^3} hl(a+bx) \)

19. \( \int \frac{x^3 dx}{(a+bx)^3} = \left( \frac{x^3}{b} - \frac{6a^2 x}{b^3} - \frac{9a^3}{2b^4} \right) \frac{1}{(a+bx)^2} - \frac{3a}{b^4} hl(a+bx) \)

d. \( \frac{dx}{x^M (a+bx)} \)

20. \( \int \frac{dx}{x^M (a+bx)} = -\frac{1}{(M-1)ax^{M-1}} + \frac{b}{(M-2)a^2 x^{M-2}} - \frac{b^2}{(M-3)a^3 x^{M-3}} + \ldots \)

\( \frac{b^{M-1}}{a^M} hl \frac{a+bx}{x} \)

Examples.

21. \( \int \frac{dx}{x(a+bx)} = -\frac{1}{a} hl \frac{a+bx}{x} \)

22. \( \int \frac{dx}{x^2 (a+bx)} = -\frac{1}{ax} + \frac{b}{x^2} hl \frac{a+bx}{x} \)

23. \( \int \frac{dx}{x^3 (a+bx)} = -\frac{1}{2ax^2} + \frac{b}{a^2 x} - \frac{b^2}{a^3} hl \frac{a+bx}{x} \)

e. \( \frac{dx}{x^M (a+bx^2)} \)

24. \( \int \frac{dx}{x(a+bx)^2} = \frac{1}{a(a+bx)} - \frac{1}{a^2} hl \frac{a+bx}{x} \)

25. \( \int \frac{dx}{x^2 (a+bx)^2} = \left( -\frac{1}{ax} - \frac{2b}{a^2} \right) \frac{1}{a+bx} + \frac{2b}{a^3} hl \frac{a+bx}{x} \)

26. \( \int \frac{dx}{x^3 (a+bx)^2} = \left( -\frac{1}{2ax^2} + \frac{3b}{2a^2 x} + \frac{3b^2}{a^3} \right) \frac{1}{a+bx} - \frac{3b^2}{a^4} hl \frac{a+bx}{x} \)

f. \( \frac{dx}{x^M (a+bx)^3} \)

27. \( \int \frac{dx}{x(a+bx)^3} = \left( \frac{3}{2a} + \frac{bx}{a^2} \right) \frac{1}{(a+bx)^2} - \frac{1}{a^3} hl \frac{a+bx}{x} \)

28. \( \int \frac{dx}{x^2 (a+bx)^3} = \left( -\frac{1}{ax} - \frac{9b}{2a^2} - \frac{3b^2 x}{a^3} \right) \frac{1}{(a+bx)^2} + \frac{3b}{a^4} hl \frac{a+bx}{x} \)

29. \( \int \frac{dx}{x^3 (a+bx)^3} = \left( -\frac{1}{2ax^2} + \frac{2b}{a^2 x} + \frac{9b^2}{a^3} + \frac{6b^5 x}{a^4} \right) \frac{1}{(a+bx)^2} - \frac{6b^2}{a^5} hl \frac{a+bx}{x} \)

B. \( x^m (a+bx^2)^{-n} dx \)

a. \( \frac{x^m dx}{a+bx^2} \)

30. \( \int \frac{dx}{a+bx^2} + \sqrt{\frac{1}{ab}} \text{arc tang } x \sqrt{\frac{b}{a}} = \frac{1}{2\sqrt{ab}} hl \frac{\sqrt{a+x}\sqrt{-b}}{\sqrt{a-x}\sqrt{-b}} \)

31. \( \int \frac{dx}{a+bx^2} = \frac{1}{2b} hl (a+bx^2) \)

32. \( \int \frac{x^2 dx}{a+bx^2} = \frac{x}{b} - \frac{a}{b} \int \frac{dx}{a+bx^2} \)

33. \( \int \frac{x^2 dx}{a+bx^2} = \frac{x^2}{2b} - \frac{a}{2b^2} hl (a+bx^2) \)

b. \( \frac{x^m dx}{(a+bx^2)^2} \)

34. \( \int \frac{dx}{(a+bx^2)^2} = \frac{x}{2a(a+bx^2)} + \frac{1}{2a} \int \frac{dx}{a+bx^2} \)

35. \( \int \frac{dx}{(a+bx^2)^2} = -\frac{1}{2b(a+bx^2)} \) 36. \( \int \frac{x^2 dx}{(a+bx^2)^2} = -\frac{x}{2b(a+bx^2)} + \frac{1}{2b} \int \frac{dx}{a+bx^2} \)

37. \( \int \frac{x^3 dx}{(a+bx^2)^2} = \frac{a}{2b^2(a+bx^2)} + \frac{1}{2b^2} hl (a+bx^2) \)

c. \( \frac{x^m dx}{(a+bx^2)^5} \)

38. \( \int \frac{dx}{(a+bx^2)^3} = \left( \frac{3bx^2}{8a^2} + \frac{5x}{8a} \right) \frac{1}{(a+bx^2)^2} + \frac{3}{8a^2} \int \frac{dx}{a+bx^2} \)

39. \( \int \frac{xdx}{(a+bx^2)^3} = -\frac{1}{4b(a+bx^2)^2} \)

40. \( \int \frac{x^2 dx}{(a+bx^2)^3} = \left( \frac{x^3}{8a} - \frac{x}{8b} \right) \frac{1}{(a+bx^2)^2} + \frac{1}{8ab} \int \frac{dx}{a+bx^2} \)

41. \( \int \frac{x^3 dx}{(a+bx^2)^3} = \left( -\frac{x^2}{2b} - \frac{a}{4b^2} \right) \frac{1}{(a+bx^2)^2} \)

d. \( \frac{dx}{x^m(a+bx^2)} \)

42. \( \int \frac{dx}{x(a+bx^2)} = \frac{1}{2a} hl \frac{x^2}{a+bx^2} \)

43. \( \int \frac{dx}{x^2(a+bx^2)} = -\frac{1}{ax} - \frac{b}{a} \int \frac{dx}{a+bx^2} \)

44. \( \int \frac{dx}{x^3(a+bx^2)} = -\frac{1}{2ax^2} - \frac{b}{2a^2} hl \frac{x^2}{a+bx^2} \)

e. \( \frac{dx}{x^m(a+bx^2)^2} \)

45. \( \int \frac{dx}{x(a+bx^2)^2} = \frac{1}{2a(a+bx^2)} + \frac{1}{2a^2} hl \frac{x^2}{a+bx^2} \)

46. \( \int \frac{dx}{x^2(a+bx^2)^2} = \left( -\frac{1}{ax} - \frac{3bx}{2a^2} \right) \frac{1}{a+bx^2} - \frac{3b}{2a^2} \int \frac{dx}{a+bx^2} \)

47. \( \int \frac{dx}{x^3(a+bx^2)^2} = \left( -\frac{1}{2ax^2} - \frac{b}{a^2} \right) \frac{1}{a+bx^2} - \frac{b}{a^3} hl \frac{x^2}{a+bx^2} \)

f. \( \frac{dx}{x^m(a+bx^2)^3} \)

48. \( \int \frac{dx}{x(a+bx^2)^3} = \left( \frac{3}{4a} + \frac{b x^2}{2a^2} \right) \frac{1}{(a+bx^2)^2} + \frac{1}{2a^3} hl \frac{x^2}{a+bx^2} \)

49. \( \int \frac{dx}{x^2(a+bx^2)^3} = \left( -\frac{1}{ax} - \frac{25bx}{8a^2} - \frac{15b^2 x^3}{8a^3} \right) \frac{1}{(a+bx^2)^2} - \frac{15b}{8a^3} \int \frac{dx}{a+bx^2} \)

50. \( \int \frac{dx}{x^3(a+bx^2)^3} = \left( -\frac{1}{2ax^2} - \frac{9b}{4a^2} - \frac{3b^2 x^2}{2a^3} \right) \frac{1}{(a+bx^2)^2} - \frac{3b}{2a^4} hl \frac{x^2}{a+bx^2} \)

C. \( x^m(a+bx+cx^2)^{-n} dx \)

Put \( a+bx+cx^2 = y, 4ac-b^2 = k \).

a. \( \frac{x^m dx}{a+bx+cx^2} = \frac{x^m dx}{y} \)

51. \( \int \frac{dx}{a+bx+cx^2} = \frac{2}{\sqrt{k}} \text{ arc tang } \frac{2cx+b}{\sqrt{k}} = \frac{1}{\sqrt{-k}} hl \frac{2cx+b-\sqrt{-k}}{2cx+b+\sqrt{-k}} \)

52. \( \int \frac{xdx}{a+bx+cx^2} = \frac{1}{2c} hl y - \frac{b}{2c} \int \frac{dx}{y} \)

53. \( \int \frac{x^2 dx}{a+bx+cx^2} = \frac{x}{c} - \frac{b}{2c^2} hl y + \left( \frac{b^2}{2c^2} - \frac{a}{c} \right) \int \frac{dx}{y} \)

54. \( \int \frac{x^3 dx}{a+bx+cx^2} = \frac{x^2}{2c} - \frac{bx}{c^2} + \left( \frac{b^2}{2c^3} - \frac{a}{2c^2} \right) hl y - \left( \frac{b^3}{2c^3} - \frac{3ab}{2c^2} \right) \int \frac{dx}{y} \)

b. \( \frac{x^m dx}{(a+bx+cx^2)^2} = \frac{x^m dx}{y^2} \)

55. \( \int \frac{dx}{(a+bx+cx^2)^2} = \frac{2cx+b}{ky} + \frac{2c}{k} \int \frac{dx}{y} \)

56. \( \int \frac{xdx}{(a+bx+cx^2)^2} = -\frac{1}{2cy} - \frac{b}{2c} \int \frac{dx}{y^2} \)

57. \( \int \frac{x^2 dx}{(a+bx+cx^2)^2} = -\frac{x}{cy} + \frac{a}{c} \int \frac{dx}{y^2} \)

58. \( \int \frac{x^3 dx}{(a+bx+cx^2)^2} = \left( \frac{bx}{c^2} + \frac{b}{2c^2} \right) \frac{1}{y} + \frac{1}{2c^2} hl y - \frac{ab}{2c^2} \int \frac{dx}{y^2} - \frac{b}{2c^2} \int \frac{dx}{y} \)

c. \( \frac{x^m dx}{(a+bx+cx^2)^3} = \frac{x^m dx}{y^3} \)

59. \( \int \frac{dx}{(a+bx+cx^2)^3} = \left( \frac{1}{2ky^2} + \frac{3c}{k^2 y} \right) (2cx+b) + \frac{6c^2}{k^2} \int \frac{dx}{y} \)

60. \( \int \frac{xdx}{(a+bx+cx^2)^3} = -\frac{1}{4cy^2} - \frac{b}{2c} \int \frac{dx}{y^3} \)

61. \( \int \frac{x^2 dx}{(a+bx+cx^2)^3} = \left( -\frac{x}{3c} + \frac{b}{12c^2} \right) \frac{1}{y^2} + \left( \frac{b^2}{6c^2} + \frac{a}{3c} \right) \int \frac{dx}{y^3} \)

62. \( \int \frac{x^3 dx}{(a+bx+cx^2)^3} = \left( -\frac{x^2}{2c} - \frac{a}{4c^2} \right) \frac{1}{y^2} - \frac{ab}{2c^2} \int \frac{dx}{y^3} \) d. \( \frac{dx}{x^m(a+bx+cx^2)} = \frac{dx}{x^my} \)

63. \( \int \frac{dx}{x(a+bx+cx^2)} = \frac{1}{2a} hl \frac{x^2}{y} - \frac{b}{2a} \int \frac{dx}{y} \)

64. \( \int \frac{dx}{x^2(a+bx+cx^2)} = \frac{1}{ax} - \frac{b}{2a^2} hl \frac{x^2}{y} + \left( \frac{b^2}{2a^2} - \frac{c}{a} \right) \int \frac{dx}{y} \)

65. \( \int \frac{dx}{x^3(a+bx+cx^2)} = -\frac{1}{2ax^2} + \frac{b}{a^2x} + \left( \frac{b^2}{2a^2} - \frac{c}{2a^2} \right) hl \frac{x^2}{y} - \left( \frac{b^3}{2a^3} - \frac{3bc}{2a^3} \right) \int \frac{dx}{y} \)

e. \( \frac{dx}{x^m(a+bx+cx^2)^2} = \frac{dx}{x^my^2} \)

66. \( \int \frac{dx}{x(a+bx+cx^2)^2} = \frac{1}{2ay} + \frac{1}{2a^2} hl \frac{x^2}{y} - \frac{b}{2a} \int \frac{dx}{y^2} - \frac{b}{2a^2} \int \frac{dx}{y} \)

67. \( \int \frac{dx}{x^2(a+bx+cx^2)^2} = \left( -\frac{1}{ax} - \frac{b}{a^2} \right) \frac{1}{y} - \frac{b}{a^3} hl \frac{x^2}{y} + \left( \frac{b^2}{a^3} - \frac{3c}{a} \right) \int \frac{dx}{y^2} + \frac{b^2}{a^3} \int \frac{dx}{y} \)

68. \( \int \frac{dx}{x^3(a+bx+cx^2)^2} = \left( -\frac{1}{2ax^2} + \frac{3b}{2a^2x} + \frac{3b^2}{2a^3} - \frac{c}{a^2} \right) \frac{1}{y} + \left( \frac{3b^2}{2a^3} - \frac{c}{a^2} \right) hl \frac{x^2}{y} - \left( \frac{3b^3}{2a^3} - \frac{11bc}{2a^2} \right) \int \frac{dx}{y^2} - \left( \frac{3b^5}{2a^5} - \frac{bc}{a^3} \right) \int \frac{dx}{y} \)

f. \( \frac{dx}{x^m(a+bx+cx^2)^3} = \frac{dx}{x^my^3} \)

69. \( \int \frac{dx}{x(a+bx+cx^2)^3} = \frac{1}{4ay^2} + \frac{1}{2a^2y} + \frac{1}{2a^3} hl \frac{x^2}{y} - \frac{b}{2a} \int \frac{dx}{y^3} - \frac{b}{2a^2} \int \frac{dx}{y^2} - \frac{b}{2a^3} \int \frac{dx}{y} \)

70. \( \int \frac{dx}{x^2(a+bx+cx^2)^3} = -\frac{1}{ax^2y} - \frac{3b}{a^2y} - \frac{5c}{a} \int \frac{dx}{xy^3} \)

71. \( \int \frac{dx}{x^3(a+bx+cx^2)^3} = \left( -\frac{1}{2ax^2} + \frac{2b}{a^2x} \right) \frac{1}{y^2} + \left( \frac{6b^2}{a^2} - \frac{3c}{a} \right) \int \frac{dx}{xy^3} + \frac{10bc}{a^2} \int \frac{dx}{y^3} \)

D. \( x^m(a+bx^3)^{-n} dx \)

a. \( \frac{x^m dx}{x+bx^3} \)

Put \( \frac{a}{b} = k^3 \).

72. \( \int \frac{dx}{a+bx^3} = \frac{1}{3bk^2} \left( \frac{1}{2} hl \frac{(x+k)^2}{x^2-kx+k^2} + \sqrt{3} \cdot \text{arc tang} \frac{\sqrt{3}x}{2k-x} \right) \)

73. \( \int \frac{xdx}{a+bx^3} = \frac{-1}{3bk} \left( \frac{1}{2} hl \frac{(x+k)^2}{x^2-kx+k^2} - \sqrt{3} \cdot \text{arc tang} \frac{\sqrt{3}x}{2k-x} \right) \)

74. \( \int \frac{x^2 dx}{a+bx^3} = \frac{1}{3b} hl (a+bx^3) \)

75. \( \int \frac{x^3 dx}{a+bx^3} = \frac{x}{b} - \frac{a}{b} \int \frac{dx}{a+bx^3} \)

b. \( \frac{x^m dx}{(a+bx^3)^2} \)

76. \( \int \frac{dx}{(a+bx^3)^2} = \frac{x}{3a(a+bx^3)} + \frac{2}{3a} \int \frac{dx}{a+bx^3} \)

77. \( \int \frac{xdx}{(a+bx^3)^2} = \frac{x^2}{3a(a+bx^3)} + \frac{1}{3a} \int \frac{xdx}{a+bx^3} \)

78. \( \int \frac{x^2 dx}{(a+bx^3)^2} = -\frac{1}{3b(a+bx^3)} \)

79. \( \int \frac{x^3 dx}{(a+bx^3)^2} = -\frac{x}{3b(a+bx^3)} + \frac{1}{3b} \int \frac{dx}{x+bx^3} \)

c. \( \frac{x^m dx}{(a+bx^3)^3} \)

80. \( \int \frac{dx}{(a+bx^3)^3} = \left( \frac{5bx^4}{18a^2} + \frac{4x}{9a} \right) \frac{1}{(x+bx^3)^2} + \frac{5}{9a^2} \int \frac{dx}{a+bx^3} \)

81. \( \int \frac{xdx}{(a+bx^3)^3} = \left( \frac{2bx^5}{9a^2} + \frac{7x^2}{18a} \right) \frac{1}{(a+bx^3)^2} + \frac{2}{9a^2} \int \frac{xdx}{a+bx^3} \)

82. \( \int \frac{x^2 dx}{(a+bx^3)^3} = -\frac{1}{6b(a+bx^3)^2} \)

83. \( \int \frac{x^3 dx}{(a+bx^3)^3} = \left( \frac{x^4}{18a} - \frac{x}{9b} \right) \frac{1}{(a+bx^3)^2} + \frac{1}{9ab} \int \frac{dx}{a+bx^3} \)

d. \( \frac{dx}{x^m(a+bx^3)} \)

84. \( \int \frac{dx}{x(a+bx^3)} = \frac{1}{3a} hl \frac{x^3}{a+bx^3} \)

85. \( \int \frac{dx}{x^2(a+bx^3)} = -\frac{1}{ax} - \frac{b}{a} \int \frac{xdx}{a+bx^3} \)

86. \( \int \frac{dx}{x^3(a+bx^3)} = -\frac{1}{2ax^2} - \frac{b}{a} \int \frac{dx}{x+bx^3} \)

e. \( \frac{dx}{x^m(a+bx^3)^2} \)

87. \( \int \frac{dx}{x(a+bx^3)^2} = \frac{1}{3a(a+bx^3)} - \frac{1}{3a^2} hl \frac{a+bx^5}{x^2} \) 88. \( \int \frac{dx}{x^2(a+bx^3)^2} = \left( -\frac{1}{ax} - \frac{4bx^2}{3a^2} \right) \frac{1}{a+bx^3} - \frac{4b}{3a^2} \)

\( \int \frac{xdx}{a+bx^3} \)

89. \( \int \frac{dx}{x^5(a+bx^5)^2} = \left( -\frac{1}{2ax^2} - \frac{5bx}{ba^2} \right) \frac{1}{a+bx^5} - \frac{5b}{3a^2} \int \frac{dx}{a+bx^5} \)

E. \( x^{N-M}(a+bx^N)^{-1} dx \)

a. \( N = 2p + 1 \), an odd number; putting \( k^N = \frac{a}{b} \),

\( \theta = \frac{180^\circ}{N}, \frac{540^\circ}{N}, \frac{900^\circ}{N}, \ldots \frac{(N-2)180^\circ}{N} \), p values.

90. \( \int \frac{x^{2p+1-N} dx}{a+bx^{2p+1}} = \frac{1}{nb(-k)^{M-1}} hl (x+k) + \frac{1}{nbk^{M-1}} \)

\( \Sigma (\cos (M-1) \theta hl (x^2-2kx \cos \theta + k^2) + 2 \sin (M-1) \theta \arctan \frac{x \sin \theta}{k-x \cos \theta}) \); the characteristic \( \Sigma \) implying the sum of the r values depending on those of \( \theta \). From Cotes's discoveries.

91. \( \int \frac{x^{2p-N} dx}{x+bx^{2p}} \); 1, when \( \frac{a}{b} \) is negative, putting \( k^N = -\frac{a}{b} \);

\( = \frac{1}{nbk^{M-1}} hl (x-k) + \frac{1}{nb(-k)^{M-1}} hl (x+k) + \frac{1}{nbk^{M-1}} \)

\( \Sigma (\cos (M-1) \theta hl (x^2-2kx \cos \theta + k^2) + 2 \sin (M-1) \theta \arctan \frac{x \sin \theta}{k-x \cos \theta}) \); \( \Sigma \) relating to the p-1 values of \( \theta, \frac{360^\circ}{N}, \frac{720^\circ}{N}, \frac{1080^\circ}{N}, \ldots \frac{(N-2)180^\circ}{N} \); 2, when \( \frac{a}{b} \) is positive, putting \( k^N = \frac{a}{b} = \frac{1}{nbk^{M-1}} \)

\( \Sigma (\cos (M-1) \theta hl (x^2-2kx \cos \theta + k^2) + 2 \sin (M-1) \theta \arctan \frac{x \sin \theta}{k-x \cos \theta}) \); \( \Sigma \) relating to the p values of \( \theta, \frac{180^\circ}{N}, \frac{540^\circ}{N}, \frac{900^\circ}{N}, \ldots \)

F. \( x^{2N-M}(a+bx^N+cx^{2N})^{-1} dx \)

92. \( \int \frac{x^{2N-M} dx}{a+bx^N+cx^{2N}} \); first, \( 4ac \) being \( > b^2 \), and

putting \( \cos x = -\frac{b}{2\sqrt{ac}} \) and \( \varphi = \frac{x}{N} \),

\( \frac{360^\circ + x}{N}, \ldots \frac{(N-1)360^\circ + x}{N} = \frac{\cosec x}{2Nch^{M-1}} \)

\( \Sigma (-\sin (M-N-1) \theta hl (x^2-2kx \cos \theta + k^2) \)

\( + 2 \cos (M-N-1) \theta \arctan \frac{x \sin \theta}{k-x \cos \theta}) \);

secondly, \( 4ac \) being \( < b^2 \), and putting \( \sqrt{(b^2-4ac)} = h, \frac{b-h}{2} = f, \) and \( \frac{b+h}{2} = g \),

\( \int \frac{x^M dx}{a+bx^N+cx^{2N}} = \frac{c}{h} \left( \int \frac{x^M dx}{cx^N+f} - \int \frac{x^M dx}{cx^N+g} \right) \)

G. \( x^m(x+f)^{-1}(x+g)^{-1} \ldots (x^2+ax+b)^{-1} \ldots dx \)

93. \( \int \frac{dx}{(x+f)(x+g)} = \frac{1}{g-f} \frac{hl^{x+f}}{x+g} \)

94. \( \int \frac{xdx}{(x+f)(x+g)} = \frac{1}{g-f} (g hl (x+g) - f hl [x+f]) \)

95. \( \int \frac{dx}{(x+f)(x+g)(x+h)} = \frac{1}{(g-f)(h-f)} hl \)

\( (x+f) + \frac{1}{(f-g)(h-g)} hl (x+g) + \frac{1}{(f-h)(g-h)} hl (x+h) \)

96. \( \int \frac{dx}{(x+f)(x^2+a)} = \frac{1}{f^2+a} \left( hl \frac{x+f}{\sqrt{x^2+a}} + f \int \frac{dx}{x^2+a} \right) \)

97. \( \int \frac{dx}{(x^2+a)(x^2+b)} = \frac{1}{b-a} \left( \int \frac{dx}{x^2+a} - \int \frac{dx}{x^2+b} \right) \)

98. \( \int \frac{dx}{(x+f)(x^2+ax+b)} = \frac{1}{f^2-af+b} \left( \frac{1}{2} hl \frac{x+f^2}{x^2+ax+b} + c - \frac{1}{2} a \right) \int \frac{dx}{x^2+ax+b} \)

H. \( x^m(A+Bx+Cx^2 \ldots)(a+bx+cx^2 \ldots)^{-1} dx \)

99. \( \int \frac{A+Bx+Cx^2 \ldots}{a+bx+cx^2 \ldots} dx = \Sigma hl (x-\xi)^{\nu} ; \xi \) being successively each of the roots of the equation \( a+bx+cx^2 \ldots = 0, v = A+B\xi+C\xi^2 \ldots \), and \( \xi = b+2c\xi+3d\xi^2 \ldots \); provided, however, that the denominator contain higher powers of \( x \) than the numerator, and that all the values of \( \xi \) be different; a limitation first laid down by Newton.

100. \( \int \frac{A+Bx+Cx^2 \ldots}{a+bx+cx^2 \ldots} x^m dx = \Sigma \int \frac{x^m dx}{x-\xi} \)

101. \( \int \frac{A+Bx+Cx^2 \ldots}{a+bx+cx^2 \ldots} \frac{dx}{x^m} = \Sigma \int \frac{dx}{x^m(x-\xi)} \) SECT. IV.—Irrational Fluxions.

A. \( x^m(a+bx)^n dx \)

a. \( \frac{x^m dx}{\sqrt{a+bx}} = \frac{x^m dx}{\sqrt{y}} \)

102. \( \int \frac{dx}{\sqrt{a+bx}} = \frac{2}{b} \sqrt{a+bx} = \frac{2}{b} \sqrt{y} \)

103. \( \int \frac{xdx}{\sqrt{a+bx}} = \left( \frac{1}{3} y - a \right) \frac{2 \sqrt{y}}{b^2} \)

104. \( \int \frac{x^2 dx}{\sqrt{a+bx}} = \left( \frac{1}{5} y^2 - \frac{2}{3} ay + y^2 \right) \frac{2 \sqrt{y}}{b^5} \)

105. \( \int \frac{x^3 dx}{\sqrt{a+bx}} = \left( \frac{1}{7} y^3 - \frac{3}{5} ay^2 + a^2 y - a^3 \right) \frac{2 \sqrt{y}}{b^7} \)

b. \( \frac{dx}{x^m \sqrt{a+bx}} = \frac{dx}{x^m \sqrt{y}} \)

106. \( \int \frac{dx}{x \sqrt{a+bx}} = \frac{1}{\sqrt{a}} \ln \frac{\sqrt{y} - \sqrt{a}}{\sqrt{y} + \sqrt{a}} = \frac{2}{\sqrt{-a}} \arctan \frac{\sqrt{y}}{\sqrt{-a}} \). The ambiguity of the roots being decided by the conditions of the problem.

107. \( \int \frac{dx}{x^2 \sqrt{a+bx}} = - \frac{\sqrt{y}}{ax} - \frac{b}{2a} \int \frac{dx}{x \sqrt{y}} \)

108. \( \int \frac{dx}{x^3 \sqrt{a+bx}} = \left( - \frac{1}{2ax^2} + \frac{3b}{4a^2x} \right) \sqrt{y} + \frac{3b^2}{8a^2} \int \frac{dx}{x \sqrt{y}} \)

c. \( \frac{x^m dx}{(a+bx)^{\frac{3}{2}}} = \frac{x^m dx}{y^{\frac{3}{2}}} \)

109. \( \int \frac{dx}{(a+bx)^{\frac{3}{2}}} = - \frac{2}{b \sqrt{a+bx}} = - \frac{2}{b \sqrt{y}} \)

110. \( \int \frac{xdx}{(a+bx)^{\frac{3}{2}}} = (y+a) \frac{2}{b^2 \sqrt{y}} \)

111. \( \int \frac{x^2 dx}{(a+bx)^{\frac{3}{2}}} = (\frac{1}{3} y^2 - 2ay - a^2) \frac{2}{b^3 \sqrt{y}} \)

112. \( \int \frac{x^3 dx}{(a+bx)^{\frac{3}{2}}} = (\frac{1}{7} y^3 - ay^2 + 3a^2 y + a^3) \frac{2}{b^7 \sqrt{y}} \)

d. \( \frac{dx}{x^m (a+bx)^{\frac{3}{2}}} = \frac{dx}{x^m y^{\frac{3}{2}}} \)

113. \( \int \frac{dx}{x(a+bx)^{\frac{3}{2}}} = \frac{2}{a \sqrt{y}} + \frac{1}{a} \int \frac{dx}{x \sqrt{y}} \)

114. \( \int \frac{dx}{x^2 (a+bx)^{\frac{3}{2}}} = \left( - \frac{1}{ax} - \frac{3b}{a^2} \right) \frac{1}{\sqrt{y}} - \frac{3b}{2a^2} \int \frac{dx}{x \sqrt{y}} \)

115. \( \int \frac{dx}{x^3 (a+bx)^{\frac{3}{2}}} = \left( - \frac{1}{2ax^2} + \frac{5b}{4a^2x} + \frac{15b^2}{4a^3} \right) \frac{1}{\sqrt{y}} + \frac{15b^2}{8a^3} \int \frac{dx}{x \sqrt{y}} \)

e. \( \frac{x^m dx}{(a+bx)^{\frac{5}{2}}} \)

116. \( \int \frac{dx}{(a+bx)^{\frac{5}{2}}} = - \frac{2}{3by \sqrt{y}} \)

117. \( \int \frac{xdx}{(a+bx)^{\frac{5}{2}}} = \left( -y + \frac{1}{3} a \right) \frac{2}{b^2 y \sqrt{y}} \)

118. \( \int \frac{x^2 dx}{(a+bx)^{\frac{5}{2}}} = \left( y^2 + 2ay - \frac{1}{3} a^2 \right) \frac{2}{b^3 y \sqrt{y}} \)

119. \( \int \frac{x^3 dx}{(a+bx)^{\frac{5}{2}}} = \left( \frac{1}{3} y^3 - 3ay^2 - 3a^2 y + \frac{1}{3} a^3 \right) \frac{2}{b^5 y \sqrt{y}} \)

f. \( \frac{dx}{x^m (a+bx)^{\frac{5}{2}}} \)

120. \( \int \frac{dx}{x(a+bx)^{\frac{5}{2}}} = \left( \frac{8}{3a} + \frac{2bx}{a^2} \right) \frac{1}{y \sqrt{y}} + \frac{1}{a^2} \int \frac{dx}{x \sqrt{y}} \)

121. \( \int \frac{dx}{x^2 (a+bx)^{\frac{5}{2}}} = \left( - \frac{1}{ax} - \frac{20b}{3a^2} - \frac{5b^2 x}{a^3} \right) \frac{1}{y \sqrt{y}} \)

\[ - \frac{5b}{2a^3} \int \frac{dx}{x \sqrt{y}} \]

122. \( \int \frac{dx}{x^3 (a+bx)^{\frac{5}{2}}} = \left( - \frac{1}{2ax^2} + \frac{7b}{4a^2x} + \frac{35b^2}{3a^3} + \frac{35b^3 x}{4a^4} \right) \frac{1}{y \sqrt{y}} + \frac{35b^2}{8a^4} \int \frac{dx}{x \sqrt{y}} \)

g. \( x^m \sqrt{(a+bx)dx} = x^m \sqrt{y} dx \)

123. \( \int \sqrt{(a+bx)dx} = \frac{2y \sqrt{y}}{3b} \)

124. \( \int x \sqrt{(a+bx)dx} = \left( \frac{1}{5} y - \frac{1}{3} a \right) \frac{2y \sqrt{y}}{b^2} \)

125. \( \int x^2 \sqrt{(a+bx)dx} = \left( \frac{1}{7} y^2 - \frac{2}{5} ay + \frac{1}{3} a^2 \right) \frac{2y \sqrt{y}}{b^3} \)

126. \( \int x^3 \sqrt{(a+bx)dx} = \left( \frac{1}{9} y^3 - \frac{3}{7} ay^2 + \frac{3}{5} a^2 y - \frac{1}{3} a^3 \right) \frac{2y \sqrt{y}}{b^7} \)

h. \( x^{-m} \sqrt{(a+bx)dx} = x^{-m} \sqrt{y} dx \)

127. \( \int \frac{\sqrt{(a+bx)dx}}{x} = 2 \sqrt{y} + a \int \frac{dx}{x \sqrt{y}} \)

128. \( \int \frac{\sqrt{(a+bx)dx}}{x^2} = - \frac{\sqrt{y}}{x} + \frac{b}{2a} \int \frac{dx}{x \sqrt{y}} \)

129. \( \int \frac{\sqrt{(a+bx)dx}}{x^3} = - \frac{y \sqrt{y}}{2ax^2} + \frac{b \sqrt{y}}{4ax} - \frac{b^2}{8a} \int \frac{dx}{x \sqrt{y}} \)

i. \( x^m (a+bx)^{\frac{3}{2}} dx = x^m y^{\frac{3}{2}} dx \)

130. \( \int (a+bx)^{\frac{3}{2}} dx = \frac{2y^2 \sqrt{y}}{5b} \)

131. \( \int x (a+bx)^{\frac{3}{2}} dx = \left( \frac{1}{7} y - \frac{1}{5} a \right) \frac{2y^2 \sqrt{y}}{b^2} \)

132. \( \int x^2 (a+bx)^{\frac{3}{2}} dx = \left( \frac{1}{9} y^2 - \frac{2}{7} ay + \frac{1}{5} a^2 \right) \frac{2y^2 \sqrt{y}}{b^3} \) 133. \( \int x^3(a+bx)^{\frac{2}{3}} dx = \left( \frac{1}{11} y^5 - \frac{1}{3} ay^2 + \frac{3}{7} a^2 y - \frac{1}{5} a^3 \right) \frac{2y^2 \sqrt{y}}{b^3} \)

k. \( x^{-m}(a+bx)^{\frac{2}{3}} dx = x^{-m} y^{\frac{2}{3}} dx \)

134. \( \int (a+bx)^{\frac{2}{3}} dx = \left( \frac{1}{3} y + a \right) 2\sqrt{y} + a^2 \int \frac{dx}{x \sqrt{y}} \)

135. \( \int \frac{(a+bx)^{\frac{2}{3}} dx}{x^2} = -\frac{y^2 \sqrt{y}}{ax} + \frac{3b}{2a} \int \frac{y^{\frac{5}{2}} dx}{x} \)

136. \( \int \frac{(a+bx)^{\frac{2}{3}} dx}{x^3} = \left( -\frac{1}{2ax^2} - \frac{b}{4a^2 x} \right) y^2 \sqrt{y} + \frac{3b^2}{8a^2} \int \frac{y^{\frac{5}{2}} dx}{x} \)

l. \( x^m(a+bx)^{\frac{2}{3}} dx = x^m y^{\frac{2}{3}} dx \)

137. \( \int (a+bx)^{\frac{2}{3}} dx = \frac{2y^2 \sqrt{y}}{7b} \)

138. \( \int x(a+bx)^{\frac{2}{3}} dx = \left( \frac{1}{9} y - \frac{1}{7} a \right) \frac{2y^2 \sqrt{y}}{b^2} \)

139. \( \int x^2(a+bx)^{\frac{2}{3}} dx = \left( \frac{1}{11} y^2 - \frac{2}{9} ay + \frac{1}{7} a^2 \right) \frac{2y^2 \sqrt{y}}{b^3} \)

140. \( \int x^3(a+bx)^{\frac{2}{3}} dx = \left( \frac{1}{13} y^5 - \frac{3}{11} ay^2 + \frac{1}{3} a^2 y - \frac{1}{7} a^3 \right) \frac{2y^2 \sqrt{y}}{b^3} \)

m. \( x^{-m}(a+bx)^{\frac{2}{3}} dx = x^{-m} y^{\frac{2}{3}} dx \)

141. \( \int \frac{(a+bx)^{\frac{2}{3}} dx}{x} = \left( \frac{1}{5} y^2 + \frac{1}{3} ay + a^2 \right) 2\sqrt{y} + a^3 \int \frac{dx}{x \sqrt{y}} \)

142. \( \int \frac{(a+bx)^{\frac{2}{3}} dx}{x^2} = -\frac{y^2 \sqrt{y}}{ax} + \frac{5b}{2a} \int \frac{y^{\frac{5}{2}} dx}{x} \)

143. \( \int \frac{(a+bx)^{\frac{2}{3}} dx}{x^3} = \left( -\frac{1}{2ax^2} - \frac{3b}{4a^2 x} \right) y^2 \sqrt{y} + \frac{15b^2}{8a^2} \int \frac{y^{\frac{5}{2}} dx}{x} \)

C. \( x^m(a+bx)^{\frac{n}{3}} dx \)

a. \( \frac{x^m dx}{(a+bx)^{\frac{1}{3}}} = x^m y^{-\frac{1}{3}} dx \)

144. \( \int \frac{dx}{(a+bx)^{\frac{1}{3}}} = \frac{3y^{\frac{4}{3}}}{2b} \)

145. \( \int \frac{xdx}{(a+bx)^{\frac{1}{3}}} = \left( \frac{1}{5} y - \frac{1}{2} a \right) \frac{3y^{\frac{5}{3}}}{b^2} \)

146. \( \int \frac{x^2 dx}{(a+bx)^{\frac{1}{3}}} = \left( \frac{1}{8} y^2 - \frac{2}{5} ay + \frac{1}{2} a^2 \right) \frac{3y^{\frac{8}{3}}}{b^3} \)

147. \( \int \frac{x^3 dx}{(a+bx)^{\frac{1}{3}}} = \left( \frac{1}{11} y^5 - \frac{3}{8} ay^2 + \frac{3}{5} a^2 y - \frac{1}{2} a^3 \right) \frac{3y^{\frac{8}{3}}}{b^4} \)

b. \( \frac{dx}{x^m(a+bx)^{\frac{1}{3}}} = \frac{dx}{x^m y^{\frac{1}{3}}} \)

148. \( \int \frac{dx}{x(a+bx)^{\frac{1}{3}}} = \frac{1}{a^{\frac{1}{3}}} \left( \frac{3}{2} hl \frac{y^{\frac{1}{3}} - a^{\frac{1}{3}}}{x^{\frac{1}{3}}} + \sqrt{3} \text{ arc tang} \frac{\sqrt{3} y^{\frac{1}{3}}}{y^{\frac{1}{3}} + 2a^{\frac{1}{3}}} \right) \)

149. \( \int \frac{dx}{x^2(a+bx)^{\frac{1}{3}}} = -\frac{y^{\frac{2}{3}}}{ax} - \frac{b}{3a} \int \frac{dx}{xy^{\frac{1}{3}}} \)

150. \( \int \frac{dx}{x^3(a+bx)^{\frac{1}{3}}} = \left( \frac{1}{2ax^2} + \frac{2b}{3a^2 x} \right) y^{\frac{2}{3}} + \frac{2b^2}{9a^2} \int \frac{dx}{xy^{\frac{1}{3}}} \)

c. \( x^m(a+bx)^{-\frac{2}{3}} dx = x^m y^{-\frac{2}{3}} dx \)

151. \( \int \frac{dx}{(a+bx)^{\frac{2}{3}}} = \frac{3y^{\frac{1}{3}}}{b} \)

152. \( \int \frac{xdx}{(a+bx)^{\frac{2}{3}}} = \left( \frac{1}{4} y - a \right) \frac{3y^{\frac{1}{3}}}{b^{\frac{4}{3}}} \)

153. \( \int \frac{x^2 dx}{(a+bx)^{\frac{2}{3}}} = \left( \frac{1}{7} y^2 - \frac{1}{2} ay + a^2 \right) \frac{3y^{\frac{1}{3}}}{b^{\frac{5}{3}}} \)

154. \( \int \frac{x^3 dx}{(a+bx)^{\frac{2}{3}}} = \left( \frac{1}{10} y^5 - \frac{3}{7} ay^2 + \frac{3}{4} a^2 y - a^2 \right) \frac{3y^{\frac{1}{3}}}{b^{\frac{8}{3}}} \)

d. \( \frac{dx}{x^m(a+bx)^{\frac{2}{3}}} = \frac{dx}{x^m y^{\frac{2}{3}}} \)

155. \( \int \frac{dx}{x(a+bx)^{\frac{2}{3}}} = \frac{1}{a^{\frac{2}{3}}} \left( \frac{3}{2} hl \frac{y^{\frac{1}{3}} - a^{\frac{1}{3}}}{x^{\frac{1}{3}}} + \sqrt{3} \text{ arc tang} \frac{\sqrt{3} y^{\frac{1}{3}}}{y^{\frac{1}{3}} + 2a^{\frac{1}{3}}} \right) \)

156. \( \int \frac{dx}{x^2(a+bx)^{\frac{2}{3}}} = -\frac{y^{\frac{1}{3}}}{ax} - \frac{2b}{3a} \int \frac{dx}{xy^{\frac{2}{3}}} \)

157. \( \int \frac{dx}{x^3(a+bx)^{\frac{2}{3}}} = \left( -\frac{1}{2ax^2} + \frac{5b}{6a^2 x} \right) y^{\frac{1}{3}} + \frac{5b^2}{9a^2} \int \frac{dx}{xy^{\frac{2}{3}}} \)

e. \( x^m(a+bx)^{\frac{1}{3}} dx = x^m y^{\frac{1}{3}} dx \)

158. \( \int (a+bx)^{\frac{1}{3}} dx = \frac{3y^{\frac{4}{3}}}{4b} \)

159. \( \int x(a+bx)^{\frac{1}{3}} dx = \left( \frac{1}{7} y - \frac{1}{4} a \right) \frac{3y^{\frac{5}{3}}}{b^2} \) 160. \( \int x^2(a+bx)^{\frac{3}{2}} dx = \left( \frac{1}{10} y^5 - \frac{2}{7} ay + \frac{1}{4} a^2 \right) \frac{3y^{\frac{4}{3}}}{b^{\frac{4}{3}}} \)

161. \( \int x^3(a+bx)^{\frac{1}{2}} dx = \left( \frac{1}{15} y^5 - \frac{3}{10} ay^2 + \frac{3}{7} a^2 y - \frac{1}{4} a^3 \right) \frac{3y^{\frac{2}{3}}}{b^{\frac{2}{3}}} \)

f. \( x^{-m}(a+bx)^{\frac{3}{2}} dx = x^{-m} y^{\frac{3}{2}} dx \)

162. \( \int \frac{(a+bx)^{\frac{3}{2}} dx}{x} = 3y^{\frac{1}{2}} + a \int \frac{dx}{xy^{\frac{1}{2}}} \)

163. \( \int \frac{(a+bx)^{\frac{3}{2}} dx}{x^2} = -\frac{y^{\frac{3}{2}}}{ax} + \frac{b}{3a} \int \frac{y^{\frac{1}{2}} dx}{x} \)

164. \( \int \frac{(a+bx)^{\frac{3}{2}} dx}{x^3} = \left( -\frac{1}{2ax^{\frac{3}{2}}} + \frac{b}{3a^2 x} \right) y^{\frac{3}{2}} - \frac{b^2}{9a^2} \)

\( \int \frac{y^{\frac{3}{2}} dx}{x} \)

g. \( x^m (a+bx)^{\frac{3}{2}} dx = x^m y^{\frac{3}{2}} dx \)

165. \( \int (a+bx)^{\frac{3}{2}} dx = \frac{3y^{\frac{5}{2}}}{5b} \)

166. \( \int x(a+bx)^{\frac{3}{2}} dx = \left( \frac{1}{8y^{\frac{1}{2}}} - \frac{1}{3a} \right) \frac{3y^{\frac{5}{2}}}{b^{\frac{5}{2}}} \)

167. \( \int x^2(a+bx)^{\frac{3}{2}} dx = \left( \frac{1}{11y^{\frac{1}{2}}} - \frac{1}{4} ay + \frac{1}{5} a^2 \right) \frac{3y^{\frac{5}{2}}}{b^{\frac{5}{2}}} \)

168. \( \int x^3(a+bx)^{\frac{3}{2}} dx = \left( \frac{1}{14y^{\frac{1}{2}}} - \frac{3}{11} ay^2 + \frac{2}{8} a^2 y - \frac{1}{5} a^3 \right) \frac{3y^{\frac{5}{2}}}{b^{\frac{5}{2}}} \)

h. \( x^{-m}(a+bx)^{\frac{3}{2}} dx = x^{-m} y^{\frac{3}{2}} dx \)

169. \( \int \frac{(a+bx)^{\frac{3}{2}} dx}{x} = \frac{3}{2y^{\frac{1}{2}}} + a \int \frac{dx}{xy^{\frac{1}{2}}} \)

170. \( \int \frac{(a+bx)^{\frac{3}{2}} dx}{x^2} = \frac{y^{\frac{5}{2}}}{ax} + \frac{2b}{3a} \int \frac{y^{\frac{1}{2}} dx}{x} \)

171. \( \int \frac{(a+bx)^{\frac{3}{2}} dx}{x^3} = \left( -\frac{1}{2ax^{\frac{3}{2}}} + \frac{b}{6a^2 x} \right) y^{\frac{5}{2}} - \frac{b^2}{9a^2} \)

\( \int \frac{y^{\frac{5}{2}} dx}{x} \)

D. \( x^m (a+bx^2)^{\frac{n}{2}} dx \)

a. \( x^m (a+bx^2)^{-\frac{1}{2}} dx = x^m y^{-\frac{1}{2}} dx \)

172. \( \int \frac{dx}{\sqrt{(a+bx^2)}} = \frac{1}{\sqrt{b}} \text{ hl } (\sqrt{x} b + \sqrt{y}) = \frac{1}{\sqrt{-b}} \text{ arc sin } x \sqrt{\frac{-b}{a}} \); thus \( \int \frac{dx}{\sqrt{(1-x^2)}} = \text{arc sin } x \)

173. \( \int \frac{xdx}{\sqrt{(a+bx^2)}} = \frac{\sqrt{y}}{b} \)

\( \int \frac{xdx}{\sqrt{(1-x^2)}} = -\sqrt{y} = \cos \text{ arc sin } x \)

174. \( \int \frac{x^2 dx}{\sqrt{(a+bx^2)}} = \frac{x \sqrt{y}}{2b} - \frac{a}{2b} \int \frac{dx}{\sqrt{y}} \)

\( \int \frac{x^2 dx}{\sqrt{(1-x^2)}} = -\frac{1}{2} x \sqrt{y} + \frac{1}{2} \text{ arc sin } x \)

175. \( \int \frac{x^2 dx}{\sqrt{(a+bx^2)}} = \left( \frac{x^2}{3b} - \frac{2a}{3b^2} \right) \sqrt{y} \)

\( \int \frac{x^3 dx}{\sqrt{(1-x^2)}} = -\left( \frac{1}{3} x^2 + \frac{2}{3} \right) \sqrt{y} \)

176. \( \int \frac{x^4 dx}{\sqrt{(a+bx^2)}} = \left( \frac{x^3}{4b} - \frac{3ax}{8b^2} \right) \sqrt{y} + \frac{3a^2}{8b^2} \int \frac{dx}{\sqrt{y}} \)

\( \int \frac{x^4 dx}{\sqrt{(1-x^2)}} = -\left( \frac{1}{4} x^3 + \frac{3}{8} x \right) \sqrt{y} + \frac{3}{8} \text{ arc sin } x \)

b. \( \frac{dx}{x^m \sqrt{(a+bx^2)}} = \frac{dx}{x^m \sqrt{y}} \)

177. \( \int \frac{dx}{x \sqrt{(a+bx^2)}} = \frac{1}{2\sqrt{a}} \text{ hl } \frac{\sqrt{y}-\sqrt{a}}{\sqrt{y}+\sqrt{a}} = \frac{1}{\sqrt{-a}} \text{ arc sec } \left( x \sqrt{-\frac{b}{a}} \right) \); thus

\( \int \frac{dx}{x \sqrt{(1+x^2)}} \text{ hl} = \frac{\sqrt{y}-1}{x} \)

\( \int \frac{dx}{x \sqrt{(1-x^2)}} = \text{hl} \frac{\sqrt{y}-1}{x} = \text{hl} \frac{1-\sqrt{y}}{x} \)

\( \int \frac{dx}{x \sqrt{(x^2-1)}} = \text{arc sec } x \)

178. \( \int \frac{dx}{x^2 \sqrt{(a+bx^2)}} = -\frac{\sqrt{y}}{ax} \)

179. \( \int \frac{dx}{x^5 \sqrt{(a+bx^2)}} = -\frac{\sqrt{y}}{2ax^2} - \frac{b}{2a} \int \frac{dx}{x \sqrt{y}} \)

180. \( \int \frac{dx}{x^4 \sqrt{(a+bx^2)}} = \left( -\frac{1}{3ax^3} + \frac{2b}{3a^2 x^2} \right) \sqrt{y} \)

Remark. In some of these cases, the signs of the roots, being ambiguous, require to be determined by the conditions of the problem; but we must adhere to the same root in the same solution.

c. \( x^m (a+bx^2)^{-\frac{3}{2}} dx = x^m y^{-\frac{3}{2}} dx \)

181. \( \int \frac{dx}{(a+bx^2)^{\frac{3}{2}}} = \frac{x}{a \sqrt{y}} \)

182. \( \int \frac{x dx}{(a+bx^2)^{\frac{3}{2}}} = -\frac{1}{b \sqrt{y}} \)

183. \( \int \frac{x^2 dx}{(a+bx^2)^{\frac{3}{2}}} = -\frac{x}{b \sqrt{y}} + \frac{1}{b} \int \frac{dx}{\sqrt{y}} \)

184. \( \int \frac{x^3 dx}{(a+bx^2)^{\frac{3}{2}}} = \left( \frac{x^2}{b} + \frac{2a}{b^2} \right) \frac{1}{\sqrt{y}} \)

d. \( \frac{dx}{x^m (a+bx^2)^{\frac{3}{2}}} = \frac{dx}{x^m y^{\frac{3}{2}}} \)

185. \( \int \frac{dx}{x (a+bx^2)^{\frac{3}{2}}} = \frac{1}{a \sqrt{y}} + \frac{1}{a} \int \frac{dx}{x \sqrt{y}} \) (177.) 186. \( \int \frac{dx}{x^2(a+bx^2)^{\frac{3}{2}}} = \left( -\frac{1}{ax} - \frac{2bx}{a^2} \right) \frac{1}{\sqrt{y}} \)

187. \( \int \frac{dx}{x^3(a+bx^2)^{\frac{3}{2}}} = \left( -\frac{1}{2ax^2} - \frac{3b}{2a^2} \right) \frac{1}{\sqrt{y}} - \frac{3b}{2a^2} \)

\( \int \frac{dx}{x \sqrt{y}} \)

e. \( x^m (a+bx^2)^{-\frac{3}{2}} dx = x^m y^{-\frac{5}{2}} dx \)

188. \( \int \frac{dx}{(a+bx^2)^{\frac{5}{2}}} = \left( \frac{2bx^3}{3a^2} + \frac{x}{a} \right) \frac{1}{y \sqrt{y}} \)

189. \( \int \frac{xdx}{(a+bx^2)^{\frac{5}{2}}} = -\frac{1}{3by \sqrt{y}} \)

190. \( \int \frac{x^2dx}{(a+bx^2)^{\frac{5}{2}}} = \frac{x^3}{3ay \sqrt{y}} \)

191. \( \int \frac{x^3dx}{(a+bx^2)^{\frac{5}{2}}} = \left( -\frac{x^2}{b} - \frac{2a}{3b^2} \right) \frac{1}{y \sqrt{y}} \)

f. \( \frac{dx}{x^m (a+bx^2)^{\frac{5}{2}}} dx = \frac{dx}{x^m y^{\frac{5}{2}}} \)

192. \( \int \frac{dx}{x(a+bx^2)^{\frac{5}{2}}} = \left( \frac{4}{3a} + \frac{bx^2}{a^2} \right) \frac{1}{y \sqrt{y}} + \frac{1}{a^2} \)

\( \int \frac{dx}{y \sqrt{y}} \)

193. \( \int \frac{dx}{x^2(a+bx^2)^{\frac{5}{2}}} = -\frac{1}{axy \sqrt{y}} - \frac{4b}{a} \int \frac{dx}{y} \)

194. \( \int \frac{dx}{x^3(a+bx^2)^{\frac{5}{2}}} = -\frac{1}{2ax^2y \sqrt{y}} - \frac{5b}{2a} \int \frac{dx}{xy^{\frac{3}{2}}} \)

g. \( x^m \sqrt{a+bx^2} dx = x^m y^{\frac{1}{2}} dx \)

195. \( \int \sqrt{a+bx^2} dx = \frac{x \sqrt{y}}{2} + \frac{a}{2} \int \frac{dx}{\sqrt{y}} \)

196. \( \int x \sqrt{a+bx^2} dx = \frac{y \sqrt{y}}{3b} \)

197. \( \int x^2 \sqrt{a+bx^2} dx = \frac{xy \sqrt{y}}{4b} - \frac{a}{4b} \int \sqrt{y} dx \)

198. \( \int x^5 \sqrt{a+bx^2} dx = \left( \frac{x^2}{5b} - \frac{2a}{15b^2} \right) y \sqrt{y} \)

Particular values, from \( x = 0 \) to \( x = a \); putting \( \pi = 3.14159 \).

i. \( \int' (a^2-x^2)^{\frac{3}{2}} dx = \frac{3\pi a^4}{16} \)

ii. \( \int' x(a^2-x^2)^{\frac{3}{2}} dx = \frac{a^5}{5} \)

iii. \( \int' x^2(a^2-x^2)^{\frac{3}{2}} dx = \frac{1}{6} \cdot \frac{3\pi a^6}{16} \)

iv. \( \int' x^3(a^2-x^2)^{\frac{3}{2}} dx = \frac{2}{7} \cdot \frac{a^7}{5} \)

v. \( \int' x^4(a^2-x^2)^{\frac{3}{2}} dx = \frac{1}{6} \cdot \frac{3\pi a^8}{16} \)

vi. \( \int' x^5(a^2-x^2)^{\frac{3}{2}} dx = \frac{2}{7} \cdot \frac{a^9}{5} \)

k. \( x^{-m}(a+bx^2)^{\frac{5}{2}} dx = x^{-m} y^{\frac{5}{2}} dx \)

206. \( \int \frac{(a+bx^2)^{\frac{3}{2}} dx}{x} = \left( \frac{y}{3} + a \right) \sqrt{y} + a^2 \int \frac{dx}{x \sqrt{y}} \)

207. \( \int \frac{(a+bx^2)^{\frac{3}{2}} dx}{x^2} = -\frac{y^{\frac{3}{2}}}{ax} + \frac{4b}{a} \int \frac{y^{\frac{3}{2}} dx}{x} \)

208. \( \int \frac{(a+bx^2)^{\frac{3}{2}} dx}{x^3} = -\frac{y^{\frac{3}{2}}}{2ax^2} + \frac{3b}{2a} \int \frac{y^{\frac{3}{2}} dx}{x} \)

l. \( x^m (a+bx^2)^{\frac{5}{2}} dx = x^m y^{\frac{5}{2}} dx \)

209. \( \int (a+bx^2)^{\frac{5}{2}} dx = \left( \frac{y^2}{6} + \frac{5ay}{24} + \frac{5a^2}{16} \right) x \sqrt{y} + \frac{5a^3}{16} \int \frac{dx}{\sqrt{y}} \)

210. \( \int x(a+bx^2)^{\frac{5}{2}} dx = \frac{y^{\frac{7}{2}}}{7b} \)

211. \( \int x^2(a+bx^2)^{\frac{5}{2}} dx = \frac{xy^{\frac{7}{2}}}{8b} - \frac{a}{8b} \int y^{\frac{5}{2}} dx \) 212. \( \int x^3(a+bx^2)^{\frac{5}{3}} dx = \left( \frac{x^2}{9b} - \frac{2a}{63b^2} \right) y^{\frac{7}{3}} \)

Particular values, from \( x = 0 \) to \( x = a \); putting \( \sigma = 3.14159 \).

i. \( \int' (a^2-x^2)^{\frac{5}{3}} dx = \frac{5\pi a^6}{32} \)

ii. \( \int' x(a^2-x^2)^{\frac{5}{3}} dx = \frac{a^7}{7} \)

iii. \( \int' x^2(a^2-x^2)^{\frac{5}{3}} dx = \frac{1}{8} \cdot \frac{5\pi a^8}{32} \)

iv. \( \int' x^3(a^2-x^2)^{\frac{5}{3}} dx = \frac{2}{9} \cdot \frac{a^9}{7} \)

v. \( \int' x^4(a^2-x^2)^{\frac{5}{3}} dx = \frac{1.3}{8.10} \cdot \frac{5\pi a^{10}}{32} \)

vi. \( \int' x^5(a^2-x^2)^{\frac{5}{3}} dx = \frac{2.4}{9.11} \cdot \frac{a^{11}}{7} \)

m. \( x^{-M}(a+bx^2)^{\frac{5}{3}} dx = x^{-M} y^{\frac{5}{3}} dx \)

213. \( \int \frac{(a+bx^2)^{\frac{5}{3}}}{x} = \left( \frac{y^2}{5} + \frac{ay}{3} + a^2 \right) \lambda y + a^3 \int \frac{dx}{x \sqrt{y}} \)

214. \( \int \frac{(a+bx^2)^{\frac{5}{3}}}{x^2} = -\frac{y^{\frac{7}{3}}}{ax} + \frac{6b}{a} \int \frac{y^{\frac{5}{3}} dx}{x} \)

215. \( \int \frac{(a+bx^2)^{\frac{5}{3}}}{x^3} = -\frac{y^{\frac{7}{3}}}{2ax^2} + \frac{5b}{2a} \int \frac{y^{\frac{5}{3}} dx}{x} \)

E. \( x^M(ax+bx^2)^{\frac{n}{3}} dx \)

a. \( x^M(ax+bx^2)^{-\frac{1}{3}} dx = x^M y^{-\frac{1}{3}} dx \)

216. \( \int \frac{dx}{\sqrt{(ax+bx^2)}} = \frac{1}{\sqrt{b}} \ln \frac{\sqrt{y} + x \sqrt{b}}{\sqrt{y} - x \sqrt{b}} = \frac{2}{\sqrt{-b}} \arctan \frac{x \sqrt{b}}{\sqrt{y}} \): thus

\( \int \frac{dx}{\sqrt{(x^2+x)}} = \pm \ln (2x+1 \pm 2\sqrt{y}) \)

\( \int \frac{dx}{\sqrt{(x^2-x)}} = \pm \ln (1-2x \pm 2\sqrt{y}) \)

217. \( \int \frac{xdx}{\sqrt{(ax+bx^2)}} = \frac{\sqrt{y}}{b} - \frac{a}{2b} \int \frac{dx}{\sqrt{y}} \)

218. \( \int \frac{x^2 dx}{\sqrt{(ax+bx^2)}} = \left( \frac{x}{2b} - \frac{3a}{4b^2} \right) \sqrt{y} + \frac{3a^2}{8b^2} \int \frac{dx}{\sqrt{y}} \)

219. \( \int \frac{x^3 dx}{\sqrt{(ax+bx^2)}} = \left( \frac{x^2}{3b} - \frac{5ax}{12b^2} + \frac{5a^2}{8b^3} \right) \sqrt{y} - \frac{5a^3}{16b^3} \int \frac{dx}{\sqrt{y}} \)

b. \( \frac{dx}{x^M \sqrt{(ax+bx^2)}} = \frac{dx}{x^M \sqrt{y}} \)

220. \( \int \frac{dx}{x \sqrt{(ax+bx^2)}} = \frac{2\sqrt{y}}{ax} \)

221. \( \int \frac{dx}{x^2 \sqrt{(ax+bx^2)}} = \left( -\frac{1}{3ax^2} + \frac{2b}{3a^2x} \right) 2\sqrt{y} \)

222. \( \int \frac{dx}{x^5 \sqrt{(ax+bx^2)}} = \left( -\frac{1}{5ax^5} + \frac{4b}{15a^2x^2} - \frac{8b^2}{15a^3x} \right) 2\sqrt{y} \)

c. \( x^M(ax+bx^2)^{-\frac{1}{3}} dx \)

223. \( \int \frac{dx}{(ax+bx^2)^{\frac{3}{5}}} = -\frac{2(2bx+a)}{a^2 \sqrt{y}} \)

224. \( \int \frac{xdx}{(ax+bx^2)^{\frac{3}{5}}} = \frac{2x}{a \sqrt{y}} \)

225. \( \int \frac{x^2 dx}{(ax+bx^2)^{\frac{3}{5}}} = -\frac{2x}{\sqrt{y}} + \frac{1}{b} \int \frac{dx}{\sqrt{y}} \)

226. \( \int \frac{x^5 dx}{(ax+bx^2)^{\frac{3}{5}}} = \left( \frac{x^2}{b} + \frac{3ax}{b^2} \right) \frac{1}{\sqrt{y}} - \frac{3a}{2b^2} \int \frac{dx}{\sqrt{y}} \)

d. \( \frac{dx}{x^M(ax+bx^2)^{\frac{3}{5}}} = \frac{dx}{x^M y^{\frac{3}{5}}} \)

227. \( \int \frac{dx}{x(x+bx^2)^{\frac{3}{5}}} = -\frac{2}{3ax \sqrt{y}} - \frac{4b}{3a} \int \frac{dx}{y^{\frac{3}{5}}} \)

228. \( \int \frac{dx}{x^2(ax+bx^2)^{\frac{3}{5}}} = \left( -\frac{1}{5ax^2} + \frac{2b}{5a^2x} \right) \frac{2}{\sqrt{y}} \)

\[ + \frac{8b^2}{5a^2} \int \frac{dx}{y^{\frac{3}{5}}} \]

229. \( \int \frac{dx}{x^5(ax+bx^2)^{\frac{3}{5}}} = \left( -\frac{1}{7ax^5} + \frac{8b}{35a^2x^2} - \frac{16b^2}{35a^3x} \right) \frac{2}{\sqrt{y}} - \frac{64b^3}{35a^3} \int \frac{dx}{y^{\frac{3}{5}}} \)

e. \( x^M(ax+bx^2)^{-\frac{5}{3}} dx = x^M y^{-\frac{5}{3}} dx \)

220. \( \int \frac{dx}{(ax+bx^2)^{\frac{5}{3}}} = \left( -\frac{2}{3y} + \frac{16b}{3a^2} \right) \frac{2bx+a}{a^2 \sqrt{y}} \)

231. \( \int \frac{xdx}{(ax+bx^2)^{\frac{5}{3}}} = \frac{2x}{3ay \sqrt{y}} - \frac{8(2bx+a)}{3a^2 \sqrt{y}} = \left( \frac{1}{a+bx} - \frac{4(2bx+a)}{a^2} \right) \frac{2}{a \sqrt{y}} \)

232. \( \int \frac{x^2 dx}{(ax+bx^2)^{\frac{5}{3}}} = \left( \frac{2x^2}{3ay} + \frac{4x}{3a^2} \right) \frac{1}{\sqrt{y}} \)

233. \( \int \frac{x^3 dx}{(ax+bx^2)^{\frac{5}{3}}} = \frac{2x^3}{3ay \sqrt{y}} \)

f. \( \frac{dx}{x^M(ax+bx^2)^{\frac{5}{3}}} = \frac{dx}{x^M y^{\frac{5}{3}}} \)

234. \( \int \frac{dx}{x(ax+bx^2)^{\frac{5}{3}}} = -\frac{2}{5axy \sqrt{y}} - \frac{8b}{5a} \int \frac{dx}{y^{\frac{5}{3}}} \)

235. \( \int \frac{dx}{x^2(ax+bx^2)^{\frac{5}{3}}} = \left( -\frac{1}{7ax^2} + \frac{2b}{7a^2x} \right) \frac{2}{y \sqrt{y}} \)

\[ + \frac{16b^2}{7a^2} \int \frac{dx}{y^{\frac{5}{3}}} \]

236. \( \int \frac{dx}{x^5(ax+bx^2)^{\frac{5}{3}}} = \left( -\frac{1}{9ax^5} + \frac{4b}{21a^2x^2} - \frac{8b^2}{21a^3x} \right) \frac{2}{y \sqrt{y}} - \frac{64b^3}{21a^3} \int \frac{dx}{y^{\frac{5}{3}}} \) g. \( x^m \sqrt{(ax+bx^2)} dx = x^m y^{1/3} dx \)

237. \( \int \sqrt{(ax+bx^2)} dx = \left( \frac{x}{2} + \frac{a}{4b} \right) \sqrt{y} - \frac{a^2}{8b} \)

\( \int \frac{dx}{\sqrt{y}} \); thus

\( \int \sqrt{(ax-x^2)} dx = \frac{1}{2} \text{circ. segm diam } a \text{ vers sin } x. \)

238. \( \int x \sqrt{(ax+bx^2)} dx = \frac{y \sqrt{y}}{3b} - \frac{a}{2b} \int \sqrt{y} dx \)

239. \( \int x^2 \sqrt{(ax+bx^2)} dx = \left( \frac{x}{4b} - \frac{5a}{24b^2} \right) y \sqrt{y} + \frac{5a^2}{16b^2} \int \sqrt{y} dx \)

240. \( \int x^5 \sqrt{(ax+bx^2)} dx = \left( \frac{x^2}{5b} - \frac{7ax}{40b^2} + \frac{7a^2}{48b^3} \right) y \sqrt{y} - \frac{7a^3}{30b^3} \int \sqrt{y} dx \)

h. \( x^{-m} \sqrt{(ax+bx^2)} dx = x^{-m} y^{1/3} dx \)

241. \( \int \frac{\sqrt{(ax+bx^2)}}{x} dx = \sqrt{y} + \frac{a}{2} \int \frac{dx}{\sqrt{y}} \)

242. \( \int \frac{\sqrt{(ax+bx^2)}}{x^2} dx = - \frac{2\sqrt{y}}{x} + b \int \frac{dx}{\sqrt{y}} \)

243. \( \int \frac{\sqrt{(ax+bx^2)}}{x^3} dx = - \frac{2y \sqrt{y}}{3ax^3} \)

i. \( x^m (ax+bx^2)^{3/2} dx = x^m y^{5/3} dx \)

244. \( (ax+bx^2)^{3/2} dx = \left( \frac{y}{b} - \frac{3a^2}{8b^2} \right) \frac{2bx+a}{8} \sqrt{y} + \frac{3a^4}{128b^2} \int \frac{dx}{\sqrt{y}} \)

245. \( \int x (ax+bx^2)^{3/2} dx = \frac{y^{5/2}}{5b} - \frac{a}{2b} \int y^{3/2} dx \)

246. \( \int x^2 (ax+bx^2)^{3/2} dx = \left( \frac{x}{6b} - \frac{7a}{60b^2} \right) y^{5/2} + \frac{7a^2}{24b^2} \int y^{3/2} dx \)

247. \( \int x^5 (ax+bx^2)^{3/2} dx = \left( \frac{x^2}{7b} - \frac{3ax}{28b^2} + \frac{3a^2}{40b^3} \right) y^{5/2} + \frac{3a^5}{16b^5} \int y^{3/2} dx \)

k. \( x^{-m} (ax+bx^2)^{3/2} dx = x^{-m} y^{5/3} dx \)

248. \( \int \frac{(ax+bx^2)^{3/2} dx}{x} = \frac{y \sqrt{y}}{3} + \frac{a}{2} \int \sqrt{y} dx \)

249. \( \int \frac{(ax+bx^2)^{3/2} dx}{x^2} = \left( \frac{5a}{4} + \frac{bx}{2} \right) \sqrt{y} + \frac{3a^2}{8} \int \frac{dx}{\sqrt{y}} \)

250. \( \int \frac{(ax+bx^2)^{3/2} dx}{x^3} = \left( b - \frac{2a}{x} \right) \sqrt{y} + \frac{3ab}{2} \int \frac{dx}{\sqrt{y}} \)

l. \( x^m (ax+bx^2)^{5/3} dx = x^m y^{5/3} dx \)

251. \( \int (ax+bx^2)^{5/3} dx = \left( \frac{y^2}{b} - \frac{5a^2y}{16b^2} + \frac{15a^4}{128b^3} \right) \)

\( \frac{2bx+a}{12} \sqrt{y} - \frac{5a^6}{1024b^3} \int \frac{dx}{\sqrt{y}} \)

252. \( \int x (ax+ax^2)^{5/3} dx = \frac{y^{7/3}}{7b} - \frac{a}{2b} \int y^{5/3} dx \)

253. \( \int x^2 (ax+bx^2)^{5/3} dx = \left( \frac{x}{8b} - \frac{9a}{112b^2} \right) y^{7/3} + \frac{9a^2}{32b^2} \int y^{5/3} dx \)

254. \( \int x^3 (ax+bx^2)^{5/3} dx = \left( \frac{x^2}{9b} - \frac{11ax}{144b^2} + \frac{11a^2}{224b^3} \right) \)

\( y^{7/3} - \frac{11a^3}{64b^3} \int y^{5/3} dx \)

m. \( x^{-m} (ax+bx^2)^{5/3} dx = x^{-m} y^{5/3} dx \)

255. \( \int \frac{(ax+bx^2)^{5/3} dx}{x} = \frac{y^{5/3}}{5} + \frac{a}{2} \int y^{5/3} dx \)

256. \( \int \frac{(ax+bx^2)^{5/3} dx}{x^2} = \left( \frac{y^2}{4x} + \frac{5ay}{24} \right) \sqrt{y} + \frac{5a^2}{16} \int \sqrt{y} dx \)

257. \( \int \frac{(ax+bx^2)^{5/3} dx}{x^5} = \left( \frac{y^2}{3x^2} + \frac{5ay}{12x} + \frac{5a^2}{8} \right) \sqrt{y} + \frac{5a^3}{16} \int \frac{dx}{\sqrt{y}} \)

F. \( x^m (a+bx+cx^2)^n dx \)

a. \( x^m (a+bx+cx^2)^{-1/2} dx = x^m y^{-1/2} dx \)

258. \( \int \frac{dx}{\sqrt{(a+bx+cx^2)}} = \frac{1}{\sqrt{c}} \ln (2cx+b \pm 2\sqrt{c} \sqrt{y}) \)

\( = \frac{-1}{\sqrt{-c}} \arcsin \frac{2cx+b}{\sqrt{(b^2-4ac)}} \)

259. \( \int \frac{xdx}{\sqrt{(a+bx+cx^2)}} = \frac{\sqrt{y}}{c} - \frac{b}{2c} \int \frac{dx}{\sqrt{y}} \)

260. \( \int \frac{x^2 dx}{\sqrt{(a+bx+cx^2)}} = \left( \frac{x}{2c} - \frac{3b}{4c^2} \right) \sqrt{y} + \left( \frac{3b^2}{8c^2} - \frac{a}{2c} \right) \int \frac{dx}{\sqrt{y}} \)

261. \( \int \frac{x^3 dx}{\sqrt{(a+bx+cx^2)}} = \left( \frac{x^2}{3c} - \frac{5b^2}{12c^2} + \frac{5b^2}{8c^2} - \frac{2a}{3c^2} \right) \)

\( \sqrt{y} - \left( \frac{5b^3}{16c^3} - \frac{3ab}{4c^2} \right) \int \frac{dx}{\sqrt{y}} \) b. \( \frac{dx}{x^m \sqrt{(a+bx+cx^2)}} = \frac{dx}{x^m \sqrt{y}} \)

262. \( \int \frac{dx}{x \sqrt{(a+bx+cx)}} = \frac{1}{\sqrt{a}} \ln \frac{2a+bx \pm 2\sqrt{a} \sqrt{y}}{x} \)

\( = \frac{1}{\sqrt{-a}} \arctan \frac{2a+bx}{2\sqrt{-a} \sqrt{y}} \)

263. \( \int \frac{dx}{x^2 \sqrt{(a+bx+cx^2)}} = -\frac{\sqrt{y}}{ax} - \frac{b}{2a} \int \frac{dx}{x \sqrt{y}} \)

264. \( \int \frac{dx}{x^3 \sqrt{(a+bx+cx^2)}} = \left( -\frac{1}{2ax^2} + \frac{3b}{4a^2x} \right) \sqrt{y} \)

\( + \left( \frac{3b^2}{8a^2} - \frac{c}{2a} \right) \int \frac{dx}{x \sqrt{y}} \)

c. \( x^m (a+bx+cx^2)^{-\frac{3}{2}} dx = x^m y^{-\frac{3}{2}} dx \)

265. \( \int \frac{dx}{(a+bx+cx^2)^{\frac{3}{2}}} = \frac{4cx+2b}{(4ac-b^2) \sqrt{y}} \)

266. \( \int \frac{x dx}{(a+bx+cx^2)^{\frac{3}{2}}} = \frac{4a+2bx}{(4ac-b^2) \sqrt{y}} \)

267. \( \int \frac{x^2 dx}{(a+bx+cx^2)^{\frac{3}{2}}} = \frac{(4ac-2b^2)x-2ab}{c(4ac-b^2) \sqrt{y}} + \frac{1}{c} \)

\( \int \frac{dx}{\sqrt{y}} \)

268. \( \int \frac{x^2 dx}{(x+bx+cx^2)^{\frac{3}{2}}} = \frac{x^2}{c \sqrt{y}} - \frac{2a}{c} \int \frac{xdx}{y^{\frac{3}{2}}} - \frac{3b}{2a} \int \frac{dx}{y^{\frac{3}{2}}} \)

\( \int \frac{x^2 dx}{y^{\frac{3}{2}}} \)

d. \( \frac{dx}{x^m (a+bx+cx^2)^{\frac{5}{2}}} = \frac{dx}{x^m y^{\frac{5}{2}}} \)

269. \( \int \frac{dx}{x(a+bx+cx^2)^{\frac{5}{2}}} = \frac{1}{a \sqrt{y}} - \frac{b}{2a} \int \frac{dx}{y^{\frac{3}{2}}} + \frac{1}{a} \)

\( \int \frac{dx}{x \sqrt{y}} \)

270. \( \int \frac{dx}{x^2 (a+bx+cx^2)^{\frac{5}{2}}} = \left( -\frac{1}{ax} - \frac{3b}{2a^2} \right) \frac{1}{\sqrt{y}} + \left( \frac{3b^2}{4a^2} - \frac{2c}{a} \right) \int \frac{dx}{y^{\frac{3}{2}}} - \frac{3b}{2a^2} \int \frac{dx}{x \sqrt{y}} \)

271. \( \int \frac{dx}{(x^3 a+bx+cx^2)^{\frac{5}{2}}} = \left( -\frac{1}{2ax^2} + \frac{5b}{4a^2x} + \frac{15a^2}{8a^3} - \frac{3c}{2a^2} \right) \frac{1}{\sqrt{y}} - \left( \frac{15b^2}{16a^3} - \frac{13bc}{4a^2} \right) \int \frac{dx}{y^{\frac{3}{2}}} + \left( \frac{15b^2}{8a^3} - \frac{3c}{2a^2} \right) \int \frac{dx}{x \sqrt{y}} \)

e. \( x^m (a+bx+cx^2)^{-\frac{5}{2}} dx = x^m y^{-\frac{5}{2}} dx \)

272. \( \int \frac{dx}{(a+bx+cx^2)^{\frac{5}{2}}} = \left( \frac{1}{4ac-b^2} y + \frac{8c}{3(4ac-b^2)^2} \right) \frac{4cx+2b}{\sqrt{y}} \)

273. \( \int \frac{xdx}{(a+bx+cx^2)^{\frac{5}{2}}} = -\frac{1}{3cy \sqrt{y}} - \frac{b}{2c} \int \frac{dx}{y^{\frac{5}{2}}} \)

274. \( \int \frac{x^2 dx}{(a+bx+cx^2)^{\frac{5}{2}}} = \left( -\frac{x}{2c} + \frac{1}{12c^2} \right) \frac{1}{y \sqrt{y}} + \left( \frac{b^2}{8c^2} + \frac{a}{2c} \right) \int \frac{dx}{y^{\frac{5}{2}}} \)

275. \( \int \frac{x^3 dx}{(a+bx+cx^2)^{\frac{5}{2}}} = \left( -\frac{x^2}{c} - \frac{bx}{4c^2} + \frac{b^2}{24c^3} - \frac{2a}{3c^2} \right) \frac{1}{y \sqrt{y}} + \left( \frac{b^3}{16c^3} - \frac{3ab}{4c^2} \right) \int \frac{dx}{y^{\frac{5}{2}}} \)

f. \( \frac{dx}{x^m (a+bx+cx^2)^{\frac{5}{2}}} = \frac{dx}{x^m y^{\frac{5}{2}}} \)

276. \( \int \frac{dx}{x(a+bx+cx^2)^{\frac{5}{2}}} = \left( \frac{1}{3ay} + \frac{1}{a^2} \right) \frac{1}{\sqrt{y}} - \frac{b}{2a} \)

\( \int \frac{dx}{y^{\frac{5}{2}}} - \frac{b}{2a^2} \int \frac{dx}{y^{\frac{3}{2}}} + \frac{1}{a^2} \int \frac{dx}{x \sqrt{y}} \)

277. \( \int \frac{dx}{x^2 (a+bx+cx^2)^{\frac{5}{2}}} = -\frac{1}{axy \sqrt{y}} - \frac{5b}{2a} \int \frac{dx}{xy^{\frac{5}{2}}} \)

\( - \frac{4c}{a} \int \frac{dx}{y^{\frac{5}{2}}} \)

278. \( \int \frac{dx}{x^3 (a+bx+cx^2)^{\frac{5}{2}}} = \left( -\frac{1}{2ax^2} + \frac{7b}{4a^2} \right) \frac{1}{y \sqrt{y}} \)

\( + \left( \frac{95b^2}{8a^2} - \frac{5c}{2a} \right) \int \frac{dx}{xy^{\frac{5}{2}}} + \frac{7bc}{8a^2} \int \frac{dx}{y^{\frac{5}{2}}} \)

g. \( x^m (a+bx+cx^2) dx = x^m \sqrt{y} dx \)

279. \( \int (a+bx+cx^2) dx = \frac{2cx+b}{4c} \sqrt{y} + \frac{4ac-b^2}{8c} \)

\( \int \frac{dx}{\sqrt{y}} \)

280. \( \int x \sqrt{(a+bx+cx^2)} dx = \frac{y \sqrt{y}}{3c} - \frac{b}{2c} \int \sqrt{y} dx \)

281. \( \int x^2 \sqrt{(a+bx+cx^2)} dx = \left( \frac{x}{4c} - \frac{5b}{24c^2} \right) y \sqrt{y} \)

\( + \left( \frac{5b^2}{16c^2} - \frac{a}{4c} \right) \int \sqrt{y} dx \)

282. \( \int x^3 \sqrt{(a+bx+cx^2)} dx = \left( \frac{x^2}{5c} - \frac{7bx}{40c^2} + \frac{7b^2}{48c^3} \right. \)

\( \left. - \frac{2a}{15c^2} \right) y \sqrt{y} - \left( \frac{7b^3}{32c^3} - \frac{3ab}{8c^2} \right) \int \sqrt{y} dx \)

h. \( x^{-m} \sqrt{(a+bx+cx^2)} dx = x^{-m} \sqrt{y} dx \)

283. \( \int \frac{\sqrt{(a+bx+cx^2)} dx}{x} = \sqrt{y} + a \int \frac{dx}{x \sqrt{y}} + \frac{b}{2} \int \frac{dx}{\sqrt{y}} \)

284. \( \int \frac{\sqrt{(a+bx+cx^2)} dx}{x^2} = -\frac{\sqrt{y}}{x} + b \int \frac{dx}{x \sqrt{y}} + \)

\( \int \frac{c dx}{\sqrt{y}} \)

285. \( \int \frac{\sqrt{(a+bx+cx^2)} dx}{x^3} = -\left( \frac{1}{2x^2} + \frac{b}{4ac} \right) \sqrt{y} - \left( \frac{b^2}{8a} - \frac{c}{2} \right) \int \frac{dx}{x \sqrt{y}} \) i. \( x^m (a + bx + cx^2)^{\frac{5}{2}} dx = x^n y^{\frac{5}{2}} dx \)

286. \( \int (a + bx + cx^2)^{\frac{5}{2}} dx = \left( \frac{y}{8c} + \frac{12ac - 3b^2}{64c^2} \right) \) \[ (2cx + b)\sqrt{y} + 12ac \frac{3(4ac - b^2)}{128c^2} \int \frac{dx}{\sqrt{y}} \]

287. \( \int x(a + bx + cx^2)^{\frac{5}{2}} dx = \frac{y^{\frac{5}{2}}}{5c} - \frac{b}{2c} \int y^{\frac{5}{2}} dx \)

288. \( \int x^2(a + bx + cx^2)^{\frac{5}{2}} dx = \left( \frac{x}{6c} - \frac{7b}{60c^2} \right) y^{\frac{5}{2}} + \left( \frac{7b^2}{24c^2} - \frac{a}{6c} \right) \int y^{\frac{5}{2}} dx \)

289. \( \int x^3(a + bx + cx^2)^{\frac{5}{2}} dx = \left( \frac{x^2}{7c} - \frac{3bx}{28c^2} + \frac{3b^2}{40c^3} \right) y^{\frac{5}{2}} - \left( \frac{2a}{35c^2} \right) y^{\frac{5}{2}} - \left( \frac{3b^2}{16c^3} - \frac{ab}{4c^2} \right) \int y^{\frac{5}{2}} dx \)

k. \( x^{-m}(a + bx + cx^2)^{\frac{5}{2}} dx = x^{-n} y^{\frac{5}{2}} dx \)

290. \( \int \frac{(a + bx + cx^2)^{\frac{5}{2}} dx}{x} = \left( \frac{y}{3} + a \right) \sqrt{y} + a^2 \) \[ \int \frac{dx}{x\sqrt{y}} + \frac{ab}{2} \int \frac{dx}{\sqrt{y}} + \frac{b}{2} \int \sqrt{y} dx \]

291. \( \int \frac{(a + bx + cx^2)^{\frac{5}{2}} dx}{x^2} = -\frac{y^{\frac{5}{2}}}{ax} + \frac{3b}{2a} \int \frac{y^{\frac{5}{2}} dx}{x} \) \[ + \frac{4c}{a} \int y^{\frac{5}{2}} dx \]

292. \( \int \frac{(a + bx + cx^2)^{\frac{5}{2}} dx}{x^3} = \left( -\frac{1}{2ax^2} - \frac{b}{4a^2x} \right) y^{\frac{5}{2}} \) \[ + \left( \frac{3b^2}{8a^2} + \frac{3c}{2a} \right) \int \frac{y^{\frac{5}{2}} dx}{x} + \frac{bc}{a^2} \int y^{\frac{5}{2}} dx \]

1. \( x^m (a + bx + cx^2)^{\frac{5}{2}} dx = x^n y^{\frac{5}{2}} dx \)

293. \( \int (a + bx + cx^2)^{\frac{5}{2}} dx = \left( \frac{y^2}{12c} + \frac{5(4ac - b^2)y}{192c^2} + \frac{5(4ac - b^2)^2}{512c^3} \right) (2cx + b)\sqrt{y} + \frac{5(4ac - b^2)^3}{1024c^3} \int \frac{dx}{\sqrt{y}} \)

294. \( \int x(a + bx + cx^2)^{\frac{5}{2}} dx = \frac{y^{\frac{7}{2}}}{7c} - \frac{b}{2c} \int y^{\frac{5}{2}} dx \)

295. \( \int x^2(a + bx + cx^2)^{\frac{5}{2}} dx = \left( \frac{x}{8c} - \frac{9b}{112c^2} \right) y^{\frac{7}{2}} + \left( \frac{9b^2}{32c^2} - \frac{a}{8c} \right) \int y^{\frac{5}{2}} dx \)

296. \( \int x^3(a + bx + cx^2)^{\frac{5}{2}} dx = \left( \frac{x^2}{9c} - \frac{11bx}{144c^2} + \frac{11b^2}{224c^3} - \frac{2a}{63c^2} \right) y^{\frac{7}{2}} - \left( \frac{11b^3}{64c^3} - \frac{3ax}{16c^2} \right) \int y^{\frac{5}{2}} dx \)

m. \( x^{-m}(a + bx + cx^2)^{\frac{5}{2}} dx = x^{-n} y^{\frac{5}{2}} dx \)

297. \( \int \frac{(a + bx + cx^2)^{\frac{5}{2}} dx}{x} = \left( \frac{y^2}{5} + \frac{ay}{3} + a^2 \right) \sqrt{y} + \)

\( a^3 \int \frac{dx}{x\sqrt{y}} + \frac{a^2b}{2} \int \frac{dx}{\sqrt{y}} + \frac{ab}{2} \int \sqrt{y} dx + \frac{b}{2} \int y^{\frac{5}{2}} dx \)

298. \( \int \frac{(a + bx + cx^2)^{\frac{5}{2}} dx}{x^2} = -\frac{y^{\frac{7}{2}}}{ax} + \frac{5b}{2a} \int \frac{y^{\frac{5}{2}} dx}{x} + \frac{6c}{a} \int y^{\frac{5}{2}} dx \)

299. \( \int \frac{(a + bx + cx^2)^{\frac{5}{2}} dx}{x^3} = \left( -\frac{1}{2ax^2} - \frac{3b}{4a^2x} \right) y^{\frac{7}{2}} + \left( \frac{15b^2}{8a^2} + \frac{5c}{2a} \right) \int \frac{y^{\frac{5}{2}} dx}{x} + \frac{9bc}{2a^2} \int y^{\frac{5}{2}} dx \)

G. \( x^m (a + bx)^n dx \)

300. \( \int x^m (a + bx)^n dx = \int x^m y^n dx = \)

\[ \int \left( \frac{y - a}{b} \right)^m \frac{dy}{y^n} = \left( \frac{1}{b} \right)^m + 1 \int (-a + y)^{\frac{m}{2}} y^n dy. \] (A): thus

\[ \int \frac{dx}{\sqrt{x(a + bx)}} = \pm \frac{2}{\sqrt{(ab)}} \text{ arc tang } \sqrt{\frac{bx}{a}} = \] \[ \frac{1}{\sqrt{(-ab)}} \ln \frac{a - bx + 2\sqrt{x}\sqrt{(-ab)}}{a + bx} \]

H. \( \frac{x^{m-\frac{1}{2}} dx}{(a + bx^2)^N} \)

a. \( \frac{x^m dx}{\sqrt{x(a + bx^2)}} = \frac{x^m dx}{\sqrt{xy}} \)

301. \( \int \frac{dx}{\sqrt{x(a + bx^2)}} = \frac{1}{\sqrt{2b}} \left( \frac{b}{a} \right)^{\frac{1}{4}} \)

\[ \left( h l + x \left( \frac{a}{b} \right)^{\frac{1}{4}} \sqrt{(2x) + \left( \frac{a}{b} \right)^{\frac{1}{2}}} \right) \] \[ + \text{ arc tang } \frac{\left( \frac{a}{b} \right)^{\frac{1}{4}} \sqrt{(2x)}}{\left( \frac{a}{b} \right)^{\frac{1}{2}} - x} \] \[ = \frac{1}{2b} \left( \frac{b}{a} \right)^{\frac{1}{4}} \left( \frac{(-a)}{(-b)} \right)^{\frac{1}{4}} \sqrt{x} \left( \frac{(-a)}{(-b)} \right)^{\frac{1}{4}} + \sqrt{x} \] \[ - 2 \text{ arc tang } \frac{\sqrt{x}}{\left( \frac{a}{b} \right)^{\frac{1}{4}}} \]

302. \( \int \frac{\sqrt{x} dx}{a + bx^2} = \frac{1}{2b} \left( \frac{b}{a} \right)^{\frac{1}{4}} \)

\[ \left( h l + x \left( \frac{a}{b} \right)^{\frac{1}{4}} + \left( \frac{a}{b} \right)^{\frac{1}{4}} \sqrt{(2x)} \right) \] \[ \frac{\sqrt{y}}{\sqrt{y}} \] + arc tang \( \frac{\left( \frac{a}{b} \right)^{\frac{1}{2}} \sqrt{(2x)}}{\left( \frac{a}{b} \right)^{\frac{1}{2}} - x} \)

\( = \frac{1}{2b} \left( -\frac{b}{a} \right)^{\frac{1}{2}} \left[ \frac{\left( -\frac{a}{b} \right)^{\frac{1}{2}} - \sqrt{x}}{\left( -\frac{a}{b} \right)^{\frac{1}{2}} + \sqrt{x}} + 2 \text{ arc tang } \left( \frac{\sqrt{x}}{\left( -\frac{a}{b} \right)^{\frac{1}{2}}} \right) \right] \)

303. \( \int \frac{x \sqrt{x} dx}{a + bx^2} = \frac{2 \sqrt{x}}{b} - \frac{a}{b} \int \frac{dx}{y \sqrt{x}} \)

304. \( \int \frac{x^2 \sqrt{x} dx}{a + bx^2} = \frac{2x \sqrt{x}}{3b} - \frac{a}{b} \int \frac{\sqrt{x} dx}{y} \)

b. \( \frac{x^m dx}{\sqrt{x}(a + bx^2)^2} = \frac{x^n dx}{\sqrt{xy^2}} \)

305. \( \int \frac{dx}{\sqrt{x}(a + bx^2)^2} = \frac{\sqrt{x}}{2ay} + \frac{3}{4a} \int \frac{dx}{\sqrt{xy}} \)

306. \( \int \frac{\sqrt{x} dx}{(a + bx^2)^2} = \frac{x \sqrt{x}}{2ay} + \frac{1}{4a} \int \frac{\sqrt{x} dx}{y} \)

307. \( \int \frac{x \sqrt{x} dx}{(a + bx^2)^2} = -\frac{\sqrt{x}}{2by} + \frac{1}{4b} \int \frac{dx}{\sqrt{xy}} \)

308. \( \int \frac{x^2 \sqrt{x} dx}{(a + bx^2)^2} = -\frac{x \sqrt{x}}{2by} + \frac{3}{4b} \int \frac{\sqrt{x} dx}{y} \)

c. \( \frac{x^m dx}{\sqrt{x}(a + bx^2)^3} = \frac{a^m dx}{\sqrt{xy^3}} \)

309. \( \int \frac{dx}{\sqrt{x}(a + bx^2)^3} = \left( \frac{1}{4ay^2} + \frac{7}{16a^2y} \right) \sqrt{x} + \frac{21}{32a^2} \int \frac{dx}{\sqrt{xy}} \)

310. \( \int \frac{\sqrt{x} dx}{(a + bx^2)^3} = \left( \frac{1}{4ay^2} + \frac{5}{16a^2y} \right) x \sqrt{x} + \frac{5}{32a^2} \int \frac{\sqrt{x} dx}{y} \)

311. \( \int \frac{x \sqrt{x} dx}{(a + bx^2)^3} = \frac{(bx^2 - 3a) \sqrt{x}}{16aby^2} + \frac{3}{32ab} \int \frac{dx}{\sqrt{xy}} \)

312. \( \int \frac{x^2 \sqrt{x} dx}{(a + bx^2)^3} = -\frac{2x \sqrt{x}}{5by^2} + \frac{3a}{5b} \int \frac{\sqrt{x} dx}{y^3} \)

I. \( x^m (f + gx)^{-n}(a + bx)^{-\frac{1}{2}} dx = x^m y^{-n} z^{-\frac{1}{2}} dx \)

a. \( x^m (f + gx)^{-1}(a + bx)^{-\frac{1}{2}} dx = x^m y^{-1} z^{-\frac{1}{2}} dx \)

313. \( \int \frac{dx}{(f + gx) \sqrt{(a + bx)}} = \pm \frac{2}{\sqrt{(bf - ag^2)}} \text{ arc tang } \frac{gz}{bf - ag} = \frac{1}{\sqrt{(ag^2 - bfg)}} \)

hl \( \frac{bf - 2ag - bge + 2\sqrt{(ag^2 - bfg)} \sqrt{z}}{y} \)

314. \( \int \frac{x dx}{(f + gx) \sqrt{(a + bx)}} = \frac{1}{g} \int \frac{dx}{\sqrt{z}} - \frac{f}{g} \int \frac{dz}{y \sqrt{z}} \)

315. \( \int \frac{x^2 dx}{(f + gx) \sqrt{(a + bx)}} = \frac{1}{g} \int \frac{x dx}{\sqrt{z}} - \frac{f}{g^2} \int \frac{dx}{\sqrt{z}} + \frac{f^2}{g^2} \int \frac{dx}{y \sqrt{z}} \)

316. \( \int \frac{x^3 dx}{(f + gx) \sqrt{(a + bx)}} = \frac{1}{g} \int \frac{x^2 dx}{\sqrt{z}} - \frac{f}{g^2} \int \frac{x dx}{\sqrt{z}} + \frac{f^2}{g^3} \int \frac{dx}{\sqrt{z}} - \frac{f^3}{g^3} \int \frac{dx}{y \sqrt{z}} \)

b. \( x^m (f + gx)^{-2}(a + bx)^{-\frac{1}{2}} dx = x^m y^{-2} z^{-\frac{1}{2}} dx \)

317. \( \int \frac{dx}{(f + gx)^2 \sqrt{(a + bx)}} = \frac{\sqrt{z}}{(bf - ag)y} + \frac{b}{2bf - 2ag} \int \frac{dx}{y \sqrt{z}} \)

318. \( \int \frac{x dx}{(f + gx)^2 \sqrt{(a + bx)}} = \frac{1}{g} \int \frac{dx}{y \sqrt{z}} - \frac{f}{g^2} \int \frac{dx}{y^2 \sqrt{z}} \)

319. \( \int \frac{x^2 dx}{(f + gx)^2 \sqrt{(a + bx)}} = \frac{1}{g^2} \int \frac{dx}{\sqrt{z}} - \frac{2f}{g^2} \int \frac{dx}{y \sqrt{z}} + \frac{f^2}{g^2} \int \frac{dx}{y^2 \sqrt{z}} \)

320. \( \int \frac{x^3 dx}{(f + gx)^2 \sqrt{(a + bx)}} = \frac{1}{g^2} \int \frac{x dx}{\sqrt{z}} - \frac{2f}{g^2} \int \frac{dx}{y \sqrt{z}} + \frac{3f^2}{g^3} \int \frac{dx}{y \sqrt{z}} - \frac{f^3}{g^3} \int \frac{dx}{y^2 \sqrt{z}} \)

c. \( x^m (f + gx)^{-3}(a + bx)^{-\frac{1}{2}} dx = y^{-m} z^{-\frac{1}{2}} dx \)

321. \( \int \frac{dx}{(f + gx)^3 \sqrt{(a + bx)}} = \left( \frac{1}{2(bf - ag)y^2} + \frac{3b}{4(bf - ag)^2 y^2} \right) \sqrt{z} + \frac{3b^2}{8(bf - ag)^2} \int \frac{dx}{y \sqrt{z}} \)

322. \( \int \frac{x dx}{(f + gx)^3 \sqrt{(a + bx)}} = \frac{1}{g} \int \frac{dx}{y^2 \sqrt{z}} - \frac{f}{g^2} \int \frac{dx}{y^3 \sqrt{z}} \)

323. \( \int \frac{x^2 dx}{(f + gx)^3 \sqrt{(a + bx)}} = \frac{1}{g^2} \int \frac{dx}{y \sqrt{z}} - \frac{2f}{g^2} \int \frac{dx}{y^2 \sqrt{z}} + \frac{f^2}{g^2} \int \frac{dx}{y^3 \sqrt{z}} \)

324. \( \int \frac{x^3 dx}{(f + gx)^3 \sqrt{(a + bx)}} = \frac{1}{g^3} \int \frac{dx}{\sqrt{z}} - \frac{3f}{g^3} \int \frac{dx}{y \sqrt{z}} + \frac{3f^2}{g^3} \int \frac{dx}{y^2 \sqrt{z}} - \frac{f^3}{g^3} \int \frac{dx}{y^3 \sqrt{z}} \)

d. \( \frac{dx}{x^m (f + gx) \sqrt{(a + bx)}} = \frac{dx}{x^m y \sqrt{z}} \)

325. \( \frac{dx}{x(f + gx) \sqrt{(a + bx)}} = \frac{1}{f} \int \frac{dx}{x \sqrt{z}} - \frac{g}{f} \int \frac{dx}{y \sqrt{z}} \)

326. \( \frac{dx}{x^2(f + gx) \sqrt{(a + bx)}} = \frac{1}{f} \int \frac{dx}{x^2 \sqrt{z}} - \frac{f^2}{f^2} \int \frac{dx}{x \sqrt{z}} + \frac{g^2}{f^2} \int \frac{dx}{y \sqrt{z}} \) 827. \( \frac{dx}{x^3(f+gx)\sqrt{(a+bx)}} = \frac{1}{f} \int \frac{dx}{x^3\sqrt{z}} - \frac{g}{f^2} \int \frac{dx}{x^2\sqrt{z}} \)

\[ + \frac{g^2}{f^3} \int \frac{dx}{x\sqrt{z}} - \frac{g^3}{f^3} \int \frac{dx}{y\sqrt{x}} \]

K. \( x^m(f+gx)^{-1}(a+bx^2)^{-\frac{1}{2}} dx \)

a. \( \frac{x^m dx}{(f+gx)\sqrt{(a+bx^2)}} = \frac{x^m dx}{y\sqrt{z}} \)

328. \( \int \frac{xdx}{(f+gx)\sqrt{(a+bx^2)}} = \pm \frac{1}{\sqrt{(ag^2+bf^2)}} hl \)

\[ \frac{ag-bfx}{y} \pm \frac{\sqrt{(ag^2+bf^2)}\sqrt{z}}{y} \]

\[ = \frac{1}{\sqrt{-(ag^2+bf^2)}} \text{arc tang } \frac{ag-bfx}{\sqrt{-(ag^2+bf^2)}}\sqrt{z} \]

329. \( \int \frac{xdx}{(f+gx)\sqrt{(a+bx^2)}} = \frac{1}{g} \int \frac{dx}{\sqrt{z}} - \frac{f}{g} \int \frac{dx}{y\sqrt{z}} \)

330. \( \int \frac{xdx}{(f+gx)\sqrt{(a+bx^2)}} = \frac{1}{g} \int \frac{xdx}{\sqrt{z}} - \frac{f}{g^2} \int \frac{dx}{\sqrt{z}} \)

\[ + \frac{f^2}{g^2} \int \frac{dx}{y\sqrt{z}} \]

331. \( \int \frac{x^2dx}{(f+gx)\sqrt{(a+bx^2)}} = \frac{1}{g} \int \frac{x^2dx}{\sqrt{z}} - \frac{f}{g^2} \int \frac{xdx}{\sqrt{z}} \)

\[ + \frac{f^2}{g^2} \int \frac{dx}{y\sqrt{z}} \]

b. \( \frac{dx}{x^m(f+gx)\sqrt{(a+bx^2)}} = \frac{dx}{x^my\sqrt{z}} \)

332. \( \int \frac{dx}{x(f+gx)\sqrt{(a+bx^2)}} = \frac{1}{f} \int \frac{dx}{x\sqrt{z}} - \frac{g}{f^2} \int \frac{dx}{y\sqrt{z}} \)

333. \( \int \frac{dx}{x^2(f+gx)\sqrt{(a+bx^2)}} = \frac{1}{f} \int \frac{dx}{x^2+z} - \frac{g}{f^2} \int \frac{dx}{x\sqrt{z}} + \frac{g^2}{f^2} \int \frac{dx}{y\sqrt{z}} \)

334. \( \int \frac{dx}{x^2(f+gx)\sqrt{(a+bx^2)}} = \frac{1}{f} \int \frac{dx}{x^3\sqrt{z}} - \frac{g}{f^2} \int \frac{dx}{x^2\sqrt{z}} \)

\[ + \frac{g^2}{f^3} \int \frac{dx}{x\sqrt{z}} - \frac{g^3}{f^3} \int \frac{dx}{y\sqrt{x}} \]

L. \( x^m(f+gx^2)^{-1}(a+bx^2)^{-\frac{1}{2}} dx = x^m y^{-1} z^{-\frac{1}{2}} dx \)

335. \( \int \frac{dx}{(f+gx^2)\sqrt{(a+bx^2)}} = \frac{1}{\sqrt{(bf^2-ag^2)}} hl \)

\[ \frac{\sqrt{z}+\sqrt{(bf^2-ag^2)}}{\sqrt{y}} \]

\[ = \frac{1}{\sqrt{(ag^2-bf^2)}} \text{arc tang } \frac{x\sqrt{(ag^2-bf^2)}}{f\sqrt{z}} \]

336. \( \int \frac{xdx}{(f+gx^2)\sqrt{(a+bx^2)}} = \frac{1}{\sqrt{(ag^2-bf^2)}} hl \)

\[ \frac{g\sqrt{z}-\sqrt{(ag^2-bf^2)}}{\sqrt{y}} \]

337. \( \int \frac{x^2dx}{(f+gx^2)\sqrt{(a+bx^2)}} = \frac{1}{g} \int \frac{dx}{\sqrt{z}} - \frac{f}{g} \int \frac{dx}{y\sqrt{z}} \)

338. \( \int \frac{x^2dx}{(f+gx^2)\sqrt{(a+bx^2)}} = \frac{1}{g} \int \frac{xdx}{\sqrt{z}} - \frac{f}{g} \int \frac{xdx}{y\sqrt{z}} \)

M. \( x^m(f+gx^2)^{-1}(a+bx^2)^{\frac{1}{2}} dx = x^m y^{-1} z^{\frac{1}{2}} dx \)

339. \( \int \frac{\sqrt{(a+bx^2)}dx}{f+gx^2} = \frac{b}{g} \int \frac{dx}{\sqrt{z}} + \left(a-\frac{bf}{g}\right) \int \frac{dx}{y\sqrt{z}} \)

340. \( \int \frac{x\sqrt{(a+bx^2)}dx}{f+gx^2} = \frac{b}{g} \int \frac{xdx}{\sqrt{z}} + \left(a-\frac{bf}{g}\right) \int \frac{xdx}{y\sqrt{z}} \)

341. \( \int \frac{x^2\sqrt{(a+bx^2)}dx}{f+gx^2} = \frac{b}{g} \int \frac{x^2dx}{\sqrt{z}} + \left(\frac{a}{g}-\frac{bf}{g^2}\right) \int \frac{xdx}{y\sqrt{z}} \)

\[ \int \frac{xdx}{\sqrt{z}} - \left(\frac{af}{g}-\frac{bf^2}{g^2}\right) \int \frac{xdx}{y\sqrt{z}} \]

342. \( \int \frac{x^3\sqrt{(a+bx^2)}dx}{f+gx^2} = \frac{b}{g} \int \frac{x^3dx}{\sqrt{z}} + \left(\frac{a}{g}-\frac{bf}{g^2}\right) \int \frac{xdx}{y\sqrt{z}} \)

\[ \int \frac{xdx}{\sqrt{z}} - \left(\frac{af}{g}-\frac{bf^2}{g^2}\right) \int \frac{xdx}{y\sqrt{z}} \]

N. \( x^m(f+gx)^{-1}(a+bx+cx^2)^{-\frac{1}{2}} dx = \frac{x^m}{y} \frac{1}{z} \frac{1}{z^{\frac{1}{2}}} dx \)

Putting \( ag^2-bfg+cf^2=k \)

343. \( \int \frac{dx}{(f+gx)\sqrt{(a+bx+cx^2)}} = \pm \frac{1}{\sqrt{k}} hl \)

\[ \frac{2ag-bf+(bg-2cf)x}{y} \]

\[ = \frac{1}{\sqrt{-k}} \text{arc tang } \frac{2ag-bf+(bg-2cf)x}{2\sqrt{-k}y} \]

344. \( \int \frac{xdx}{(f+gx)\sqrt{(a+bx+cx^2)}} = \frac{1}{g} \int \frac{dx}{\sqrt{z}} - \frac{f}{g} \int \frac{dx}{y\sqrt{z}} \)

345. \( \int \frac{x^2dx}{(f+gx)\sqrt{(a+bx+cx^2)}} = \frac{1}{g} \int \frac{xdx}{\sqrt{z}} - \frac{f}{g^2} \int \frac{dx}{\sqrt{z}} \)

\[ + \frac{f^2}{g^2} \int \frac{dx}{y\sqrt{z}} \]

346. \( \int \frac{x^3dx}{(f+gx)\sqrt{(a+bx+cx^2)}} = \frac{1}{g} \int \frac{x^2dx}{\sqrt{z}} - \frac{f}{g^2} \int \frac{xdx}{\sqrt{z}} \)

\[ + \frac{f^2}{g^2} \int \frac{dx}{y\sqrt{z}} - \frac{f^3}{g^3} \int \frac{dx}{y\sqrt{x}} \]

O. \( (a^4-x^4)^{-\frac{1}{2}} dx \)

Particular values, from \( x=0 \) to \( x=a \).

347. \( \int \frac{dx}{\sqrt{(a^4-x^4)}} = \frac{3.14159}{2a} \left(1-\left(\frac{1}{2}\right)^2+\left(\frac{1.3}{2.4}\right)^2-\left(\frac{1.3.5}{2.4.6}\right)^2+\ldots\right) \) 348. \( \int' \sqrt{a^4 - x^4} \, dx = \frac{3.14159\, a^5}{4} \left( 1 + \frac{1}{2} \cdot \frac{1}{4} - \frac{1.1\, 1.3}{2.4\, 4.6} + \frac{1.1\, 1.3\, 1.3\, 5}{2.4\, 6.4\, 6.8} - \ldots \right) \)

P. \( x^M (1 - x^{2N})^{-\frac{1}{2}} dx \)

Relation of particular values, from \( x = 0 \) to \( x = 1 \)

349. \( \int' \frac{x^M dx}{\sqrt{(1 - x^{2N})}} \times \int' \frac{x^{M+N} dx}{\sqrt{(1 - x^{2N})}} = \frac{3.14159}{2N(M+1)} \)

Q. \( x^M (1 - x^{M+N})^{-\frac{M+1}{M+N}} dx \)

Particular value, from \( x = 0 \) to \( x = 1 \)

350. \( \int' x^M (1 - x^{M+N})^{-\frac{M+1}{M+N}} dx = \frac{3.14159}{M+N} \cosec \frac{M+1}{M+N} 180^\circ \)

SECTION V.—Circular Fluxions.

A. \( \sin^m \varphi d\varphi \)

351. \( \int \sin \varphi d\varphi = -\cos \varphi \)

352. \( \int \sin^2 \varphi d\varphi = -\frac{1}{2} \sin \varphi \cos \varphi + \frac{1}{2} \varphi \)

\( = -\frac{1}{4} \sin^2 \varphi + \frac{1}{2} \varphi \)

353. \( \int \sin^3 \varphi d\varphi = \left( -\frac{1}{3} \sin^2 \varphi - \frac{2}{3} \right) \cos \varphi \)

\( = \frac{1}{12} \cos^3 \varphi - \frac{3}{4} \cos \varphi \)

354. \( \int \sin^4 \varphi d\varphi = \left( -\frac{1}{4} \sin^3 \varphi - \frac{3}{8} \sin \varphi \right) \cos \varphi + \frac{3}{8} \varphi \)

\( = \frac{1}{32} \sin 4\varphi - \frac{1}{4} \sin 2\varphi + \frac{3}{8} \varphi \)

355. \( \int \sin^5 \varphi d\varphi = \left( -\frac{1}{5} \sin^4 \varphi - \frac{4}{15} \sin^2 \varphi - \frac{8}{15} \right) \cos \varphi \)

\( = \frac{1}{80} \cos 5\varphi + \frac{5}{48} \cos 3\varphi - \frac{5}{8} \cos \varphi \)

356. \( \int \sin^6 \varphi d\varphi = \left( -\frac{1}{6} \sin^5 \varphi - \frac{5}{24} \sin^3 \varphi - \frac{5}{16} \sin \varphi \right) \cos \varphi + \frac{5}{16} \varphi \)

\( = -\frac{1}{192} \sin 6\varphi + \frac{3}{64} \sin 4\varphi - \frac{15}{64} \sin 2\varphi + \frac{5}{16} \varphi \)

B. \( \cos^m \varphi d\varphi \)

357. \( \int \cos \varphi d\varphi = \sin \varphi \)

358. \( \int \cos^2 \varphi d\varphi = \frac{1}{2} \sin \varphi \cos \varphi + \frac{1}{2} \varphi \)

\( = \frac{1}{4} \sin 2\varphi + \frac{1}{2} \varphi \)

359. \( \int \cos^3 \varphi d\varphi = \left( \frac{1}{3} \cos^2 \varphi + \frac{2}{3} \right) \sin \varphi \)

\( = \frac{1}{12} \sin 3\varphi + \frac{3}{4} \sin \varphi \)

360. \( \int \cos^4 \varphi d\varphi = \left( \frac{1}{4} \cos^3 \varphi + \frac{3}{8} \cos \varphi \right) \sin \varphi + \frac{3}{8} \varphi \)

\( = \frac{1}{32} \sin 4\varphi + \frac{1}{4} \sin 2\varphi + \frac{3}{8} \varphi \)

361. \( \int \cos^5 \varphi d\varphi = \left( \frac{1}{5} \cos^4 \varphi + \frac{4}{15} \cos^2 \varphi + \frac{8}{15} \right) \sin \varphi \)

\( = \frac{1}{80} \sin 5\varphi + \frac{5}{48} \sin 3\varphi + \frac{5}{8} \sin \varphi \)

362. \( \int \cos^6 \varphi d\varphi = \left( \frac{1}{6} \cos^5 \varphi + \frac{5}{24} \cos^3 \varphi + \frac{5}{16} \cos \varphi \right) \sin \varphi + \frac{5}{16} \varphi \)

\( = \frac{1}{192} \sin 6\varphi + \frac{3}{64} \sin 4\varphi + \frac{15}{64} \sin^2 \varphi + \frac{5}{16} \varphi \)

C. \( \sin^m \varphi \cos^n \varphi d\varphi \)

a. \( \sin \varphi \cos^n \varphi d\varphi \)

363. \( \int \sin \varphi \cos^n \varphi d\varphi = -\frac{1}{n+1} \cos^{n+1} \varphi \)

It may be remarked that \( \cos^x \varphi = \frac{1}{2^{x-1}} \)

\( (\cos n\varphi + n \cos (n-2)\varphi + n \frac{n-1}{2} \cos (n-4)\varphi + \ldots ) \); continuing the series through all positive angles, and putting \( \frac{1}{2} \) instead of \( \cos 0 \).

b. \( \sin^n \varphi \cos \varphi d\varphi \)

364. \( \int \sin^n \varphi \cos \varphi d\varphi = \frac{1}{n+1} \sin^{n+1} \varphi \)

We have for the powers of \( \sin \varphi \), \( \sin^n \varphi = \pm \frac{1}{2^{n-1}} (\cos n\varphi - n \cos (n-2)\varphi + n \frac{n-1}{2} \cos (n-4)\varphi - \ldots ) \); + when \( n = 4p \), — when \( n = 4p + 2 \); and \( \sin^n \varphi = \pm \frac{1}{2^{n-1}} (\sin n\varphi \). —n sin (n—2) φ + n·\( \frac{n-1}{2} \) sin (n—4) φ — ...), + when n = 4r + 1, and —when n = 4r + 3; the last term, when it becomes cos 0, being altered to \( \frac{1}{2} \).

c. \( \sin^2 \varphi \cos^n \varphi \) dφ

365. \( \int \sin^2 \varphi \cos \varphi \, d\varphi = \frac{1}{3} \sin^3 \varphi \)

\[ = - \frac{1}{4} \left( \frac{1}{3} \sin 3\varphi - \sin \varphi \right) \]

366. \( \int \sin^2 \varphi \cos^2 \varphi \, d\varphi = \frac{1}{4} \sin^3 \varphi \cos \varphi - \frac{1}{8} \sin \varphi \cos \varphi \)

\[ + \frac{1}{8} \varphi \]

\[ = - \frac{1}{8} \left( \frac{1}{4} \sin 4\varphi - \varphi \right) \]

367. \( \int \sin^2 \varphi \cos^2 \varphi \, d\varphi = \left( \frac{1}{5} \cos^2 \varphi + \frac{2}{15} \right) \sin^5 \varphi \)

\[ = - \frac{1}{16} \left( \frac{1}{5} \sin 5\varphi + \frac{1}{3} \sin 3\varphi - 2 \sin \varphi \right) \]

368. \( \int \sin^2 \varphi \cos^4 \varphi \, d\varphi = \frac{1}{6} \sin^3 \varphi \cos^3 \varphi + \frac{1}{2} \int \sin^2 \varphi \cos^2 \varphi \, d\varphi \)

\[ = - \frac{1}{32} \left( \frac{1}{6} \sin 6\varphi + \frac{1}{2} \sin 4\varphi - \frac{1}{2} \sin 2\varphi - 2\varphi \right) \]

369. \( \int \sin^2 \varphi \cos^5 \varphi \, d\varphi = \left( \frac{1}{7} \cos^4 \varphi + \frac{4}{35} \cos^2 \varphi + \frac{8}{105} \right) \sin^5 \varphi \)

\[ = - \frac{1}{64} \left( \frac{1}{7} \sin 7\varphi + \frac{3}{5} \sin 5\varphi + \frac{1}{3} \sin 3\varphi - 5 \sin \varphi \right) \]

370. \( \int \sin^2 \varphi \cos^6 \varphi \, d\varphi = - \frac{1}{128} \left( \frac{1}{8} \sin 8\varphi + \frac{2}{3} \sin 6\varphi + \sin 4\varphi - 2 \sin 2\varphi - 5\varphi \right) \)

d. \( \sin^3 \varphi \cos^n \varphi \) dφ

371. \( \int \sin^3 \varphi \cos \varphi \, d\varphi = \frac{1}{4} \sin^4 \varphi \)

\[ = \frac{1}{8} \left( \frac{1}{4} \cos 4\varphi - \cos 2\varphi \right) \]

372. \( \int \sin^3 \varphi \cos^2 \varphi \, d\varphi = \left( \frac{1}{5} \sin^4 \varphi - \frac{1}{15} \sin^2 \varphi - \frac{2}{15} \right) \cos \varphi \)

\[ = \frac{1}{16} \left( \frac{1}{5} \cos 5\varphi - \frac{1}{3} \cos 3\varphi - 2 \cos \varphi \right) \]

373. \( \int \sin^3 \varphi \cos^3 \varphi \, d\varphi = \left( \frac{1}{6} \cos^2 \varphi + \frac{1}{12} \right) \sin^4 \varphi \)

\[ = \frac{1}{32} \left( \frac{1}{6} \cos 6\varphi - \frac{3}{2} \cos 2\varphi \right) \]

374. \( \int \sin^3 \varphi \cos^4 \varphi \, d\varphi = \frac{1}{7} \sin^4 \varphi \cos^3 \varphi - \frac{3}{7} \int \sin^3 \varphi \cos^2 \varphi \, d\varphi \)

\[ = \frac{1}{64} \left( \frac{1}{7} \cos 7\varphi + \frac{1}{5} \cos 5\varphi - \cos^3 \varphi - 3 \cos \varphi \right) \]

375. \( \int \sin^3 \varphi \cos^5 \varphi \, d\varphi = \frac{1}{128} \left( \frac{1}{8} \cos 8\varphi + \frac{1}{3} \cos 6\varphi - \frac{1}{2} \cos 4\varphi - 3 \cos 2\varphi \right) \)

376. \( \int \sin^3 \varphi \cos^6 \varphi \, d\varphi = \frac{1}{256} \left( \frac{1}{9} \cos 9\varphi + \frac{3}{7} \cos 7\varphi - \frac{8}{3} \cos 8\varphi - 6 \cos \varphi \right) \)

e. \( \sin^4 \varphi \cos^n \varphi \) dφ

377. \( \int \sin^4 \varphi \cos \varphi \, d\varphi = \frac{1}{5} \sin 5\varphi \)

\[ = \frac{1}{16} \left( \frac{1}{5} \sin 5\varphi - \sin 3\varphi + 2 \sin \varphi \right) \]

378. \( \int \sin^4 \varphi \cos^2 \varphi \, d\varphi = \left( \frac{1}{6} \sin^5 \varphi - \frac{1}{24} \sin^3 \varphi - \frac{1}{16} \sin \varphi \right) \cos \varphi + \frac{1}{16} \varphi \)

\[ = \frac{1}{32} \left( \frac{1}{6} \sin 6\varphi - \frac{1}{2} \sin 4\varphi - \frac{1}{2} \sin 2\varphi + 2\varphi \right) \]

379. \( \int \sin^4 \varphi \cos^3 \varphi \, d\varphi = \left( \frac{1}{7} \cos^2 \varphi + \frac{2}{35} \right) \sin 5\varphi \)

\[ = \frac{1}{64} \left( \frac{1}{7} \sin 7\varphi - \frac{1}{5} \sin 5\varphi - \sin 3\varphi + 3 \sin \varphi \right) \]

380. \( \int \sin^4 \varphi \cos^4 \varphi \, d\varphi = \frac{1}{128} \left( \frac{1}{8} \sin 8\varphi - \sin 4\varphi + 3\varphi \right) \)

381. \( \int \sin^4 \varphi \cos^5 \varphi \, d\varphi = \frac{1}{256} \left( \frac{1}{9} \sin 9\varphi + \frac{1}{7} \sin 7\varphi - \frac{4}{5} \sin 5\varphi - \frac{4}{3} \sin 3\varphi + 6 \sin \varphi \right) \)

382. \( \int \sin^4 \varphi \cos^6 \varphi \, d\varphi = \frac{1}{512} \left( \frac{1}{10} \sin 10\varphi + \frac{1}{4} \sin 8\varphi - \frac{1}{2} \sin 6\varphi - 2 \sin 4\varphi + \sin 2\varphi + 6\varphi \right) \) f. \( \sin^5 \varphi \cos^n \varphi d\varphi \)

383. \( \int \sin^5 \varphi \cos \varphi d\varphi = \frac{1}{6} \sin^6 \varphi \)

\[ = - \frac{1}{32} \left( \frac{1}{6} \cos 6\varphi - \cos 4\varphi + \frac{5}{2} \cos 2\varphi \right) \]

384. \( \int \sin^5 \varphi \cos^2 \varphi d\varphi = \frac{1}{7} \sin^6 \varphi \cos \varphi + \frac{1}{7} \int \sin^5 \varphi d\varphi \)

\[ = - \frac{1}{64} \left( \frac{1}{7} \cos 7\varphi - \frac{3}{5} \cos 5\varphi + \frac{1}{3} \cos 3\varphi + 5 \cos \varphi \right) \]

385. \( \int \sin^5 \varphi \cos^3 \varphi d\varphi = \left( \frac{1}{8} \cos^2 \varphi + \frac{1}{24} \right) \sin^6 \varphi \)

\[ = - \frac{1}{128} \left( \frac{1}{8} \cos 8\varphi - \frac{1}{3} \cos 6\varphi - \frac{1}{2} \cos 4\varphi + 3 \cos 2\varphi \right) \]

386. \( \int \sin^5 \varphi \cos^4 \varphi d\varphi = - \frac{1}{256} \left( \frac{1}{9} \cos 9\varphi - \frac{1}{7} \cos 7\varphi - \frac{4}{5} \cos 5\varphi + \frac{4}{3} \cos 3\varphi + 6 \cos \varphi \right) \)

387. \( \int \sin^5 \varphi \cos^5 \varphi d\varphi = - \frac{1}{512} \left( \frac{1}{10} \cos 10\varphi - \frac{5}{6} \cos 6\varphi + 5 \cos 2\varphi \right) \)

388. \( \int \sin^5 \varphi \cos^6 \varphi d\varphi = - \frac{1}{1024} \left( \frac{1}{11} \cos 11\varphi + \frac{1}{9} \cos 9\varphi - \frac{5}{7} \cos 7\varphi - \cos 5\varphi + \frac{10}{3} \cos 3\varphi + 10 \cos \varphi \right) \)

g. \( \sin^6 \varphi \cos^n \varphi d\varphi \)

389. \( \int \sin^6 \varphi \cos \varphi d\varphi = \frac{1}{7} \sin 7\varphi \)

\[ = \frac{1}{64} \left( \frac{1}{7} \sin 7\varphi - \sin 5\varphi + 3 \sin 3\varphi - 5 \sin 5\varphi \right) \]

390. \( \int \sin^6 \varphi \cos^2 \varphi d\varphi = \left( \frac{1}{8} \sin^7 \varphi - \frac{1}{48} \sin^5 \varphi - \frac{5}{192} \sin^3 \varphi - \frac{5}{128} \sin \varphi \right) \cos \varphi + \frac{5}{128} \varphi \)

\[ = - \frac{1}{128} \left( \frac{1}{8} \sin 8\varphi - \frac{2}{3} \sin 6\varphi + \sin 4\varphi + 2 \sin 2\varphi - 5\varphi \right) \]

391. \( \int \sin^6 \varphi \cos^5 \varphi d\varphi = \left( \frac{1}{9} \cos^2 \varphi + \frac{2}{63} \right) \sin 7\varphi \)

\[ = - \frac{1}{256} \left( \frac{1}{9} \sin 9\varphi - \frac{3}{7} \sin 7\varphi + \frac{3}{8} \sin 3\varphi - 6 \sin \varphi \right) \]

392. \( \int \sin^6 \varphi \cos^4 \varphi d\varphi = - \frac{1}{512} \left( \frac{1}{10} \sin 10\varphi - \frac{1}{4} \sin 8\varphi - \frac{1}{2} \sin 6\varphi + 2 \sin 4\varphi + \sin 2\varphi - 6\varphi \right) \)

393. \( \int \sin^6 \varphi \cos^3 \varphi d\varphi = - \frac{1}{1024} \left( \frac{1}{11} \sin 11\varphi - \frac{1}{9} \sin 9\varphi - \frac{5}{7} \sin 7\varphi + \sin 5\varphi + \frac{10}{3} \sin 3\varphi - 10 \sin \varphi \right) \)

394. \( \int \sin^6 \varphi \cos^6 \varphi d\varphi = - \frac{1}{2048} \left( \frac{1}{12} \sin 12\varphi - \frac{3}{4} \sin 8\varphi + \frac{15}{4} \sin 4\varphi - 10\varphi \right) \)

D. \( \sin^{-m} \varphi d\varphi \)

395. \( \int \frac{d\varphi}{\sin \varphi} = \mathrm{hl} \tan \frac{\varphi}{2} \)

396. \( \int \frac{d\varphi}{\sin^2 \varphi} = - \frac{\cos \varphi}{\sin \varphi} = - \cot \varphi \)

397. \( \int \frac{d\varphi}{\sin^3 \varphi} = - \frac{\cos \varphi}{2 \sin^2 \varphi} + \frac{1}{2} \mathrm{hl} \tan \frac{\varphi}{2} \)

398. \( \int \frac{d\varphi}{\sin^4 \varphi} = \left( - \frac{1}{3 \sin^3 \varphi} - \frac{2}{3 \sin \varphi} \right) \cos \varphi = - \cot \varphi - \frac{1}{3} \cot^3 \varphi \)

399. \( \int \frac{d\varphi}{\sin^5 \varphi} = \left( - \frac{1}{4 \sin^4 \varphi} - \frac{3}{8 \sin^2 \varphi} \right) \cos \varphi + \frac{3}{8} \mathrm{hl} \tan \frac{\varphi}{2} \)

400. \( \int \frac{d\varphi}{\sin^6 \varphi} = \left( - \frac{1}{5 \sin^5 \varphi} - \frac{4}{15 \sin^3 \varphi} - \frac{8}{15 \sin \varphi} \right) \cos \varphi \)

E. \( \cos^{-m} \varphi d\varphi \)

401. \( \int \frac{d\varphi}{\cos \varphi} = \mathrm{hl} \tan \left( 45^\circ + \frac{\varphi}{2} \right) \)

402. \( \int \frac{d\varphi}{\cos^2 \varphi} = \frac{\sin \varphi}{\cos \varphi} = \tan \varphi \)

403. \( \int \frac{d\varphi}{\cos^3 \varphi} = \frac{\sin \varphi}{2 \cos^2 \varphi} + \frac{1}{2} \mathrm{hl} \tan \left( 45^\circ + \frac{\varphi}{2} \right) \)

404. \( \int \frac{d\varphi}{\cos^4 \varphi} = \left( \frac{1}{3 \cos^3 \varphi} + \frac{2}{3 \cos \varphi} \right) \sin \varphi = \tan \varphi + \frac{1}{3} \tan^3 \varphi \) 405. \( \int \frac{d\varphi}{\cos^5 \varphi} = \left( \frac{1}{4 \cos^4 \varphi} + \frac{3}{8 \cos^2 \varphi} \right) \sin \varphi + \frac{3}{8} \)

hl tang \( (45^\circ + \frac{\varphi}{2}) \)

406. \( \int \frac{d\varphi}{\cos^6 \varphi} = \left( \frac{1}{5 \cos^5 \varphi} + \frac{4}{15 \cos^3 \varphi} + \frac{8}{15 \cos \varphi} \right) \)

\( \sin \varphi \)

F. \( \sin^m \varphi \cos^{-n} \varphi d\varphi \)

a. \( \sin^m \varphi \cos^{-1} \varphi d\varphi \)

407. \( \int \frac{\sin \varphi d\varphi}{\cos \varphi} = - hl \cos \varphi = hl \sec \varphi \)

408. \( \int \frac{\sin^2 \varphi d\varphi}{\cos \varphi} = - \sin \varphi + hl \tang \left(45^\circ + \frac{\varphi}{2}\right) \)

409. \( \int \frac{\sin^3 \varphi d\varphi}{\cos \varphi} = - \frac{\sin^2 \varphi}{2} - hl \cos \varphi \)

410. \( \int \frac{\sin^4 \varphi d\varphi}{\cos \varphi} = - \frac{\sin^3 \varphi}{3} - \sin \varphi + hl \tang \left(45^\circ + \frac{\varphi}{2}\right) \)

411. \( \int \frac{\sin^5 \varphi d\varphi}{\cos \varphi} = - \frac{\sin^4 \varphi}{4} - \frac{\sin^2 \varphi}{2} - hl \cos \varphi \)

412. \( \int \frac{\sin^6 \varphi d\varphi}{\cos \varphi} = - \frac{\sin^5 \varphi}{5} - \frac{\sin^3 \varphi}{3} - \sin \varphi + hl \tang \left(45^\circ + \frac{\varphi}{2}\right) \)

b. \( \sin^m \varphi \cos^{-2} \varphi d\varphi \)

413. \( \int \frac{\sin \varphi d\varphi}{\cos^2 \varphi} = \frac{1}{\cos \varphi} = \sec \varphi \)

414. \( \int \frac{\sin^2 \varphi d\varphi}{\cos^2 \varphi} = \frac{\sin \varphi}{\cos \varphi} = \tang \varphi - \varphi \)

415. \( \int \frac{\sin^3 \varphi d\varphi}{\cos^2 \varphi} = (- \sin^2 \varphi + 2) \frac{1}{\cos \varphi} = \cos \varphi + \sec \varphi \)

416. \( \int \frac{\sin^4 \varphi d\varphi}{\cos^2 \varphi} = \left( - \frac{1}{2} \sin^3 \varphi + \frac{3}{2} \sin \varphi \right) \frac{1}{\cos \varphi} \frac{3}{2} \varphi \)

417. \( \int \frac{\sin^5 \varphi d\varphi}{\cos^2 \varphi} = \left( - \frac{1}{3} \sin^4 \varphi - \frac{4}{3} \sin^2 \varphi + \frac{8}{3} \right) \frac{1}{\cos \varphi} \)

418. \( \int \frac{\sin^6 \varphi d\varphi}{\cos^2 \varphi} = \left( - \frac{1}{4} \sin^5 \varphi - \frac{5}{8} \sin^3 \varphi + \frac{15}{8} \sin \varphi \right) \)

\( \frac{1}{\cos \varphi} - \frac{15}{8} \varphi \)

c. \( \sin^m \varphi \cos^{-3} \varphi d\varphi \)

419. \( \int \frac{\sin \varphi d\varphi}{\cos^3 \varphi} = \frac{1}{2 \cos^2 \varphi} \)

420. \( \int \frac{\sin^2 \varphi d\varphi}{\cos^3 \varphi} = \frac{\sin \varphi}{2 \cos^2 \varphi} - \frac{1}{2} hl \tang \left(45^\circ + \frac{\varphi}{2}\right) \)

421. \( \int \frac{\sin^3 \varphi d\varphi}{\cos^3 \varphi} = \frac{1}{2 \cos^2 \varphi} + hl \cos \varphi \)

422. \( \int \frac{\sin^4 \varphi d\varphi}{\cos^5 \varphi} = (- \sin^5 \varphi + \frac{3}{2} \sin \varphi) \frac{1}{\cos^5 \varphi} - \frac{3}{2} hl \tang \left(45^\circ + \frac{\varphi}{2}\right) \)

423. \( \int \frac{\sin^5 \varphi d\varphi}{\cos^5 \varphi} = \left( - \frac{1}{2} \sin^4 \varphi + 1 \right) \frac{1}{\cos^5 \varphi} + 2 hl \cos \varphi \)

424. \( \int \frac{\sin^6 \varphi d\varphi}{\cos^5 \varphi} = \left( - \frac{1}{3} \sin^5 \varphi - \frac{5}{3} \sin^3 \varphi + \frac{5}{2} \sin \varphi \right) \)

\( \frac{1}{\cos^5 \varphi} - \frac{5}{2} hl \tang \left(45^\circ + \frac{\varphi}{2}\right) \)

d. \( \sin^m \varphi \cos^{-4} \varphi d\varphi \)

425. \( \int \frac{\sin \varphi d\varphi}{\cos^4 \varphi} = \frac{1}{-3 \cos^3 \varphi} \)

426. \( \int \frac{\sin^2 \varphi d\varphi}{\cos^4 \varphi} = \frac{\sin^5 \varphi}{3 \cos^3 \varphi} = \frac{1}{3} \tang^3 \varphi \)

427. \( \int \frac{\sin^3 \varphi d\varphi}{\cos^4 \varphi} = \left( \sin^2 \varphi - \frac{2}{3} \right) \frac{1}{\cos^3 \varphi} \)

428. \( \int \frac{\sin^4 \varphi d\varphi}{\cos^4 \varphi} = \left( \frac{4}{3} \sin^3 \varphi - \sin \varphi \right) \frac{1}{\cos^3 \varphi} + \varphi = \)

\( \frac{1}{3} \tang^5 \varphi - \tang \varphi + \varphi \)

429. \( \int \frac{\sin^5 \varphi d\varphi}{\cos^5 \varphi} = \left( - \sin^4 \varphi + 4 \sin^2 \varphi - \frac{8}{3} \right) \frac{1}{\cos^5 \varphi} \)

430. \( \int \frac{\sin^6 \varphi d\varphi}{\cos^5 \varphi} = \left( - \frac{1}{2} \sin^5 \varphi + \frac{10}{3} \sin^3 \varphi - \frac{5}{2} \sin \varphi \right) \)

\( \frac{1}{\cos^5 \varphi} + \frac{5}{2} \varphi \)

e. \( \sin^m \varphi \cos^{-5} \varphi d\varphi \)

431. \( \int \frac{\sin \varphi d\varphi}{\cos^5 \varphi} = \frac{1}{4 \cos^4 \varphi} \)

432. \( \int \frac{\sin^2 \varphi d\varphi}{\cos^5 \varphi} = \left( \frac{1}{8} \sin^3 \varphi + \frac{1}{8} \sin \varphi \right) \frac{1}{\cos^4 \varphi} - \frac{1}{8} hl \tang \left(45^\circ + \frac{\varphi}{2}\right) \)

433. \( \int \frac{\sin^3 \varphi d\varphi}{\cos^5 \varphi} = \frac{\sin^4 \varphi}{4 \cos^4 \varphi} = \frac{1}{4} \tang^4 \varphi \)

434. \( \int \frac{\sin^4 \varphi d\varphi}{\cos^5 \varphi} = \left( \frac{5}{8} \sin^3 \varphi - \frac{3}{8} \sin \varphi \right) \frac{1}{\cos^4 \varphi} + \frac{3}{8} hl \tang \left(45^\circ + \frac{\varphi}{2}\right) \)

435. \( \int \frac{\sin^5 \varphi d\varphi}{\cos^5 \varphi} = \left( \frac{3}{4} \sin^4 \varphi - \frac{1}{2} \sin^2 \varphi \right) \frac{1}{\cos^4 \varphi} - hl \cos \varphi = \)

\( \cos \varphi = \frac{1}{4} \tang^4 \varphi - \frac{1}{2} \tang^2 \varphi - hl \cos \varphi \)

436. \( \int \frac{\sin^6 \varphi d\varphi}{\cos^5 \varphi} = \left( - \sin^5 \varphi + \frac{25}{8} \sin^3 \varphi - \frac{15}{8} \sin \varphi \right) \)

\( \frac{1}{\cos^5 \varphi} + \frac{15}{8} hl \tang \left(45^\circ + \frac{\varphi}{2}\right) \)

f. \( \sin^m \varphi \cos^{-6} \varphi d\varphi \). 437. \( \int \frac{\sin \varphi d\varphi}{\cos^5 \varphi} = \frac{1}{5 \cos^5 \varphi} \)

438. \( \int \frac{\sin^2 \varphi d\varphi}{\cos^5 \varphi} = \left( -\frac{2}{15} \sin^5 \varphi + \frac{1}{3} \sin^3 \varphi \right) \frac{1}{\cos^5 \varphi} \)

439. \( \int \frac{\sin^3 \varphi d\varphi}{\cos^5 \varphi} = \left( \frac{1}{3} \sin^5 \varphi - \frac{2}{15} \right) \frac{1}{\cos^5 \varphi} \)

440. \( \int \frac{\sin^4 \varphi d\varphi}{\cos^5 \varphi} = \frac{1}{5} \tan^5 \varphi \)

441. \( \int \frac{\sin^5 \varphi d\varphi}{\cos^5 \varphi} = \left( \sin^4 \varphi - \frac{4}{3} \sin^2 \varphi + \frac{8}{15} \right) \frac{1}{\cos^5 \varphi} \)

442. \( \int \frac{\sin^6 \varphi d\varphi}{\cos^5 \varphi} = \frac{1}{5} \tan^5 \varphi - \frac{1}{3} \tan^3 \varphi + \tan \varphi - \varphi \)

G. \( \sin^{-m} \varphi \cos^n \varphi d\varphi \)

a. \( \sin^{-1} \varphi \cos^n \varphi d\varphi \)

443. \( \int \frac{\cos \varphi d\varphi}{\sin \varphi} = \mathrm{hl} \sin \varphi \)

444. \( \int \frac{\cos^2 \varphi d\varphi}{\sin \varphi} = \cos \varphi + \mathrm{hl} \tan \frac{\varphi}{2} \)

445. \( \int \frac{\cos^3 \varphi d\varphi}{\sin \varphi} = \frac{\cos^2 \varphi}{2} + \mathrm{hl} \sin \varphi \)

446. \( \int \frac{\cos^4 \varphi d\varphi}{\sin \varphi} = \frac{\cos^3 \varphi}{3} + \cos \varphi + \mathrm{hl} \tan \frac{\varphi}{2} \)

447. \( \int \frac{\cos^5 \varphi d\varphi}{\sin \varphi} = \frac{\cos^4 \varphi}{4} + \frac{\cos^2 \varphi}{2} + \mathrm{hl} \sin \varphi \)

448. \( \int \frac{\cos^6 \varphi d\varphi}{\sin \varphi} = \frac{\cos^5 \varphi}{5} + \frac{\cos^3 \varphi}{3} + \cos \varphi + \mathrm{hl} \tan \frac{\varphi}{2} \)

b. \( \sin^{-2} \varphi \cos^n \varphi d\varphi \)

449. \( \int \frac{\cos \varphi d\varphi}{\sin^2 \varphi} = -\frac{1}{\sin \varphi} = -\cosec \varphi \)

450. \( \int \frac{\cos^2 \varphi d\varphi}{\sin^2 \varphi} = -\frac{\cos \varphi}{\sin \varphi} = -\cot \varphi - \varphi \)

451. \( \int \frac{\cos^3 \varphi d\varphi}{\sin^2 \varphi} = (\cos^2 \varphi - 2) \frac{1}{\sin \varphi} = -\sin \varphi - \cosec \varphi \)

452. \( \int \frac{\cos^4 \varphi d\varphi}{\sin^2 \varphi} = \left( \frac{1}{2} \cos^3 \varphi - \frac{3}{2} \cos \varphi \right) \frac{1}{\sin \varphi} - \frac{3}{2} \varphi \)

453. \( \int \frac{\cos^5 \varphi d\varphi}{\sin^2 \varphi} = \left( \frac{1}{3} \cos^4 \varphi + \frac{4}{3} \cos^2 \varphi - \frac{8}{3} \right) \frac{1}{\sin \varphi} \)

454. \( \int \frac{\cos^6 \varphi d\varphi}{\sin^2 \varphi} = \left( \frac{1}{4} \cos^5 \varphi - \frac{5}{8} \cos^3 \varphi - \frac{15}{8} \cos \varphi \right) \frac{1}{\sin \varphi} - \frac{15}{8} \varphi \)

c. \( \sin^{-3} \varphi \cos^n \varphi d\varphi \)

455. \( \int \frac{\cos \varphi d\varphi}{\sin^3 \varphi} = -\frac{1}{2 \sin^2 \varphi} \)

456. \( \int \frac{\cos^2 \varphi d\varphi}{\sin^3 \varphi} = -\frac{\cos \varphi}{2 \sin^2 \varphi} - \frac{1}{2} \mathrm{hl} \tan \frac{\varphi}{2} \)

457. \( \int \frac{\cos^3 \varphi d\varphi}{\sin^3 \varphi} = -\frac{1}{2 \sin^2 \varphi} - \mathrm{hl} \sin \varphi \)

458. \( \int \frac{\cos^3 \varphi d\varphi}{\sin^3 \varphi} = \left( \cos^3 \varphi - \frac{3}{2} \cos \varphi \right) \frac{1}{\sin^3 \varphi} - \frac{3}{2} \mathrm{hl} \tan \frac{\varphi}{2} \)

459. \( \int \frac{\cos^5 \varphi d\varphi}{\sin^3 \varphi} = \left( \frac{1}{2} \cos^4 \varphi - 1 \right) \frac{1}{\sin^3 \varphi} - 2 \mathrm{hl} \sin \varphi \)

460. \( \int \frac{\cos^6 \varphi d\varphi}{\sin^3 \varphi} = \left( \frac{1}{3} \cos^6 \varphi + \frac{5}{3} \cos^3 \varphi - \frac{5}{2} \cos \varphi \right) \frac{1}{\sin^3 \varphi} - \frac{5}{2} \mathrm{hl} \tan \frac{\varphi}{2} \)

d. \( \sin^{-4} \varphi \cos^n \varphi d\varphi \)

461. \( \int \frac{\cos \varphi d\varphi}{\sin^4 \varphi} = -\frac{1}{3 \sin^3 \varphi} \)

462. \( \int \frac{\cos^2 \varphi d\varphi}{\sin^4 \varphi} = -\frac{\cos^3 \varphi}{3 \sin^3 \varphi} = -\frac{1}{3} \cot^3 \varphi \)

463. \( \int \frac{\cos^3 \varphi d\varphi}{\sin^4 \varphi} = -\left( \cos^2 \varphi + \frac{2}{3} \right) \frac{1}{\sin^3 \varphi} \)

464. \( \int \frac{\cos^4 \varphi d\varphi}{\sin^4 \varphi} = \left( -\frac{4}{3} \cos^3 \varphi + \cos \varphi \right) \frac{1}{\sin^4 \varphi} + \varphi = -\frac{1}{3} \cot^3 \varphi + \cot \varphi + \varphi \)

465. \( \int \frac{\cos^5 \varphi d\varphi}{\sin^4 \varphi} = \left( \cos^4 \varphi - 4 \cos^3 \varphi + \frac{8}{3} \right) \frac{1}{\sin^4 \varphi} \)

466. \( \int \frac{\cos^6 \varphi d\varphi}{\sin^4 \varphi} = \left( \frac{1}{2} \cos^5 \varphi - \frac{10}{3} \cos^5 \varphi + \frac{5}{2} \cos \varphi \right) \frac{1}{\sin^4 \varphi} + \frac{5}{2} \varphi \)

e. \( \sin^{-5} \varphi \cos^n \varphi d\varphi \)

467. \( \int \frac{\cos \varphi d\varphi}{\sin^5 \varphi} = -\frac{1}{4 \sin^4 \varphi} \)

468. \( \int \frac{\cos^2 \varphi d\varphi}{\sin^5 \varphi} = \left( -\frac{1}{8} \cos^3 \varphi - \frac{1}{8} \cos \varphi \right) \frac{1}{\sin^4 \varphi} - \frac{1}{8} \mathrm{hl} \tan \frac{\varphi}{2} \)

469. \( \int \frac{\cos^3 \varphi d\varphi}{\sin^5 \varphi} = -\frac{\cos^4 \varphi}{4 \sin^4 \varphi} = -\cot^4 \varphi \)

470. \( \int \frac{\cos^4 \varphi d\varphi}{\sin^5 \varphi} = \left( -\frac{5}{8} \cos^3 \varphi + \frac{3}{8} \cos \varphi \right) \frac{1}{\sin^4 \varphi} + \frac{3}{8} \mathrm{hl} \tan \frac{\varphi}{2} \)

471. \( \int \frac{\cos^5 \varphi d\varphi}{\sin^5 \varphi} = \left( -\frac{3}{4} \cos^4 \varphi + \frac{1}{2} \cos^2 \varphi \right) \frac{1}{\sin^4 \varphi} + \mathrm{hl} \sin \varphi = -\frac{1}{4} \cot^4 \varphi + \frac{1}{2} \cot^2 \varphi + \mathrm{hl} \sin \varphi \)

472. \( \int \frac{\cos^6 \varphi d\varphi}{\sin^5 \varphi} = \left( \cos^5 \varphi - \frac{25}{8} \cos^3 \varphi + \frac{15}{8} \cos \varphi \right) \frac{1}{\sin^4 \varphi} + \frac{15}{8} \mathrm{hl} \tan \frac{\varphi}{2} \)

f. \( \sin^{-6} \varphi \cos^n \varphi d\varphi \)

473. \( \int \frac{\cos \varphi d\varphi}{\sin^6 \varphi} = -\frac{1}{5 \sin^5 \varphi} \) 474. \( \int \frac{\cos^2 \varphi d\varphi}{\sin^6 \varphi} = \left( \frac{2}{15} \cos^6 \varphi - \frac{1}{3} \cos^3 \varphi \right) \frac{1}{\sin^5 \varphi} \)

475. \( \int \frac{\cos^3 \varphi d\varphi}{\sin^6 \varphi} = \left( -\frac{1}{3} \cos^2 \varphi + \frac{2}{15} \right) \frac{1}{\sin^5 \varphi} \)

476. \( \int \frac{\cos^4 \varphi d\varphi}{\sin^6 \varphi} = -\frac{1}{5} \cot^3 \varphi \)

477. \( \int \frac{\cos^5 \varphi d\varphi}{\sin^6 \varphi} = \left( -\cos^4 \varphi + \frac{4}{3} \cos^2 \varphi - \frac{8}{15} \right) \frac{1}{\sin^5 \varphi} \)

478. \( \int \frac{\cos^6 \varphi d\varphi}{\sin^6 \varphi} = \left( -\frac{1}{5} \cot^3 \varphi + \frac{1}{3} \cot^3 \varphi - \cot \varphi - 2 \right) \)

H. \( \sin^{-m} \varphi \cos^{-n} \varphi d\varphi \)

a. \( \sin^{-1} \varphi \cos^{-n} \varphi d\varphi \)

479. \( \int \frac{d\varphi}{\sin \varphi \cos \varphi} = \text{hl tang } \varphi \)

480. \( \int \frac{d\varphi}{\sin \varphi \cos \varphi} = \frac{1}{\cos \varphi} + \text{hl tang } \frac{\varphi}{2} \)

481. \( \int \frac{d\varphi}{\sin \varphi \cos^3 \varphi} = \frac{1}{2 \cos^2 \varphi} + \text{hl tang } \varphi \)

482. \( \int \frac{d\varphi}{\sin \varphi \cos^3 \varphi} = \frac{1}{3 \cos^3 \varphi} + \frac{1}{\cos \varphi} + \text{hl tang } \frac{\varphi}{2} \)

483. \( \int \frac{d\varphi}{\sin \varphi \cos^5 \varphi} = \frac{1}{4 \cos^5 \varphi} + \frac{1}{2 \cos^3 \varphi} + \text{hl tang } \varphi \)

484. \( \int \frac{d\varphi}{\sin \varphi \cos^5 \varphi} = \frac{1}{5 \cos^5 \varphi} + \frac{1}{3 \cos^3 \varphi} + \frac{1}{\cos \varphi} + \text{hl tang } \frac{\varphi}{2} \)

b. \( \sin^{-2} \varphi \cos^{-n} \varphi d\varphi \)

485. \( \int \frac{d\varphi}{\sin^2 \varphi \cos \varphi} = -\frac{1}{\sin \varphi} + \text{hl tang } (45^\circ + \frac{\varphi}{2}) \)

486. \( \int \frac{d\varphi}{\sin^2 \varphi \cos^3 \varphi} = -2 \cot^2 \varphi \)

487. \( \int \frac{d\varphi}{\sin^2 \varphi \cos^5 \varphi} = \left( \frac{1}{2 \cos^5 \varphi} - \frac{3}{2} \right) \frac{1}{\sin \varphi} + \frac{3}{2} \text{ hl tang } (45^\circ + \frac{\varphi}{2}) \)

488. \( \int \frac{d\varphi}{\sin^2 \varphi \cos^7 \varphi} = \frac{1}{3 \sin \varphi \cos^3 \varphi} - \frac{8}{3} \cot 2\varphi \)

489. \( \int \frac{d\varphi}{\sin^2 \varphi \cos^9 \varphi} = \left( \frac{1}{4 \cos^9 \varphi} + \frac{5}{8 \cos^7 \varphi} - \frac{15}{8} \right) \frac{1}{\sin \varphi} + \frac{15}{8} \text{ hl tang } (45^\circ + \frac{\varphi}{2}) \)

490. \( \int \frac{d\varphi}{\sin^2 \varphi \cos^{11} \varphi} = \frac{1}{5 \cos^{11} \varphi} + \frac{1}{5 \cos^9 \varphi} \frac{1}{\sin \varphi} - \frac{16}{5} \cot 2\varphi \)

c. \( \sin^{-3} \varphi \cos^{-n} \varphi d\varphi \)

491. \( \int \frac{d\varphi}{\sin^3 \varphi \cos \varphi} = -\frac{1}{2 \sin^2 \varphi} + \text{hl tang } \varphi \)

492. \( \int \frac{d\varphi}{\sin^3 \varphi \cos^3 \varphi} = \frac{1}{\sin^2 \varphi \cos \varphi} - \frac{3 \cos \varphi}{2 \sin^2 \varphi} + \frac{3}{2} \text{ hl tang } \frac{\varphi}{2} \)

493. \( \int \frac{d\varphi}{\sin^3 \varphi \cos^5 \varphi} = -\frac{2 \cos 2\varphi}{\sin^2 2\varphi} + 2 \text{ hl tang } \varphi \)

494. \( \int \frac{d\varphi}{\sin^3 \varphi \cos^7 \varphi} = \left( \frac{1}{3 \cos^5 \varphi} + \frac{5}{3 \cos \varphi} \right) \frac{1}{\sin^2 \varphi} - \frac{5 \cos \varphi}{2 \sin^2 \varphi} + \frac{5}{2} \text{ hl tang } \frac{\varphi}{2} \)

495. \( \int \frac{d\varphi}{\sin^3 \varphi \cos^9 \varphi} = \frac{1}{4 \sin^2 \varphi \cos^4 \varphi} - \frac{3 \cos^2 2\varphi}{\sin^2 2\varphi} + 3 \text{ hl tang } \varphi \)

496. \( \int \frac{d\varphi}{\sin^3 \varphi \cos^{11} \varphi} = \left( \frac{1}{5 \cos^9 \varphi} + \frac{7}{15 \cos^7 \varphi} + \frac{7}{3 \cos \varphi} \right) \frac{1}{\sin^2 \varphi} - \frac{7 \cos \varphi}{2 \sin^2 \varphi} + \frac{7}{2} \text{ hl tang } \frac{\varphi}{2} \)

d. \( \sin^{-4} \varphi \cos^{-n} \varphi d\varphi \)

497. \( \int \frac{d\varphi}{\sin^4 \varphi \cos \varphi} = -\frac{1}{3 \sin^3 \varphi} - \frac{1}{\sin \varphi} + \text{hl tang } \left( 45^\circ + \frac{\varphi}{2} \right) \)

498. \( \int \frac{d\varphi}{\sin^4 \varphi \cos^3 \varphi} = -\frac{1}{3 \cos \varphi \sin^3 \varphi} - \frac{8}{3} \cot 2\varphi \)

499. \( \int \frac{d\varphi}{\sin^4 \varphi \cos^5 \varphi} = \frac{1}{2 \cos^2 \varphi \sin^2 \varphi} + \frac{5}{2} \int \frac{d\varphi}{\sin^4 \varphi \cos \varphi} \)

500. \( \int \frac{d\varphi}{\sin^4 \varphi \cos^7 \varphi} = \left( -\frac{8}{3 \sin^2 2\varphi} - \frac{16}{3 \sin 2\varphi} \right) \cos 2\varphi \)

501. \( \int \frac{d\varphi}{\sin^4 \varphi \cos^9 \varphi} = \left( \frac{1}{4 \cos^5 \varphi} + \frac{7}{8 \cos^3 \varphi} \right) \frac{1}{\sin^3 \varphi} + \frac{35}{8} \int \frac{d\varphi}{\sin^4 \varphi \cos \varphi} \)

502. \( \int \frac{d\varphi}{\sin^4 \varphi \cos^{11} \varphi} = \frac{1}{5 \cos^9 \varphi \sin^3 \varphi} + \frac{8}{5} \int \frac{d\varphi}{\sin^4 \varphi \cos^3 \varphi} \)

e. \( \sin^{-5} \varphi \cos^{-n} \varphi d\varphi \)

503. \( \int \frac{d\varphi}{\sin^5 \varphi \cos \varphi} = -\frac{1}{4 \sin^4 \varphi} - \frac{1}{2 \sin^2 \varphi} + \text{hl tang } \varphi \)

504. \( \int \frac{d\varphi}{\sin^5 \varphi \cos^3 \varphi} = \left( -\frac{1}{4 \sin^4 \varphi} - \frac{1}{8 \sin^2 \varphi} + \frac{15}{8} \right) \frac{1}{\cos \varphi} + \frac{15}{8} \text{ hl tang } \frac{\varphi}{2} \)

505. \( \int \frac{d\varphi}{\sin^5 \varphi \cos^5 \varphi} = -\frac{1}{4 \cos^2 \varphi \sin^4 \varphi} - \frac{3 \cos 2\varphi}{\sin^2 2\varphi} + 3 \text{ hl tang } \varphi \)

506. \( \int \frac{d\varphi}{\sin^5 \varphi \cos^7 \varphi} = 3 \sin^4 \varphi \cos^3 \varphi + \frac{7}{3} \int \frac{d\varphi}{\sin^5 \varphi \cos^3 \varphi} \)

507. \( \int \frac{d\varphi}{\sin^5 \varphi \cos^9 \varphi} = \left( -\frac{4}{\sin^2 2\varphi} - \frac{6}{\sin^2 2\varphi} \right) \cos 2\varphi + 6 \text{ hl tang } \varphi \)

508. \( \int \frac{d\varphi}{\sin^5 \varphi \cos^{11} \varphi} = \left( \frac{1}{5 \cos^9 \varphi} + \frac{3}{5 \cos^7 \varphi} \right) \frac{1}{\sin^4 \varphi} + \frac{21}{5} \int \frac{d\varphi}{\sin^5 \varphi \cos^3 \varphi} \) 509. \( \int \frac{d\varphi}{\sin^6 \varphi \cos \varphi} = -\frac{1}{5 \sin^3 \varphi} + \frac{1}{3 \sin^2 \varphi} - \frac{1}{\sin \varphi} + \) hl tang \( (45^\circ + \frac{\varphi}{2}) \)

510. \( \int \frac{d\varphi}{\sin^6 \varphi \cos^2 \varphi} = \left( -\frac{1}{5 \sin^3 \varphi} - \frac{2}{5 \sin^3 \varphi} \right) \frac{1}{16} \cot 2\varphi \)

511. \( \int \frac{d\varphi}{\sin^6 \varphi \cos^3 \varphi} = \left( -\frac{1}{5 \sin^3 \varphi} - \frac{7}{15 \sin^5 \varphi} - \frac{7}{3 \sin \varphi} \right) \frac{1}{\cos^2 \varphi} + \frac{7 \sin \varphi}{2 \cos^2 \varphi} + \frac{7}{2} \) hl tang \( (45^\circ + \frac{\varphi}{2}) \)

512. \( \int \frac{d\varphi}{\sin^6 \varphi \cos^4 \varphi} = -\frac{1}{5 \sin^3 \varphi \cos^3 \varphi} - \frac{8}{5} \left( \frac{8}{3 \sin^3 2\varphi} + \frac{16}{3 \sin 2\varphi} \right) \cos 2\varphi \)

512. \( \int \frac{d\varphi}{\sin^6 \varphi \cos^5 \varphi} = \left( -\frac{1}{5 \sin^3 \varphi} - \frac{3}{5 \sin^3 \varphi} \right) \frac{1}{\cos^4 \varphi} + \frac{21}{5} \int \frac{d\varphi}{\sin^2 \varphi \cos^5 \varphi} \)

513. \( \int \frac{d\varphi}{\sin^6 \varphi \cos^6 \varphi} = \left( -\frac{32}{5 \sin^3 2\varphi} - \frac{128}{15 \sin^3 2\varphi} - \frac{256}{15 \sin 2\varphi} \right) \cos 2\varphi \)

I. sin \((a+b\varphi)\) sin \((c+d\varphi)\)d\(\varphi\),

514. \( \int \sin (a+b\varphi) \sin (c+d\varphi) d\varphi = \frac{1}{2(b-d)} \sin (a-c+(b-d)\varphi) - \frac{1}{2(b+d)} \sin (a+c+(b+d)\varphi) \)

K. sin \((a+b\varphi)\) cos \((c+d\varphi)\)d\(\varphi\).

515. \( \int \sin (a+b\varphi) \cos (c+d\varphi) d\varphi = -\frac{1}{2(b+d)} \cos (a+c+(b+d)\varphi) - \frac{1}{2(b-d)} \cos (a-c+(b-d)\varphi) \)

L. cos \((a+b\varphi)\) cos \((c+d\varphi)\)d\(\varphi\).

516. \( \int \cos (a+b\varphi) \cos (c+d\varphi) d\varphi = \frac{1}{2(b+d)} \sin (a+c+(b+d)\varphi) + \frac{1}{2(b-d)} \sin (a-c+(b-d)\varphi) \)

M. \( \varphi^m \sin \varphi d\varphi \). For all values of \( m \).

517. \( \int \varphi^m \sin \varphi d\varphi = -\varphi^m \cos \varphi + m \varphi^{m-1} \sin \varphi + m(m-1) \varphi^{m-2} \cos \varphi - m(m-1)(m-2) \varphi^{m-3} \sin \varphi - m(m-3) \varphi^{m-4} \cos \varphi + \ldots + \ldots \ldots + \ldots \)

N. \( \varphi^m \cos \varphi d\varphi \).

518. \( \int \varphi^m \cos \varphi d\varphi = \varphi^m \sin \varphi + m \varphi^{m-1} \cos \varphi - m(m-1) \varphi^{m-2} \sin \varphi - m(m-1)(m-2) \varphi^{m-3} \cos \varphi + \ldots + \ldots \ldots \)

O. \( \varphi y dx \).

\( \int \varphi y dx = \varphi \int y dx - \int d\varphi \int y dx \)

519. \( \int \varphi \sin^m \varphi d(\sin \varphi) = \frac{1}{m+1} (\varphi \sin^{m+1} \varphi - \int \sin^{m+1} \varphi d\varphi) \); or

\( \int \arcsin x \cdot x^m dx = \frac{1}{m+1} (\arcsin x \cdot x^{m+1} - \int \frac{x^{m+1} dx}{\sqrt{(1-x^2)}}) \)

520. \( \int \varphi d\varphi = \frac{1}{2} \varphi^2 \); or

\( \int \arcsin x \frac{dx}{\sqrt{(1-x^2)}} = \frac{1}{2} (\arcsin x)^2 \)

\( \int \arccos x \frac{dx}{\sqrt{(1-x^2)}} = -\frac{1}{2} (\arccos x)^2 \)

\( \int \arctan x \frac{dx}{1+x^2} = \frac{1}{2} (\arctan x)^2 \)

\( \int \operatorname{arc cot} x \frac{dx}{1+x^2} = -\frac{1}{2} (\operatorname{arc cot} x)^2 \)

\( \int \operatorname{arc vsin} x \frac{dx}{\sqrt{(2x-x^2)}} = \frac{1}{2} (\operatorname{arc vsin} x)^2 \)

521. \( \int \varphi \sin \varphi d\varphi = -\varphi \cos \varphi + \sin \varphi \); or

\( \int \arcsin x \frac{xdx}{\sqrt{(1-x^2)}} = -\arcsin x \cdot \sqrt{(1-x^2)} + x \)

522. \( \int \varphi \sin^2 \varphi d\varphi = \left( -\frac{1}{2} \sin \varphi \cos \varphi + \frac{1}{4} \varphi \right) \varphi + \frac{1}{4} \sin^2 \varphi \); or

\( \int \arcsin x \frac{x^2 dx}{\sqrt{(1-x^2)}} = \left( -\frac{1}{2} x \sqrt{(1-x^2)} + \frac{1}{4} \arcsin x \right) \arcsin x + \frac{1}{4} x^2 \)

523. \( \int \varphi \sin^3 \varphi d\varphi = -\left( \frac{1}{3} \sin^2 \varphi + \frac{2}{3} \right) \cos \varphi + \frac{1}{9} \sin^5 \varphi + \frac{2}{3} \sin \varphi \); or

\( \int \arcsin x \frac{x^3 dx}{\sqrt{(1-x^2)}} = -\left( \frac{1}{3} x^2 + \frac{2}{3} \right) \sqrt{(1-x^2)} \cdot \arcsin x + \frac{1}{9} x^5 + \frac{2}{3} x^3 \)

524. \( \int \frac{\varphi d\varphi}{\cos^2 \varphi} = \frac{\varphi \sin \varphi}{\cos \varphi} + \text{hl} \cos \varphi \); or

\[ \int \arcsin x \frac{dx}{(1-x^2)^{\frac{3}{2}}} = \frac{x \arcsin x}{\sqrt{1-x^2}} + \frac{1}{2} \ln (1-x^2) \]

525. \( \int \frac{\varphi \sin \varphi d\varphi}{\cos^2 \varphi} = \frac{\varphi}{\cos \varphi} + \frac{1}{2} \ln \frac{1-\sin \varphi}{1+\sin \varphi} \); or

\[ \int \arcsin x \frac{x dx}{(1-x^2)^{\frac{3}{2}}} = \frac{\arcsin x}{\sqrt{1-x^2}} + \frac{1}{2} \ln \frac{1-x}{1+x} \]

526. \( \int \varphi \cos^m \varphi d(\cos \varphi) = \frac{1}{m+1} \left( \varphi \cos^{m+1} \varphi - \int \cos^{m+1} \varphi d\varphi \right) \); or

\[ \int \arccos x \cdot x^m dx = \frac{1}{m+1} \left( \arccos x \cdot x^{m+1} + \int \frac{x^{m+1} dx}{\sqrt{1-x^2}} \right) \]

527. \( \int \varphi \tang^m \varphi d(\tang \varphi) = \frac{1}{m+1} \left( \varphi \tang^{m+1} \varphi - \int \tang^{m+1} \varphi d\varphi \right) \); or

\[ \int \arctan x \cdot x^m dx = \frac{1}{m+1} \left( \arctan x \cdot x^{m+1} - \int \frac{x^{m+1} dx}{1+x^2} \right) \]

528. \( \int \varphi \cot^m \varphi d(\cot \varphi) = \frac{1}{m+1} \left( \varphi \cot^{m+1} \varphi - \int \frac{\cos^{m+1} \varphi d\varphi}{\sin^{m+1} \varphi} \right) \); or

\[ \int \arccot x \cdot x^m dx = \frac{1}{m+1} \left( \arccot x \cdot x^{m+1} + \int \frac{x^{m+1} dx}{1+x^2} \right) \]

529. \( \int \varphi \sec^m \varphi d(\sec \varphi) = \frac{1}{m+1} \left( \varphi \sec^{m+1} \varphi - \int \frac{d\varphi}{\cos^{m+1} \varphi} \right) \); or

\[ \int \arcsec x \cdot x^m dx = \frac{1}{m+1} \left( \arcsec x \cdot x^{m+1} - \int \frac{x^m dx}{\sqrt{x^2-1}} \right) \]

530. \( \int \varphi \cosec^m \varphi d(\cosec \varphi) = \frac{1}{m+1} \left( \varphi \cosec^{m+1} \varphi - \int \frac{d\varphi}{\sin^{m+1} \varphi} \right) \); or

\[ \int \arcosec x \cdot x^m dx = \frac{1}{m+1} \left( \arcosec x \cdot x^{m+1} + \int \frac{x^m d\varphi}{\sqrt{x^2-1}} \right) \]

531. \( \int \varphi \vsin^m \varphi d(\vsin \varphi) = \frac{1}{m+1} \left( \varphi \vsin^{m+1} \varphi - \int \frac{\vsin^{m+1} \varphi d\varphi}{\vsin^m \varphi} \right) \)

\[ \int (1-\cos \varphi)^{m+1} d\varphi; \text{ or } \]

\[ \int \arcsin x \cdot x^m dx = \frac{1}{m+1} \left( \arcsin x \cdot x^{m+1} - \int \frac{x^{m+1} dx}{\sqrt{(2x-x^2)}} \right) \]

532. \( \int \varphi \tang^2 \varphi d\varphi = \left( \tang \varphi - \frac{1}{2} \varphi \right)^2 - \ln \sec \varphi \); or

\[ \int \arctan x \frac{x^2 dx}{1+x^2} = \left( x - \frac{1}{2} \arctan x \right) \arctan x - \frac{1}{2} \ln (1+x^2) \]

533. \( \int \varphi \cos^2 \varphi d\varphi = \left( \frac{1}{2} \tang \varphi \cos^2 \varphi + \frac{1}{4} \varphi \right) \varphi + \frac{1}{4} \cos^2 \varphi \); or

\[ \int \arctan x \frac{dx}{(1+x^2)^2} = \left( \frac{x}{2(1+x^2)} + \frac{1}{4} \arctan x \right) \arctan x + \frac{1}{4(1+x^2)} \]

P. \( (a+b \cos \varphi)^{-m} (f+g \cos \varphi) d\varphi \)

534. \( \int \frac{d\varphi}{a+b \cos \varphi} = \frac{1}{\sqrt{(a^2-b^2)}} \arccos \frac{b+a \cos \varphi}{a+b \cos \varphi} = \frac{1}{\sqrt{(b^2-a^2)}} \ln \frac{b+a \cos \varphi + \sin \varphi \sqrt{(b^2-a^2)}}{a+b \cos \varphi} \); or, for \( a=b \)

\[ \int \frac{d\varphi}{a+a \cos \varphi} = \frac{1}{a} \tang \frac{1}{2} \varphi \]

535. \( \int \frac{\cos \varphi d\varphi}{a+b \cos \varphi} = \frac{\varphi}{b} - \frac{a}{b} \int \frac{d\varphi}{a+b \cos \varphi} \)

536. \( \int \frac{d\varphi}{(a+b \cos \varphi)^2} = \frac{1}{(a^2-b^2)} \left( \frac{a \sin \varphi}{a+b \cos \varphi} + a \int \frac{d\varphi}{a+b \cos \varphi} \right) \)

537. \( \int \frac{\cos \varphi d\varphi}{(a+b \cos \varphi)^2} = \frac{1}{(a^2-b^2)} \left( \frac{a \sin \varphi}{a+b \cos \varphi} - b \int \frac{d\varphi}{a+b \cos \varphi} \right) \)

Q. \( (a+b \cos \varphi)^{-1} \sin \varphi d\varphi \)

538. \( \int \frac{\sin \varphi d\varphi}{a+b \cos \varphi} = -\frac{1}{b} \ln (a+b \cos \varphi) \)

R. \( (1+a \cos \varphi)^m d\varphi \). For fractional powers see Méc. Cél.; also Ivory and Wallace, Ed. Trans. 1798, 1805.

539. \( \int (1+a \cos \varphi) d\varphi = \varphi + a \sin \varphi \)

540. \( \int (1+a \cos \varphi)^2 d\varphi = \left( 1+\frac{1}{2} a^2 \right) \varphi + 2a \sin \varphi + \frac{1}{4} a^2 \sin 2\varphi \) 541. \( \int (1 + a \cos \varphi)^3 d\varphi = \left(1 + \frac{3}{2} a^2\right) \varphi + \left(3a + \frac{3}{4} a^3\right) \sin \varphi + \frac{3}{4} a^2 \sin 2\varphi + \frac{1}{12} a^3 \sin 3\varphi \)

542. \( \int (1 + a \cos \varphi)^4 d\varphi = \left(1 + 3a^2 + \frac{3}{8} a^4\right) \varphi + (4a + 3a^3) \sin \varphi + \left(\frac{3}{2} a^2 + \frac{1}{4} a^4\right) \sin 2\varphi + \frac{1}{3} a^3 \sin 3\varphi + \frac{1}{32} a^4 \sin 4\varphi \)

Sect. VI.—Logarithmic Fluxions.

A. hl xydz

\( \int \text{hl } x y dz = \text{hl } x \int y dz - \int d \text{hl } x \int y dz = \text{hl } x \int y dz - \int \frac{dx}{x} \int y dz \)

543. \( \int x^m \text{hl } x dx = \frac{x^{m+1}}{m+1} \left( \text{hl } x - \frac{1}{m+1} \right) \)

544. \( \int x^{-1} \text{hl } x dx = \frac{1}{2} \text{hl}^2 x \)

545. \( \int \text{hl} \left(a + bx\right) \frac{dx}{x} = \text{hl} a \text{hl} x + \frac{bx}{a} - \frac{b^2 x^2}{2a^2} + \frac{b^5 x^5}{3^2 a^5} ... \) \( = \frac{1}{2} \text{hl}^2 b x - \frac{a}{bx} + \frac{a^2}{2^2 b^2 x^2} - \frac{a^3}{3^2 b^3 x^3} + ... \)

546. \( \int \frac{\text{hl} x dx}{a + bx} = \frac{1}{b} \text{hl} x \text{hl} (a + bx) - \frac{1}{b} \int \frac{dx}{x} \text{hl} (a + bx) \) \( = \frac{1}{b} \text{hl} x \text{hl} \frac{a + bx}{a} - \frac{x}{a} + \frac{bx^2}{2^2 a^2} - \frac{b^2 x^3}{3^2 a^3} + ... \) \( = \frac{1}{b} \text{hl} x \text{hl} (a + bx) - \frac{1}{2b} \text{hl}^2 b x + \frac{a}{b^2 x} - \frac{a^2}{2^2 b^2 x^2} + \frac{a^3}{3^2 b^3 x^3} - ... \)

B. hl^n xy dx

Since \( \int Y dZ = \frac{dY}{dX} \int Z dX - \frac{d^2 Y}{dX^2} \int^2 Z dX^2 + ... \) (n5), taking the dX of this theorem \( = \frac{dx}{x}, Y = \text{hl}^n x \) and dZ = y dx, we have

\( \int Y dZ = \int \text{hl}^n x y dx = \frac{dY}{dX} \int \left( \int y dx \right) \frac{dx}{x} ... \) \( = d \text{hl}^n x \frac{x}{dx} \int \left( \int y dx \right) \frac{dx}{x} ... \); thus, \( y = x^m \)

547. \( \int x^m \text{hl}^n x dx = \frac{x^{m+1}}{m+1} \left( \text{hl}^n x - \frac{n}{m+1} \text{hl}^{n-1} x + \frac{n(n-1)}{(m+1)^2} \text{hl}^{n-2} x - ... \right) \).

When n is a negative whole number = -N, we may obtain a finite series by making Y = yx, and dZ = \( \frac{dx}{x} \) hl^{-N} x, but the last term still contains a fluent.

548. \( \int x^{-1} \text{hl}^n x dx = \int \frac{dx}{x} \text{hl}^n x = \frac{1}{n+1} \text{hl}^{n+1} x \)

549. \( \int \frac{dx}{\text{hl} x} = \text{hl} \ln x + \frac{1}{1} + \frac{1}{2!} \frac{1}{2} + \frac{1}{3!} \frac{1}{2.3} + ... \)

550. \( \int \frac{x^m dx}{\text{hl} x} = \int \frac{dy}{\text{hl} y}, \) when \( y = x^{m+1} \)

551. \( \int \frac{x^m dx}{\text{hl}^2 x} = -\frac{x^{m+1}}{\text{hl} x} + (m+1) \int \frac{x^m dx}{\text{hl} x} \)

552. \( \int \frac{x^m dx}{\text{hl}^3 x} = -\frac{x^{m+1}}{2 \text{hl}^2 x} - \frac{(m+1)x^{m+1}}{2.1 \text{hl} x} + \frac{(m+1)^2}{2.1} \int \frac{x^m dx}{\text{hl} x} \)

553. \( \int \frac{dx}{\text{hl}^2 x} = \text{hl} \ln x - \frac{\text{hl} x}{1} + \frac{1}{2!} \frac{\text{hl}^2 x}{2} - \frac{1}{3!} \frac{\text{hl}^3 x}{3.1.2.3} + ... \)

Particular values, from x=0 to x=1.

554. \( \int' \frac{dx}{\sqrt{\text{hl}^2 x}} = \sqrt{3.141592} \). Euler, Comm. Ac. Petr. XVI.

555. \( \int' dx \left( \frac{\text{hl} \left( \frac{1}{x} \right)}{} \right)^{2m+1} = \frac{1.3.5.7...(2m+1)}{2^{m+1}} \sqrt{3.141592} \). Ibid.

Sect. VII.—Exponential Fluxions.

A. a^x y dx

In the theorem \( \int Y dZ = \frac{dY}{dX} \int Z dX - \int^2 Z dX^2 \), we may put either dX = dx, Y = y, and dZ = a^x dx = d_{hl a} a^x, or dX = dx, Y = a^x, and dZ = y dx; and, in the former manner, we obtain,

556. \( \int a^x x^m dx = \frac{a^x x^m}{\text{hl} a} - \frac{ma^x x^{m-1}}{\text{hl}^2 a} + \frac{m(m-1)a^x x^{m-2}}{\text{hl}^3 a} - ... \); thus,

557. \( \int a^x dx = \frac{a^x}{\text{hl} a} \)

558. \( \int a^x x dx = \frac{a^x x}{\text{hl} a} - \frac{a^x}{\text{hl}^2 a} \)

559. \( \int a^x x^2 dx = \frac{a^x x^2}{\text{hl} a} - \frac{2a^x x}{\text{hl}^2 a} + \frac{2.1 a^x}{\text{hl}^3 a} \) 560. \( \int a^x x^2 dx = \frac{a^x x^5}{\text{hl}\,a} - \frac{3a^x x^2}{\text{hl}^2 a} + \frac{3.2a^x x}{\text{hl}^3 a} - \frac{3.2.1a^x}{\text{hl}^4 a} \).

561. \( \int \frac{a^x dx}{x} = \text{hl}\,x + \frac{x \text{hl}\,a}{1} + \frac{x^2 \text{hl}^2 a}{1.2.2} + \frac{x^3 \text{hl}^3 a}{1.2.3.3} + \ldots \)

562. \( \int \frac{a^x dx}{x^2} = -\frac{a^x}{x} + \text{hl}\,a \int \frac{a^x dx}{x} \)

563. \( \int \frac{a^x dx}{x^3} = -\frac{a^x}{2x^2} - \frac{a^x \text{hl}\,a}{2.1x} + \frac{\text{hl}^2 a}{2.1} \int \frac{a^x dx}{x} \)

564. \( \int \frac{a^x dx}{x^4} = -\frac{a^x}{3x^3} - \frac{a^x \text{hl}\,a}{3.2x^2} - \frac{a^x \text{hl}^2 a}{3.2.1} + \frac{\text{hl}^3 a}{3.2.1} \int \frac{a^x dx}{x} \)

565. \( \int \frac{a^x dx}{\sqrt{x}} = \frac{a^x}{\sqrt{x}} \left( \frac{1}{\text{hl}\,a} - \frac{1}{2x \text{hl}^2 a} + \frac{1.3}{(2x)^2 \text{hl}^3 a} + \frac{1.3.5}{(2x)^3 \text{hl}^4 a} + \ldots \right) \) \( = \frac{a^x}{\sqrt{x}} \left( \frac{2x}{1} - \frac{(2x)^2 \text{hl}\,a}{1.3} + \frac{(2x)^3 \text{hl}^2 a}{1.3.5} - \ldots \right) \)

566. \( \int \frac{a^x dx}{1-x} = a^x \left( \frac{1}{(1-x)\text{hl}\,a} - \frac{1}{(1-x)^2 \text{hl}^2 a} + \frac{1.2}{(1-x)^3 \text{hl}^3 a} - \frac{1.2.5}{(1-x)^4 \text{hl}^4 a} + \ldots \right) \)

B. \( a^{mx} x^n dx \)

567. \( \int a^{mx} x^n dx = \frac{1}{m^{n+1}} \int a^y y^n dy \); making \( y=mx \)

C. \( x^{m+nx} dx \)

568. \( \int x^{m+nx} dx = \int \left( 1 + \frac{nx \text{hl}\,x}{1} + \frac{n^2 x^2 \text{hl}^2 x}{1.2} + \frac{n^3 x^3 \text{hl}\,x}{1.2.3} + \ldots \right) x^m dx \)

Particular value, from \( x=0 \) to \( x=1 \)

569. \( \int x^n + nx dx = \frac{1}{m+1} - \frac{n}{(m+2)^2} + \frac{n^2}{(m+3)^3} - \ldots \)

D. \( e^{-xx} dx \)

Putting, in the Taylorian theorem, \( \int Y dZ = \frac{dY}{dX} \int Z dX - \ldots \) (n.5) \( Y = e^{-xx} \), \( Z=x \), and \( dX=d(-xx)=2xdx \), we have,

570. \( \int e^{-xx} dx = e^{-xx} \left( -\frac{2}{3} x^3 \right) - e^{-xx} \frac{4}{3.5} x^5 + e^{-xx} \left( -\frac{8}{3.5.7} x^7 \right) - \ldots \) \( = -e^{-xx} \left( \frac{2}{3} x^3 + \frac{4}{3.5} x^5 + \frac{8}{3.5.7} x^7 + \ldots \right) \)

Particular value, from \( x=-\infty \) to \( x=+\infty \)

571. \( \int' e^{-xx} dx = \sqrt{3.14159} \), Laplace, Méc. Cél. X.; or thus, \( \int e^{-xx} dx = \int y dx, -xx=\text{hl}\,y, xx=\text{hl}\,\frac{1}{y}, x= \sqrt{\text{hl}\,\frac{1}{y}}; dy = -e^{-xx} 2xdx; \frac{dy}{\sqrt{\text{hl}\,\frac{1}{y}}} = -2e^{-xx} dx \), and \( \int e^{-xx} dx = -\frac{1}{2} \int \frac{dy}{\sqrt{\text{hl}\,\frac{1}{y}}} \) (n.554): in this expression we may make \( \frac{1}{\sqrt{\text{hl}\,\frac{1}{y}}}=Y \), then \( dY= \frac{d\text{hl}\,y}{2(\text{hl}\,\frac{1}{y})^{3/2}} \), and if we wish to have \( \frac{dY}{dX} = Y \), we must take \( dX=\text{d hl}\,y \), and \( Z \) will be \( =y \), so that the series will give us \( \int \frac{dy}{\sqrt{\text{hl}\,\frac{1}{y}}} \) \( = \frac{1}{2(\text{hl}\,\frac{1}{y})^{3/2}} - \frac{3}{4(\text{hl}\,\frac{1}{y})^{5/2}} + \frac{3.5}{8(\text{hl}\,\frac{1}{y})^{7/2}} - \ldots = \frac{y}{\sqrt{(\text{hl}\,\frac{1}{y})}} \left( \frac{1}{2(\text{hl}\,\frac{1}{y})} - \frac{3}{4(\text{hl}\,\frac{1}{y})^2} + \frac{3.5}{8(\text{hl}\,\frac{1}{y})^3} - \ldots \right) \); but these series will fail in the extreme cases, although they converge with sufficient rapidity in most others.

E. \( \int e^{mx} dx \)

Putting \( e^{mx}=Y \), we have \( dY=e^{mx} mx^{m-1} dx \) and taking \( dX=x^{m-1} dx \), and \( Z=x \), we obtain the series.

572. \( \int e^{mx} dx = me^{mx} \frac{1}{m+1} x^{m+1} m^o e^{mx} \) \( = \frac{1}{(m+1)(2m+1)} x^{2m+1} + \ldots \)

F. \( e^{ax} \sin^m x dx \)

573. \( \int e^{ax} \sin x dx = \frac{e^{ax}(a \sin x - \cos x)}{a^2+1} \)

574. \( \int e^{ax} \sin^2 x dx = \frac{e^{ax} \sin x(a \sin x - 2 \cos x)}{a^2+4} + \frac{1.2 e^{ax}}{a(a^2+4)} \)

575. \( \int e^{ax} \sin^3 x dx = \frac{e^{ax} \sin^2 x(a \sin x - 3 \cos x)}{a^2+9} + \frac{2.3 e^{ax}(a \sin x - \cot x)}{(a^2+1)(a^2+9)} \) G. \( e^{ax} \cos^m x dx \)

576. \( \int e^{ax} \cos x dx = \frac{e^{ax}}{a^2+1} (a \cos x + \sin x) \)

577. \( \int e^{ax} \cos^2 x dx = \frac{e^{ax}}{a^2+4} \cos x (a \cos x + 2 \sin x) + \frac{1.2e^{ax}}{a(a^2+4)} \)

578. \( \int e^{ax} \cos^3 x dx = \frac{e^{ax}}{a^2+9} \cos^2 x (a \cos x + 3 \sin x) + \frac{2.3e^{ax}}{(a^2+1)(a^2+9)} (a \cos x + \sin x) \)

H. \( e^{ax} \sin bxdx \)

579. \( \int e^{ax} \sin bxdx = \frac{e^{ax}}{a^2+b^2} (a \sin bx - b \cos bx) \)

I. \( e^{ax} \cos bxdx \)

580. \( \int e^{ax} \cos bxdx = \frac{e^{ax}}{a^2+b^2} (a \cos bx + b \sin bx) \)

SECT. VIII.—Index of Fluxions.

1. dx 2. adx 3. \( x^n dx \) 4. ydx 5. ydz 6. 'dy 7. \( x^m (a+bx)^{-n} dx \) 30. \( x^m (a+bx^2)^{-n} dx \) 51. \( x^m (a+bx+cx^2)^{-n} dx \) 72. \( x^m (a+bx^2)^{-n} dx \) 90. \( x^{N-M} (a+bx^N)^{-1} dx \) 92. \( x^{2N-M} (a+bx^N+cx^{2N})^{-1} dx \) 93. \( x^M ([x+f'][x+g]...[x^2+ax+b])^{-1} ... dx \) 99. \( x^m \frac{A+Bx+Cx^2...}{a+bx+cx^2...} dx \) 102. \( x^m (a+bx)^{\frac{n}{2}} dx \) 144. \( x^m (a+bx)^{\frac{n}{3}} dx \) 172. \( x^m (a+bx^2)^{\frac{n}{2}} dx \) 216. \( x^m (ax+bx^2)^{\frac{n}{2}} dx \) 258. \( x^m (a+bx+cx^2)^{\frac{n}{2}} dx \) 300. \( x^{\frac{m}{2}} (a+bx)^n dx \) 301. \( x^{m-\frac{1}{2}} (a+bx^2)^{-n} dx \) 313. \( x^m (f+gx)^{-n} (a+bx)^{-\frac{1}{2}} dx \) 325. \( x^m (f+gx)^{-1} (a+bx^2)^{-\frac{1}{2}} dx \) 335. \( x^M (f+gx^2)^{-1} (a+bx^2)^{-\frac{1}{2}} dx \) 339. \( x^M (f+gx^2)^{-1} (a+bx^2)^{-\frac{1}{2}} dx \) 343. \( x^M (f+gx)^{-1} (a+bx+cx^2)^{-\frac{1}{2}} dx \) 347. \( (a'-x^4)^{-\frac{1}{2}} dx \) 349. \( x^M (1-x^{2N})^{-\frac{1}{2}} dx \) 350. \( x^M (1-x^{M+N})^{-\frac{m+1}{M+N}} dx \) 351. \( \sin^M \varphi d\varphi \) 357. \( \cos^M \varphi d\varphi \) 363. \( \sin^M \varphi \cos^N \varphi d\varphi \) 395. \( \sin^{-M} \varphi d\varphi \) 401. \( \cos^{-M} \varphi d\varphi \) 407. \( \sin^M \varphi \cos^{-N} \varphi d\varphi \) 443. \( \sin^{-M} \varphi \cos^N \varphi d\varphi \) 479. \( \sin^{-M} \varphi \cos^{-N} \varphi d\varphi \) 514. \( \sin (a+b\varphi) \cos (c+d\varphi) d\varphi \) 515. \( \sin (a+b\varphi) \sin (c+d\varphi) d\varphi \) 516. \( \cos (a+b\varphi) \cos (c+d\varphi) d\varphi \) 517. \( \varphi^n \sin \varphi d\varphi \) 518. \( \varphi^n \cos \varphi d\varphi \) 519. \( \varphi y dx \) 534. \( (a+b \cos \varphi)^{-M} (f+g \cos \varphi) d\varphi \) 538. \( (a+b \cos \varphi)^{-1} \sin \varphi d\varphi \) 539. \( (1+a \cos \varphi)^M d\varphi \) 543. \( h l n x y dz \) 547. \( h l n^x y dx \) 556. \( a^x y dx \) 567. \( a^{mx} x^n dx \) 568. \( x^{m+n} dx \) 570. \( e^{-ax} dx \) 572. \( e^{x^m} dx \) 573. \( e^{ax} \sin^M x dx \) 576. \( e^{ax} \cos^M x dx \) 579. \( e^{ax} \sin b x dx \) 580. \( e^{ax} \cos b x dx \)

Transformations. For A. \( x^{n-1} (a+bx^{2n})^{-M} dx \) B. \( x^{\frac{1}{2}} (a+bx)^{-\frac{1}{2}} dx \) C. \( x^{-\frac{1}{2}} (a+bx)^{-n} dx \) We have

\[ A = \frac{dy}{2(a+by^2)^{\frac{1}{2}}}; \text{ if } y = x^3 \]

\[ B = (ax+bx^2)^{\frac{1}{2}}dx; \text{ or } = x(ax+bx^2)^{-\frac{1}{2}}dx \]

\[ C = 2(a+by^2)^{-N}dy; \text{ if } y^2 = x \]

The Fluxions of which Mr Landen has assigned the fluents in the first volume of his Mathematical Memoirs, 4to, London, 1780, by means of arcs of the conic sections, are chiefly of some of the following forms:

Table III. \( x = 0, \frac{1}{2}, 1, \frac{3}{2} (a+bx^2)^{-\frac{1}{2}}, \frac{1}{4}, \frac{3}{4} dx \)

IV. \( x = 0, \frac{1}{2}, \frac{2}{3} (a+bx^2)^{-\frac{1}{2}}, \frac{1}{3}, \frac{2}{3} dx \)

XII. \( x = 0, \frac{1}{2}, 1, \frac{3}{2} (a+bx^2)^{-\frac{1}{2}}, \frac{1}{3}, \frac{2}{3} dx \)

\( x = \frac{1}{2}(a+bx+cx^2)^{-\frac{1}{2}}dx \)

\( x(f+gx)^{-\frac{1}{2}}(a+bx^2)^{-\frac{1}{2}}dx \)

\( (f+(a+bx^2)^{\frac{1}{2}})^{-\frac{1}{2}}dx \)

\( x^{-\frac{1}{2}}(a+bx)^{\frac{1}{2}}(b+cx)^{-\frac{1}{2}}dx \)

\( x^{-\frac{1}{2}}(a+bx+cx^2)^{\frac{1}{2}}(d+ex)^{-\frac{1}{2}}dx \)

\( x^{-\frac{1}{2}}(a+bx+cx^2)^{-\frac{1}{2}}(d+ex)^{-\frac{1}{2}}dx \)

\( -1, 2, +\frac{3}{2}(a+bx+cx^2)^{-\frac{1}{2}}(d+ex)^{-\frac{1}{2}}dx \)

\( 0, -1, 2, (a+bx+cx^2)^{-\frac{1}{2}}(d+ex+fx^2)^{-\frac{1}{2}}dx \)

(s. f.)