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I. F. Quinby Horatio Nelson Robinson.

# A new treatise on the elements of the differential and integral calculus

. (page 17 of 22)

b â€” a 2

\/6 â€” a tan. - + \/b + a

by Ex. 18.

26.

f-

J a

dx

V 6 â€” a tan. ~ â€” V6 + a

dx

4-6sin.a;" I ^, . iÂ» a;

a -j- 26 sin. - COS. -
2 ^2

c?a?

cc cc\ . X X

a ( sin.^ - + cos.^ - j + 26 sin. - cos. -

1

sec.2 - dx

A

a( 1 + tan.2 ^ j + 26 tan. ?

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S3G INTEGRAL CALCULUS.

a I a- â€” 6^ . / 03 . ft\*

â– when a"^ b; but, if a < 6, then

Ja + ism ol/ a; . bV V^^''

_ 1 J 2^a g

J atan.^ + i-V^-^ â€” a*

= VF^P ^ ^ â€¢

atan.- + 6+^6-i_aÂ«

j?0)?Â« Rationalization and integration of irrational functions.

Examples of integration by substitution have already been
given : we now proceed to show under what condition^ cer-
tain irrational differential expressions may, by proper substi-
tutions, be rendered rational, and integrated by the methods

previously investigated.

p
Let ns assume the form x"*(a + ix**)/ dx, in which m, n, p,

VLnd q are entire or fractional, positive or negative.

â€” Iâ€”)

dx = Â± â€” -.(Â»Â«â€” -d) Â» dz:
fib*

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irratiojsal functions. 337

whence

Now, if â€” -^â€” ia an integer, the binomial (2* â€” a) Â»

is rational in form, and may be expanded by th^ Binomial
Formula into a finite number of terms M^hen the exponent

â€” ^t_^ â€” 1 is positive. Each term of the expansion, being

multiplied by a^+Â«'^dz, will give rise to a series of monomial
differentials which can be immediately integrated.
We may also write

x'^(a + bx**y(ix=J X '^ ^ {b+<xx-'')9dx;

and, by comparing this with the first case, we conclude that
the substitution of 2^ for 6 + a^""" will reduce

/ np p

x'^^-iib + ax-"")-^ dx

to an expression that is immediately integrable when

m + 1 __p
n q

is a positive integer ; i. e., when 4- - is a negative

integer.

Hence Jx^{a-\-bx^)idx may bo rationalized and inte-

vn â€”1â€” 1

grated when is a positive integer by substituting 2* for

n

TYt â€” jâ€” 1 W

aA-hx*: and, when V - is a negative integer, by substi-

' ' ' n q

tuting 2Â« for h + ax~^.
It will be shown in a subsequent section (2) that the inte-

vn I "I

grals may also be found when is a negative integer in

43

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338 INTEGRAL CALCULUS.

the first case, and when [- - is a positive integer in

n q

the second.

Ex. 1. fx(a -\- bx) . Here tw = 1, n = 1, - = ^, and

wi + 1 Â« ...

= 2, a positive integer.

n

Put a + 6x=:22. . ,^ - - â€” ^ â€” /Â£aj=â€” T-J

o

Jx{a + bx)^dx = ~/(2' - a)z'dz
2 r

___^/2^ gg^ _ 2(a + 6a;)t /g + hx a\

Ex. 2. / T â€¢ In this example,

(g2 -|- x^y

m = 3, n = 2, ^ = â€” -, and â€” -^â€” = 2.

J 2' n

Put aÂ« + a?^ = 2'*: .'. x = (^^ -. a^)*, and do: = â€” :

r 1 = f{z^ â€” a'^)dz =-~ â€” a^z

J^a^ + x'-y ^ 3

/ 2 I oxi a?^ â€” 2g'
i=(g2 + a;-) ^

Ex. 3. / â€” t- In this case, m = 2, n = 2, ^ = â€” -,

and â€” â€” U P z=i â€” 1, a negative integer.

n q

(o^ + tcy

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IRRATIONAL FUNCTIONS, 339

Let 1 -f aÂ«a?-^ = 2^ â€¢ r.x = â€” j;

dx:

azdz

(zÂ«-i)*

r x^dx 1 pdz ^ 1 1

3a'(a* + a!*)*

Ex. 4. r â€” â€” i- Here m = -2, n = 2,^z=-J.

Put 1 + a;"* = 2* : .â€¢.a;= -., dx=: â€”

Functions in which the only irrational parts are monomials
can always be rationalized and integrated. Thus, suppose it
is required to find

n(l + x^ â€” x^)dx

Put x = i*', .'. dx=.W'dt; and we have
+

/.(l + a;* - x^)dx _ [- {l-lrt* -t*)U^dt

= - :5 *' + i^' + '* -*^'' + 2<Â» - 6< + 6 tan.-'<,
4 7 o

which becomes the integral in terms of x by replacing t by a?*.

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340 INTEGRAL CALCULUS.

The rule to be observed for rationalizing Biiich expressioiw is
to substitute for the quantity under the radical sign a new
variable affected with the least common multiple of the indices
of the radicals for its exponent.

Fractions in which the only radicals are the roots of the
same binomial of the first degree may be reduced to the case
just treated.

For example, required

Assume (ix-{-h-

{ax + by = t\ (ax + bf = t\
By these substitutions, the expression to be integrated be-
comes the rational fraction

Q^t^^{t'^by + aH'\dt
a^ t' â€” b + al^ *

The general method of integrating rational fractions will be
investigated in the next section.

a: + {ax

+&)*

I â€¢ u X â€”

~ a '

dx =

GPdt
a

dx . , 3 + 2a?

Vl â€” 3x - x' y/li

EXAMPLES.

1 r ^-

^- J Vl â€” 3x

2. ix^'lxdx = ; (ix V

â€¢^ n-|-l\ n + 1/

3. J 6 sin. Odd = sin. d â€” cos. d.

4. f -'^^â€” =tan.'-^6^

^ r NO 7 3 ^ . sin. 2x

5. / (1 â€” COS. xfax = -X â€” 2 sm. x -| â€” â€¢

â€¢/ J 4

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6.

MISCELLANEOUS EXAMPLES. Ml

x'^dx 1 a' -fa;'

/x^dx 1 ,

aÂ« â€” '/;Â« ~ fi^

/â€¢I + COS. a? _ â€ž . ^

J X -\- Bin. ^ ^ ' ^

/â€¢a; + sin. a; a;

8. Ivâ€” i aa!? = a;tan, -â€¢

J 1 4- COS. a? 2

9. J?(^ tfo; = ia; Z(Za;) - Ix.

10. J e **Â«iB. laa; cos. nxdx

_e^ a Bm.(m + ^)^ â€” (m + n)co8.(m + n)x

c** a sin. (m â€” n)a; â€” (w â€” n) cos. (m â€” n)a;

"'"T a^ + (w â€” w)'

Having found the indefinite integral, the definite integral

between assigned limits, except in special cases, can be at

once detennrned.

,^ r*" , wa^

11- J V a'^ â€” a;^ dc = -T-?

for j \/a^^x^dx = ^ + 2" s^^- - = V'(^),

and t/;(a) = â€” , t/;(0) = 0: .-. t/;(a) â€” 1/;(0) = â€” .

:7ra.

/â€¢-Â« ,x ^

12. / ver."^-aa; = j

â€¢'o a

By making a; = a(l â€” cos. d)j we find

J ver."^ -dx=^ j ad sin. ^c?^

= a sin. â€” ad cos. ^.

The limits 7t and for the transformed integral correspond
to the limits 2a and for the given integral.

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342 INTEGRAL CALCULUS.

n sin.^a^c^g; __ ja + 6x .^ Va tan. a? _ g^

le. /.VÂ«-+H.^=("-^-|.)(Â« + Â»Â«')'-

18. f ia + bx'')^dx ^ 4 (â€ž, ;.^n)i

In effecting this integration, transform by assuming

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SECTION II.

INTEGRATION OP RAtlONAL FRACTIONS BT DECOMPOSITION
INTO PARTIAL FRACTIONS.

203. A RATIONAL fraction is of the form

A' -{â€¢B'x-\- G'x^'\ \'N'x'''

in which the numerator and denominator are entire and alge-
braic functions of x; the co-efBcients A, B ...jA',B'...j being
constants.

Denote the numerator of such a fraction by F{x), and the
denominator by f(x). If the degree, with respect to x, of
F{x), is not less than that of/(a;), we may divide F{x) hy/{x)
until we arrive at a remainder of a degree inferior to that of
/(x). Let (p{x) be this remainder, and Q the quotient; then

As jQdx can always be found, the integration of the origi-

CD ( x\dx
nal fraction is reduced to the integration of â€” ~-tâ€” i in which

the degree of q:{x) is lower than that of/{x). The integra-

_ op (X)dx . rr ^ ^ , . . . - n

tion of â€”^ â€” r- 1^ elfected by resolving it into a series of more

simple fractions, called partial fractions ; and we will now
demonstrate the possibility of such resolution in all cases in

843

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344 INTEGRAL CALCULUS.

which y(iB) can bo separated into its factors of tlie first degree
in respect to x,

204:. Suppose the fraction , , to be in its lowest terms,

and that the degree of F{x) is inferior to that oÂ£/{x). If the
factor x-' a enter&/(x) p times, we shall have

/{x) = (x^ay(p{x),

(p{x) denoting the product of the other factors o{/{x) : whence

/{x) {x â€” ay(f{x) {x â€” ayq>{x) "^ (a? â€” a)''

But F{x) ~ q){x) =1 when x=za, and is therefore divisi-

9(a)

ble by x â€” a. Let y^Jiix) be the quotient, then

F{x)_ yp,{x) F{a) 1

fix) {x â€” ay-^ cf{x) "^ (p{a) {x â€” ay'

^ . F(a) ^ , ^

Denoting â€” -â€” by -4,, we nave
cp{a)

F{x)_ xp,{x) J, .

/{x) â€”{x^ay-^ qix) "^ (a; - ay '

that is, â€” ^ has been resolved into two parts, one of which is

f{x)

^ 5Â« In like manner, \p~ ^i / < ^^Y ^^ reduced to

{x â€” ay ' {X'-ay^ (f(x)

t?^i(a:) __ V^i i^) , _ A

{x â€” ay-'^(p{x) " {x â€” ay-\{x) "^ {x - ay-^'

and so on, until at last we should have

(x â€” a)q){x) (f{x) '^ x â€” a'

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RA TIONAL FRA CTIOICS. S 1 5

By the successive substitution of these values of

7 xo 1 / N> 7 xÂ«-'2 / >. '"? 1^ "t*^Â® order the inverse

of that in which they were deduced, we find

F{x)^ yp^{x) A^ A^ Aj,

/{xj (p{x) '^ (x â€” a)^'^ (x^uy-^'^ ^x-a

llf ( x^
Proceeding in the same way with the fraction ^ , the

F(x)
decomposition of into partial fractions will be at last

completely efi*ected.

20S* If the root a is imaginary, and equal to a + jS V â€” 1,
then ai = a â€” j^V^lis also a root of the equation /(x) = 0.
Suppose that all the partial fractions corresponding to the real
roots have been determined, and that there remains for resolu-

tion into such fractions the fraction , in which /i (x) =

giv^is rise to imaginary roots alone. Suppose, further, that the
pair of roots, x =:a-\- ^ V â€” 1, x = a â€” ^ V â€” 1, enters this
equation q times. Denote the factor of /i(x) that gives the
remaining imaginary roots by ipi{x) ; and, to abridge, make
a = a + p\/ â€” 1, ai=:a â€” i^V â€” 1: then

/,{x) {x â€” ay{x-^a^y(p,{x) (x â€” ay{x-a,y

{x â€” ay-^x â€” aiy ^ (x â€” a)Â«'(a: â€” ai)Â«

44

(1)-

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346 INTEGRAL CALCULUS.

In like manner,

X â€¢

a? â€” g (at â€” a)<Pi(ai)

The last term in the second member of Eq. 2 may be written

(ai â€” a)(a; â€” a)Â«-^(a; â€” a)^

But {a, - a) = - 2i3V- 1, and ^1^ is derived from ^^^^

9i(Â«i) g)i(Â«)

by changing the sign of \/â€” 1: hence, if -i-LJ = ^ -|- JBy^â€” l

(jPi(a)

then ^i^^ = ^ - B^^T, and

9l(Â«l) ^l{^)

Therefore

F,{a,) F,{a) B

g'llM _^i(Â«) ^

(ai â€” a){x ~ a)Â«-i(a; â€” ai)* {x â€” a)Â«-\a; â€” a,y^'
and the sum of the two partial fractions in the second mem-
bers of Eqs. 1 and 2 will become

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RATIONAL FRACTIONS. 347

B

A + Bs/-l , ^

-__f

"-(a;2-2cca; + a2 4-/^2)?

which is rational. It is also seen, that the numerator and de-
nominator of the first term in the second member of Eq. 2 is
divisible hj x â€” a. Dividing, and denoting the numerator of
the result by ^i(a;), this term may be put under the form

% (x^

râ€”r, ; . ^\ I ..â€ž,^ , â€” 7â€” T-: hence, by substitution in Eq. 1,

(oj^ â€” 2ca; + a^ + p/^)^-^qPi(a?) ' -^ ^ '

we should have B

Now, X^{x) is a rational and entire function of x, and the frac-
tion Y-,, -J r ^2 I J2\a-i â€” 7-T ^^y ^Â® treated as was

â€” iâ€” , and so on ; our result with respect to the assumed pair

/i(^)

of imaginary roots being of the form

I\{x)_X_,(x) M,x + N,

Mx)-cp,{x)-^{x'^2ax + a' + ti'y'^'"

^ __1^75J!A_

x^^2ax + a' + ^'^
The possibility is thus demonstrated of resolving a fraction,
the terms of which are rational functions of a single variable,
into a series of rational partial fractions whenever the denom-
inator of the given fraction can be separated into factors,

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34B INTEGRAL CALCULUS,

whether real or imaginary, of the first degree with respect to
the variable. The investigation also shows the form of the
partial fractions answering to the different kinds of factors of
the denominator of the given fraction.

Fix)
Thus if /(a:), the denominator of - ^/ / , contains the factors

â€¢^ ^ /(x)

x â€” a, (a; â€” 6)"Â», (a;â€” c)*, (a; â€” a â€” |5V^^)^

then

F{x) ^ _J_ B, B, , . . , -gm

/{x) a; â€” a "^ (x â€” 6)'Â« "^ (a; â€” 6)"*-^ "^ ^ x â€” b

i"r /_ ^\Â»-i "t- â€¢â€¢â€¢ -p

(x â€” cY (aj â€” c)**"' a; â€” c

, 3fia;+2V; . M^x + N^

â– +â–  (a;2 â€” 2aaj + a^ + j^^^p + (a^^ _ 2aa; + a^ + ^2)'-*

*^ "^ aj - * â€” 2aa; + a^ ^ ^'-^ ^ a;^ - 2^^ + r' + ^^'

The labor of determining the constants A^ A2.'.Bi...Mi,
-NJ..., for the partial fractions, which, by following the method
above indicated, would be very great in miany cases, may be
diminished by expedients whiph we will now investigate.
The most obvious of these is based on the consideration, that,
when the partial fractions are reduced to a common denomina-
tor, the numerator of the result is identically equal to the nu-
merator of the given fraction.

200* To determine the pwrticd fraction corresponding io the
single real/actor, x â€” a, off{x).

Assume -â€”-^ = â€” 7-7 (1),

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RATIONAL FRACTIONS, 345

in which -^ is a constant, and ^-7â€”/ is the sum of the partial

fractions answering to tlie remaining factors of /{x), and
/(x)=(a; â€” a)qp(x).
From (1) we have

F{x) = Ac,.{x) + (_x,- a^x) (2),

Jin identical equation. Make a? = a, and then

We also have the identical equation

/(x) = (x-a)g)(a;);
whence, hy differentiation,

f{x) = cp{x) + {x^a)cp^(x),
an equation also identical ; therefore, making x = a,

207. To determine the partial fradiana corresponding to the
real/actor (x â€” a) repecded n times,
"We now assume

f{x) (2: â€” a)** "^ (a; â€” a)*- 1 "^ a; â€” a "^ <ji(a;) ^ ^'

â€”;â€” f denoting the Bum of the fractions to which the other

cf{x) ^

factors oi f(x) give rise.

Multiply both members of (1) by (a? â€” a)**, and we have the
identical equation,

^J^=A, + A,{x-a)^ ... +A,{x-aY-'+'^^^^{x-ar,

observing that/(x) = <]f'(a;)(a; â€” a)". Denoting the first mem-

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I

350 INTEGRAL CALCULUS.

ber of this equation by x{^)j *"^ then, in it and its successive
diflferential equations, making x = a, we have

X{a)=A,, t'{a) = A,, ;f-(a) = 1.2^,...,
;c(Â»-^)(a) = 1.2...(7i-l)Jâ€ž,
and thus the numerators of the partial fractions are determined.
208. To find the partial fraction corresponding to a single
pair of imaginary factors.

Suppose ic â€” a â€” j3Vâ€” 1, a: â€” a + ^s/ â€” 1, to be the ima-
ginary factors oi fix). We then put *

F{x) _ Mx + N xp{x)^

f{x) "" x'^ â€” 2ax + a'^ + ii'^ "^ (p{x) '
whence F{x) = cp{x){Mx + N) + v^{x){x^ â€” 2aa; + a^ + P\
an identical equation.

Make a; = a + j3 s/ â€” 1; then

jr(cc + ^V^n) = (3p(a+i3V=^) \M{a + ^s/'=ri) ^ n^,
or, by making a; = a â€” p'Vâ€” 1,

i^(a - pV^^l) = g)(a - ^ V=~l) I il/(a - ^ V^=^) + iV^j .
These last equations may be written

J _ bV^^ = ( C' - 2? V=^) I M{a - jS V=l) + N\ ,
in which A, B^ C, and D are known functions of a and |S.
From either of these equations, the values of M and N may
be found by equating the real part of one member with the
real part of the other, and the imaginary part of one member
with the imaginary part of the other.

The values of M and N may also be found by the method
of Art. 206. Thus, for brevity, denote the imaginary factors
by a; â€” a, a? â€” a^ ; then the partial fractions are
F{a ) 1 Jf^(aO 1
f {a) X â€” a' f (P'\) ^ â€”cti

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RATIONAL FRACTIONS. 351

If ^j^^ = ^ + ^V^, then ^J = ^ - i?V^=^, since

- is derived from by changing the sign of Vâ€” 1:

hence, replacing a and a^ by their values

the fractions become

the sum of which is

2 A{x â€” cc) + 2B^
ic'i â€” 2aa; + a2 + /^'^*

j209, lb ^nrf tlie partial fractions corresponding to a pair
of imaginary factors which enters the denominator of the given
fraction several times.

Let cc â€” a â€” jSVâ€” 1, x - a -|- j3Vâ€” 1, be the imaginary fac-
tors, and, to abridge, put a = a + jSV â€” 1, aj r= a â€” jSVâ€” 1;
then, putting/(a:) = \{x â€” a){x â€” aj) j^g5(a;),
F{x) M,x + N, M,x + N,

; +

/(x) \{x^a){x-aO\'^ !(^-a)(^-ai)i*"'^

â– ^ ^{x-a){x-a,)\ '^ (p{xy

'w( x')
representing by ~/-r the sum of the partial fractions to

which the remaining factors of f{x) give rise. Multiply the
first member of this equation by/(x), and the second member
by its equal j {x â€” a){x â€” aj) j ^ <p(^); and we have
F{x) = {M,x + iVi)gp(x) + (1/20; + N,){x - a){x - a,)q){x)

+ {M,x + N,)\ix^a){x^a,)W(x)+...

+ \(x^a){x-^a,)\^cp{x) (1).

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352 INTEGRAL CALCULUS,

Now, whether we make a: = a, or a; = a^, all the terms in
the second member of this equation, except the first term,
vanish. Suppose a? = a, then

F{a) = iM,a + N,)cp{a)',

and if the real parts in the two members of this last be
equated, and also the imaginary parts, we shall have two
equations from which to find the values of Mi and Ni. Sub-
stitute these values in (1), transpose {MiX -{- Ni)q){x), and
divide through by [x â€” a){x â€” aj), denoting the first member
of the resulting equation by Fi (x) ; then

Fi{x) = (M^x + N^2)q>{x) + {M^x + N^Xx â€” a){x â€” ai)q>(x)'\

+ \{^-ct){x-ai)\^-'^{x) (2).

Proceeding with (2) as we did with (1), the values of M^ and
N2 may be found ; and, by repeating these operations, all of the
constants, M^ Ni, M2, -A/^..., will finally be determined.

210. The rational fraction, which may be decomposed into
partial fractions by the foregoing methods, being a differential
co-efficient, the resulting fractions are also differential co-effi-
cients ; and the sum of their integrals will be the integi-al of
the given fraction.

The differentials corresponding to these partial firactions are

of the form

{x â€” a)"*' {x^ -{-p^ + ^y^

1 A

The integral of the former is =- 7â€” x^â€” i, which

Â° m â€” 1 (05 â€” a)"*~*'

becomes Al{x â€” a) when m = 1 ; and that of the latter, when
n = 1, has been explained in Ex. 20, Art. 201. The integra-
tion of the second form, if n be greater than naity, is reserved
for the next section.

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nATIONAL FRACTION'S. 353

EXAMPZES.

1. H^^^ ^ - H = ~-^y^ . The factors of the denominator are

a;^ â€” a? â€” 2 /(x)

a; 4- Ij X â€” 2. We therefore put

3 â€” 2a? A , J,

a; 2 â€” a; â€” 2 "" a; + 1 ^ a? - 2

Substituting â€” 1 and + 2 successively for x in

F{x)_S ^ix
/(a:)"~2a?â€” l'

we get J = ^-, J^=z^ ^:

(3 â€” 2x)(?a;

5 dx

1 <;a;

a,-' _ a; - 2 -

&X+1

3 a; -2'

o r (â€¢Â»' â€” 3^ â€” 2)rfa?

"* J(a;'^+a;+lf(a; + l)'

In the denominator of this, the pair of imaginary factorsi,
a; -(- - â€” - Vâ€” 3, X -\ - ^-\ - ^\^â€” 3y enters twice ; and the real
factor x-\-l once. We put

x^ â€” Bx â€” 2 __ Mix + N^ M^x + Ni , jp(x) ^

(^â€¢i -f a? + 1 )Xx +l)"'{a;2-|-a;4-l)2^a;-^ + a;+l^a;+l'

a;^-3:i;-2 = (Jl^io; + J^i)(^ + 1)
+ (M,x + N,Xx' + X+ l){x + l) + (^x' + x+ lyxi^ix) (1).

Give to X one of the values which reduce a;* + ^ + 1 ^^ ^^^" j
then, for this value, (1) becomes

45

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354 INTEGRAL CALCULUS.

Prom X' -\- x-\-\=iQjyfiQ have a;* = â€” as â€” 1. Substituting
in (2),
__ 4a. â€” 3 = 3I^x^ + ilfiic + N^x + iVi

= ilf 1 ( - a; â€” 1 ) + Jlf 1 x + JVi a; + JVi
= -^M^-\-N^x + N^:
whence iifi â€” ^i = 3, -^ = â€” 4, 3/^ = â€” 1.

In (1), replacing Mi and i\^ by the values thus found, and
transposing, we have

a;' â€” 3a; â€” 2 + (a; + 4)(a; + 1) = 2{x^ + Â«?+!)
= (ilf,a; + N,)^x^ + x+ l)(x + 1) + (a;Â» + a: + l)Xx) (3).
Dividing through hj x^ -{- x -{- \j and in the result making
a:^ + a; + 1 = 0, we get

2 = {M,x + N,){x + \):
whence Ifj = â€” 2, -^ = 0.

The partial fraction corresponding to â–  j, ^^ may be found

by the method explained in Art. 206 ; or thus : After dividing
(3) by a;^ + a; -f- 1, replace M^ and N^ by their values, trans-
pose, and again divide by a;^+a;+l. We find i/;(a;) = 2:

*i. ^ r (^' â€” 3a;â€” 2)rfa; r ix + 4:)dx

''"''''" J (x' + a;+l)H^ + l) = "â€¢' ^x^ + x+lf

/ 2xdx r 2dx

x' + x + l'^J ^^+1'

J x^ â€” 5a?2 + 3a; + 9 '
By the method of equal roots, we readily discover that the
denominator may be resolved into the factors (a; â€” 3)^, a; -f- 1 :
hence wo put

9a;2 4-9a; â€” 128 _ A^ A^ , Bi

x^ - 5x^ 4- 3a; + 9 "~ (a; â€” 3)' ^ x â€” 3 ^ a; + 1 '
whence
9x^ + dX'-12S = Ai(x + l) + A^(x^S){x+l) + Bi(x'-'3y;

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RATIONAL FRACTIONS. 355

from which, by making a? = 3 and a? = â€” 1 successively, we
get -^1 = â€” 5, -Bi = â€” 8. If the second member were de-
veloped, the co-eflScient of x'^ would be A2-\- Bi] equating
this with the co-eflScient of x* in the first member, we have
w42 + -Bi = 9: .â€¢.-42 = 17; and therefore

r (9a;^ + 9a; ~ 128)c?ag _ __ r 5dx r lldx _ r Sdx

J x'-5x'' + Sx + ^ " ./ (x â€” 3)2 +J ^"^1^3 J 5^

5

a? â€” 3

+ ni{x - 3)-^8l{x+l).

211* Integration of â€” r- when m and n are positive

X â€” A

integers.

If n be an even number, the real roots of a;** â€” 1 = are

+ 1 and â€” 1 J and the imaginary roots (Art. 77) are given by

., . 2k7Z . / â€” ^ . 2Jc7t , . . . 7 .
the expression cos. ih v â€” 1 sm. â€” , by giving to k m

succession the values 1, 2, 3..., - â€” 1.

We will denote the arc - by 6^ â€¢ ^ ^ being the fraction to

n f{x)

to be resolved into partial fractions. It has |)een shown

(Art. 206), that, if a be a root of /(a;) = Q, the corresponding

F(a) 1 x^~^

partial fraction is â– ) [ : hence, for the fraction . ,

^ / (Â«) ^ â€” CL ' a;'* â€” 1'

the partial fraction for the root -|- 1 of the equation a;** â€” 1 =

is r=: z= â€”; and, for the root â€” 1, the partial

na"-^ na'* 7i(a;â€” 1)' ' * ^

fraction is -^ â€” . . The pair of imaginary roots,

COS. 2k0 db V^-if sin. 2kd,
give the partial fractions

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356 INTEGRAL CALCULUS.

{co^.^hd 4- V^ B i n. "Ocdy
n(x â€” COS. 2Jcd â€” V^^sin. ^kd)

I ( COS. 2hd â€” V^n^ sin. 2fe<?)'"
that is, (Art. 73), ^(a.-co8.2fe? + V:^sin.2fe?)5

COS. 2mA;<? + V^^ sin. 277l^-<^
n(a; â€” cos. 2i^ â€” \/^ sin. 2Jk^)

COS. 27nfc^ â€” Vâ€” 1 sin. 27/ifc^

?i(a; â€” COS. 2A;^ + V â€” 1 sin. 2kd)

__ 2 COS. 2mkd(x â€” cob. 2A;g) â€” 2 sin. 2mk0 sin. 2Ji ;<^^

n(a?2 â€” 2a; COS. 2fcd + 1) '

and for each pair of imaginary roots, that is, for each of the

Til

vahies 1, 2, 3..., - â€” 1 of A:, there will be a partial fraction of

the form of this. Let the symbol -S denote the sum of these ;
then

J x^â€”l '-J n{x - 1) +J 7

n(^x+l)

,2 n co^,2mkd{x â€” cos. 2Tcd) â€” sin. 2ywÂ£gsin. 2Tcd
nJ (x â€” cos. 2kdy + sin.2 2kd "

1 /â€” 1^"Â»

z=z_Z(a;-.l) + ^ â€” tJ-l(x+l)

-f - ^ COS. 2mkdl{x^- â€” 2a; COS. 2Jfc^ 4-1)

^v â€¢ o 7/14. ^liC â€” COS. 2&^

-S sm. 2mkd tan. * ; â€” ^ ,

n sm. 2kd

by observing that the last term under the sign of integration

can be separated into the two fractions

2 cos. 2mkO(x â€” cos. 2kd) 2 sin. 2mk0 sin. 2kd

n x' â€” 2a; cos. 2&^ + l ^^ " n {x â€” cos. 2/c^?)^ -f- sin.^ 2)fcf?'

5i^. Integration of â€” ^ =â€” , m and n being positive inte-

gers, and n an odd number.

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EXAMPLES. 357

In this case, aj" â€” 1 = has but one real root, + 1 ; and the

imaginary roots are the vahies assumed by the expression

ikn , / â€” - . Zhft .... 7 . -XT,

cos. - â€” db V â€” 1 sm. , by givmg to Ic in succession the

w â€” 1
values 1, 2, 3..., â€” ^ â€” (Art. 77). Hence, by operating as in

the preceding article, we find

//jÂ»Â»Â» â€” ^ ddi 1 1

'^^TZTi =^ - ^(^ â€” 1) + - -^ cos. 2mkdl{x^ â€” 2a? cos. 2kd + \)

2v â€¢ o In J, _i a? â€” COS. 2Xj^

1, sm. zmkd tan. ^ ; â€” â€¢

n ^ sin. 2kd

cc^ â€” ^ dx
213. Integration of ^ Â» m and n being entire and

X â€” |â€” 1

positive, and n even.

Under the supposition, none of the roots of a;* + 1 = are
real ; and the imaginary roots are found by giving to Jc, in the

expression cos. ' â€” tt dt V â€” 1 sm. â€” -^â€”- n, the values

0, 1, 2. . ., - â€” 1 in succession. Put ^ for - , then the partial

fractions corresponding to a pair of these roots will be

cos.(2fe+l)/? + \/:=Tsin.(2ifc + l)(? â–
X â€” cos.(2A; + 1)<? â€” V^^rsin.(2i: +1)Q

, COS. (2fc -f 1)0 ~ V^ sin. {2k + l)d
X â€” COS. {2k + 1)^ + V^^Tsin. (2A; + 1)^'
the sum of which is

2 COS. m(2^-+ 1)\$ j a: â€” COS. (2^â™¦-|- l)tf | â€” sin. m(2h -f l)tf fin. (2k +1)6
"~w j r â€” COS. (2X;+1)^] '+^!Â«~(2il;+ 1)^

Hence

/x^â€”^dx 1 ,

^Â»^1 =â€” -^'coQ.ffl(2fe + l)<?Z{a?^.~2xcas.(2A; + l)<? + l}

I 2 V â€¢ /o7 r i\/ix _i a?â€” cos.(2A; + 1)^
+ -2: sin.m( 2a; + 1 )<? tan. ^ ,- -â€”> Lâ€” i- .

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358 INTEGRAL CALCULUS.

In like manner, we integrate â€” ^ri~r' ^^^^ ^ ^-^ ^^^; t>y

OC â€”Jâ€” A

finding the ptirtial fractions corresponding to tho roots cf
a;* + 1 = 0. Ill tliis case, there is one real root, â€” 1 ; and t!io
other roots, which are imaginary, are the values assumed by
the expression

cos. (2k + 1) -dzx/'^^sin. (2k + 1) -,

by giving to k the values 1.2.3... â€” - â€” successively.
We should find

â€” -^cos.m{2k+l)d\x*â€”2xcoQ.{2k+l)0+l\

â€¢ 2^ . ,o7 I 1^.i* ^lO? â€” cos.(2i+l)^.

+ - Ssm.m(2AJ + Ij^tan. ^ ; â€” /r>7 . -l^^

^n V -T- / sm.(2&-{-l)^

EXAMPLES.

2, r_^^=JLtan.-i-4.â€” Z ?-Â±^.

J a^ â€” X* 2a* * a 4a' a â€” a;'

â€ž /â€¢ a;2^aj 1,05 â€” 11 ^jO?

, /â€¢ <?a; _ 1 fj x + b b _^x\

*â€¢ J (x^ + a-'){x +b)-b-' + a' [' ^xr+T* "^ a a) '

+ 1 V3 1 tan.-Â»(2a; â€” ^3) â€” tan -\2a; + V3) | .

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EXAMPLES. 359

dx

7. f 1 â€¢ This may be rationalized by putting

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