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+

+ &c.

[4] Let the equation V=0 have r equal impossible roots.

U_K,+L, (x+{c) K,+L, (x + {c)
Assume
V (20? + C#+e)" (x2 + 2x + e)k-1

K, +L,(x + }) P
XP + cac te

Q
then U2=

U-{K,+L; (v + $c)}Q,

XP + ca te
U, – {K, + Lę (x+c)}Q,
U,=

do + ca te

+

+

;

2

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the constants K, L, K2, L2, &c. may be obtained from the quantities U, UL, &c. as in the preceding case.

The resolution may in all these cases be effected by the method of indeterminate coefficients, by clearing the equation of fractions, and equating like powers of x; this method is however usually much more laborious.

(L. C. D.380~2; Hirsch, Int. Tab.)

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T

n

cos (2m 1)r cos (2m— 1) (r+1).* +

+ &c. ** — 2 cos (2m – 1)*+1 to In terms, when n is even.

n

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n

n

... +

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cos (2m - 1)r cos (2 m - 1) (r + 1)

) -2

+ &c. — 2 cos (2m- 1) = x + 1 to (n-1) terms, when n is odd.

n

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T

n

T cos 2 m (r + 1)^2 cos 2 mr

n
...+

+ &c.
T
2? — 2 cos 2m x + 1

n
to in-1 terms, when n is even.

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T

T cos 2m (r + 1) cos 2 mrn

n
... +

+'&c.
T
22 - 2 cos 2 m - x +1

n
to ] (n-1) terms, when n is odd.

1

is comThe partial fractions of which the fraction

21 +1 posed may be obtained from the preceding by putting r=0.

lastb,

100;
INTEGRATION OF IRRATIONAL FUNCTIONS.
Let R denote a rational function.
Differential coefficients of the form

[1] R{0,2*, vʻ, «, &c.}
may be rendered rational by assuming

x=yTM 98.00
[2] R{«, (a+b), (a+ba)*, &c. } ;

Assume a +bx=y".**
a + bio

a +6.00
[3] vs

&c.
az + 6,00
a + b c
Assume

=y...

a, + b,
[4] R {x, (a +bx+ca”)f};

Assume (a + b + c#?)=c(x + y).
[5] R{x, (a + bx - cx?)!};

let (a + bx-cx®)}={c(x-e) (ez - x)}",

Assume (a + bx - c«°)}=(x-e)cy. [6] R{x; (a.+bx)}, (az + bx)};

Assume a + bx=(a + bx)ye. [7] 200-1 R {v^, x, (a+bwnyi}.

Assume a +b0"=y',

m

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m

n

which renders the function rational if is a positive integer. [8] 2-1 (a + bæn). R(»).

Assume a + b&" = y', if - is a positive integer,

m

n

à x -" + b=y', if

m p

+ n

9

is a negative integer.

[9] xm-1 R {x", (a + bx" + c22n)} ; may be reduced to [5] by assuming X"=y, if - is a positive integer:

m

n

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1

(

=y98...

[10] Pen-1 R {x", (a + b* re*)], [bw* + (a + b*mon)?]}

Assume y=b0" + (a + b*ten).
q*-1 R { van, (a + bony, (a + bxa), &c. };

Assume (a + b2")=498***
201-1R 2mn

&c.
tbtoon + 67 x

a + bxon
Assume
a+b,.

(L. D. C. 215-21.)
(6.) The preceding are particular cases of the following more
general forms, in which R1, R2, &c. denote any rational functions,
and R,',R,', &c. the corresponding inverse functions.
[11] R{«, R;'(x)};

Assume R;'(x)=v. [12] R{w, R;'(x), R;'R;'(X), &c. R;R- ...R;'(x)};

Assume R'R;- ...R;"w)=v. [18] R{2, "(x), R;"()", R;"(x)]*, &c.} ;

Assume R,'(a)=vMn... [14] dR(w). {R,(w), R;'R, (x)};

Assume R,(x)=v. [15] R{x, R-R,(x)}; this formula may be rendered rational if we can determine

R; 'R,(w)=R,(w). R,'(). [16] d,R»(w).. R {Rmx, R, 'R, Rmx}; this formula

may

be rendered rational under the same condition as the preceding

[17] R{w, R;'(Q), Q, R;'>), , R;'(x), &c.} may be reduced to R {v, Q.(v), P,(v), &c.} by assuming

R;'(x)=v.

(Bromhead, Phil. Trans. 1816.)

INTEGRATION OF 09-(a + bx")”.

(7.) Let a + b = X.

m

n

If is a positive integer, for « substitute its value in terms of X obtained from the above equation.

m If

+ p is a negative integer, assume a x - * +b=2, and

n

substitute for x its value in terms of Z.

If neither of these conditions is fulfilled, the indiees of and X may be reduced by some of the following formulæ : from which however no result can be obtained if any of the coefficients become infinite.

A result may in these cases be obtained by substituting in the given formulæ for a, b, m, &c. the values which render the coefficients infinite.

CM X

pnb Simm-1XP

f.xm+=1

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m

m

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