Page images
PDF
EPUB

CHAPTER II.

CONSTRUCTION OF RULES FOR INTEGRATION OF

ALGEBRAICAL FUNCTIONS.

10.] We propose in the present and next Chapters to construct WE the rules of the Integral by inverting those of the Differential Calculus and first we shall from this point of view exhibit the forms of indefinite integrals which correspond to those theorems on definite integrals which have been proved in Article 8. THEOREM I. Since d'aF(x) = ar'(x)dx;

.'.

fax (x)dx = ar(x),

=

= a/F'(x) dx;

(1)

that is, in the integration of an infinitesimal element one of whose factors is a constant, the constant may be placed outside the sign of integration.

...

THEOREM II.-Since

d. {F(x) ±ƒ(x) ± ... } =

d. F(x) d. f(x) ± ...

= F(x) dx + f'(x) dx ± ...; ±

f{x (x)dx ±ƒ'(x)dx ± ... } = F(x) ±ƒ(x) ± ...

(2)

that is, the integral of an algebraic sum of infinitesimal elements is equal to the sum of the integrals of the several infinitesimal elements.

[ocr errors][merged small][merged small][merged small]

[{F'(x) +√/=1f'(x)}dx = F(x)+√/=1 f (x).

THEOREM III.-Since

· d.F(x) × ƒ (x) = ƒ (x) × F′(x) dx + F(x) × f'(x)dx; f(x) × F(x) dx = d.(F(x) × f(x)) − F(x) × ƒ'(x)dx ;

·. [f(x) ×x′(x)dx = x(x) × ƒ (x) – [ v(x) × f'(x) dx ;

(3)

and therefore if f(x) = u, F(x) = v,

Judo

dv = uv

- fvdu;

(4)

THEOREM IV. Since d. F{f(x)} = F' {ƒ (x)}ƒ'(x) dx ;

..

[x'{f(x)}ƒ'(x)dx = v{f(x)} ;

(5)

which is the theorem in the Integral Calculus corresponding to that of the differentiation of compound functions as explained in Art. 31 of Vol. I, and which may also thus be proved :

Let f(x) = 2;

.. f'(x) dx = dz ;

fx {f(x)} f'(x) dx = ['x' (z) dz,

= F(2),

= F{f(x)}.

(6)

Integration by this last process is evidently equivalent to the integration of a compound function, and is of great importance; for hence it follows that those formula of integration which are true for x and simple functions of a, are also true for compound functions.

SECTION I.-Integration of Fundamental Algebraical Functions. 11.] Integration of x" dx.

[blocks in formation]

Let n be substituted for m 1; then m = n + 1; and

[ocr errors][merged small][ocr errors][merged small]

Therefore to integrate "da, add unity to the index, divide by

the index so increased, and by dx.

Of this result the following are particular cases:

(1.) Let n be negative; that is, for n substitute -n;

[merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small]
[merged small][ocr errors][merged small][subsumed][ocr errors][ocr errors][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][subsumed][merged small][ocr errors]

12.] The formula (7) is true for all integral and fractional, positive and negative, values of n, with the exception of, n= −1; in which case the right-hand member becomes ∞, and the formula ceases to give an intelligible result: we must therefore return to the principles of definite integration, for they are exact, and by means of them obtain the correct integral.

[merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][ocr errors][ocr errors][merged small]

evaluating therefore the indeterminate fraction by the rules of Chapter V, Vol. I; and observing that n is the variable,

[merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small]

a result identical with that of Ex. 3, Art. 9; and therefore

[ocr errors][merged small][merged small][merged small]

(8)

(9)

13.] Extending the results of Art. 11 and 12 to Compound Functions, as we are authorized to do by Theorem IV, Art. 10, we have

[{ƒ (x)}"ƒ'′(x) dx = {f(x)}"+1
[{f(x)}"

(x) dx

f(x)

n+1

= log {f(x)}.

;

(10)

(11)

Hence the integral of a fraction, whose numerator is the differential of the denominator, is the Napierian logarithm of the denominator.

1

Ex. 1. f(a+bx)"dx = {} √(a+bx)"d(a+bx)

Ex. 2.

(a + bx)"+1 = b (n+1)

1

f(a+bx3)" x dx = (a + bx®)" d (a + bx")

=

[ocr errors]

(a + bx2) n+1
2b(n+1)

Ex. 3. √(aTM—xTM)" æxTM-1dx =

[merged small][merged small][merged small][ocr errors][merged small]
[merged small][subsumed][ocr errors][subsumed][subsumed][merged small]

+bx+cx2)” (b+2cx) dx

x2 dx
(a3 — x3) 3

[ocr errors]

=

[merged small][merged small][merged small][merged small][merged small][subsumed][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small]

((b + 2 cx) dx = √d (a + bx + ca2

[ocr errors]

Ex. 7.

Ex. 8.

[ocr errors][ocr errors]

Ex. 9.

a + bx + cx2

[merged small][merged small][ocr errors][merged small][subsumed][subsumed][ocr errors][merged small]
[ocr errors]

x2

x

+ x2 + ... + x*1} dæ

x2

+x"−1}

[merged small][ocr errors][merged small]

= x+ +
2

+

n

[blocks in formation]

a dx

2

a2 + x2

a dx

+

[ocr errors]

a2 + x2

[ocr errors][merged small]
[blocks in formation]

The following is a general form, which admits of being reduced to (12), when the roots of the denominator are impossible:

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

since 4ac-b2 is positive, when the roots of the denominator are

[merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small]
« PreviousContinue »