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79. Supposing X to be always a function of x, and that it were required to integrate X x; we must write this expression under

the form ƒ'Xl" x=Xl* x—n ƒ Xx px, by the formula for integra

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x

X', we shall have by the

same formula ƒ ̃Ñ¿11 x=X' ¿a-' x—(n−1 ) ƒX1 à

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T.

ƒ X" b−2 x=X" b122 x—(1−2 ) ƒ X _3 z, &c. Therefore

ƒX ba x=X l^ »—n X1 l−−1 x+n (n−1) X" [~~2 x— n (n−1) (n—2) X-3x+&c.; an expression which depends upon the integration of algebraic quantities only, and which will have a finite number of terms when n is a positive whole number.

For example. Let Xxx, and we shall have X=

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m+1

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Consequently fax la x =.

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n (n+1) (n+2) Įn−3 x+&c.

(m + 1 ) 3

The only case which does not come within the general formula is

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80. The foregoing formula applies equally to the case in which s is negative. But as then we obtain for the fluent an infinite series, we shall explain another mode of integration.

form Xx. flux lx

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(to which it is evidently equal, because flux lr.

Suppose the quantity were

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ing flux (Xr) = X' x, flux (X' x)=X" x, flux (X" x)=XTM ¿, we shall have by continuing the process

Xx

S

-Xx

X' x

X" *

(lx)" (n−1) l-1 x (n−1) (n−2) l−x (n−1)(n−2)(n−3) lTM3 s

-&c., to a term of the form

1

(n−1) (n−2) (n−3)...2.1°

the integration of which, if possible, will give that of the formula proposed.

For example, let X=r", we shall have X'=x" (m+1), X"=

(m+1)2x", X = (m+1)3 x", &c. Therefore

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(n-2) (n-3) (n-4)

S therefore the proposed integral

xm x
L

If we make TMTM+1 u, this quantity will

become- a fluxion which has not yet been integrated.

For this

lu

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a series, except in the case of m=— -1; for then we find by the pre

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81. Let it now be required to find the fluent of the exponential formula a Xx. I observe first that aa x la = flux (a*); therefore

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a; and since, integrating by parts, fa* Xx = X fa2 —

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= *. (lx)—*—— (Lx )—* flux lx, of which the fluent is evidently

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integral fax, which at least will be the most simple transcendental integral of its kind, if it be not susceptible of an exact integration. 82. We may observe that if e is the number whose logarithm =1, we shall have

fe Xx=e X-e2 X'+e* X"—e2 XTM+e® X1—&c.

For example, let X=x", and we shall get

X'=n*, X"=n(n−1)r, Xm=n (n−1)(n-2) &B

Therefore,

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_n (n−1) (n−2) +3+&c } &c}

and consequently also

ants to (1) anh (m-1) (mm) gas +&c.}

83. To find the integral of 4*_*, as the preceding rules do

not apply, I reduce a" into a series, and obtain ==

(1+xla+1+1^+&c.) == +la+Pa+&c. Therefore

2

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84. When the preceding rules will not apply to the integration of an exponential quantity, we must reduce it to a series by the formula x2 la x3 13 a πρα

a=1+xla+

2

and it will be easily integrated.

+

+

+&c.

2.3

2.3.4

Thus let y=xTM; by the above series = {1+mx le +

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13 x

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+ m3 3 12 = +&e } = x + mx_x lx + m2x2 à la x + &c. of

2.3

2

which the fluent may be found by that of alx (79), and we obtain

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which in the particular case of x=1, becomes the converging series

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These examples are sufficient to enable us to integrate such sorts of quantities by means of series.

Examples on Logarithmic and Exponential Fluxions.

1. Required the fluent of x

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ON THE INTEGRATION OF FLUXIONS CONTAINING SINES, COSINES, &c.

85. Since x cos x=flux sin x, and - sin x flux cos x, it is evident that st cos x= sin r, and ƒr sin x=-cos r ; as also that

Jy cos ny

n

sin z.

= === Sny cos ny =

n

1

n

sin ny, and similarly that fy sin ny

cos ny. Again it is clear that fz cos z (sin z)"=ƒ (sin x)" flux

1

(sin x)+1, and that f (x sin x) cos" z=

n+1

(cos 2)+1 n+1 Similarly if we desire to integrate y sin y cos ay, we must make sin y cos ay sin (a+1) y-sin (a-1)y, and the integral

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x

86. A similar process will serve for x sin x sin ax, or x cos x cos ax, &c. With equal facility we may integrate r sin r sin ax cos ba, &c. if we first reduce these products to simple sines or cosines, by means of the values of sin a cos b, sin a sin b, &c. given in the article trigonometry. The same treatment will serve to integrate x sin x, x sin 3x, x cos x, &c.; but it is more simple to integrate them in the following manner.

87. The formula i sin" x=x sin x. sin"-1. Consequently, integrating by parts, så sin" x= sin^-1 x ƒx sin x-ƒ{ flux (sinTM1 x). få sin x} ——cos x sin11 x + (n−1) fr sin22 * cos2 x = COS X sin21 x+(n−1) ƒ'r sin *2 x—(n−1 )ƒx sın" ; and transposing,

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process we have fi sin"-2 =

1

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