I. F. Quinby Horatio Nelson Robinson.

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

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Online LibraryI. F. Quinby Horatio Nelson RobinsonA new treatise on the elements of the differential and integral calculus → online text (page 18 of 22)
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*'(i-x»)*

^3^3.

whence aa;=: 7, (1 — cc') == ^.

(«' + If (1 + 8»)*

= -Z(2 + 1) — - Zfz^ — a + 1) — tan.-^ ^

^r

in which, if we replace z by its value 1 we have the

required integral in terms of x.

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SECTION III.

FOEMULJS FOB THE INTEGRATION OP BINOMIAL DIFFERENTIALS
BY SUCCESSIVE REDUCTION.

214:. The integration of diflerentials of the form

may be made to depend on that of other expressions of the
same form, in which the exponent of the Tariabl© without the
parenthesis, or the exponent cf the parenthesis itself, is Ipss
than in the original expression. This is accomplished by the
method of integration by parts. '\7e have

Jx'^i^a + Ix'^ydx =ifx"'-''-^\a + bx^^yx^'-'^dx =Judv
^ T m «-i.i ia + 'bx''y+^

and therefore

fx^(a + hx''ydx = x^^-"+' ^^+-^)^^
•^ ^ ' ^ 726(^9 -(-1)

The integration of a7'"(a + hx")^ is thus brought to that of
^m-n^^_|_ j^n^;>H-i^ whicli last is morc simple than the first
when m is positive, and greater than n, and when^ is negative;
for then the numerical value of p + 1 is less than^.

But wo may find a formula in which the exponent of the
variable without the parenthesis is diminished, while that of
the parenthesis itself is unchanged.

Thus we have the identical equation

x*^-»(a + bx''y-^^ =: a;'"-"(a + 6:c")^(a + Ja?")

= ax'^-^a + bx^y + bx"'{a + fta?")**;

860

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REDUCTION FORMULA. 361

therefore

+ bfx"'{a + bx''ydx.
Substituting this value in Eq. 1, we have

whence, by transposition and reduction,
jx (a + to ) ^- b{m + nj>+l)

Tlie integration of x"^{a + hx^ydx is then made to depend
on that of x^~^{a + hx^ydx; and, by another application of
the formula, the integration of this last reduces to that of
x'^-'^ia + hx^ydxj and so on: hence, if m is positive, and

greater than n, and i denote the entire part of the quotient -,

the integral to be determined after a number, t, of reductions,
would be

Jx'^-^ia + bx'^ydx.

If m — in = 71 — 1, this expression is immediately integra-
ble; for

Sx^-\a + hx^ydx= ^2^p+l) ^
but m — in = 71 — 1 leads to — ^^ z= i + 1, and the condition
of integrability (Art.* 202) is then satisfied.

46

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362 INTEGRAL CALCULUS.

Formula A cannot be applied when m -j- ??/7 -f- 1 = ^ ? ^r
then its second member takes the form oa> — oo : but iu this

*n 1 1

case — ' \-p is equal to zero, that is, an entire number: and

the original expression is therefore immediateJj rntegrable.

21S* Formula for the reduction of the exponent of the
parenthesis.

Assume

x'^ia + bx^ydx = (a + bx^yd - - ^ = udv:
then, integrating by parts,

-^Jy/a;-+"(a + &c")^-^da: (1).

In this formula, the exponent of the binomial has been
diminished by 1, while that of x without the parenthesis has
been increased n units. We may, however, diminish the for-
mer without increasing the latter exponent. In formula A,
last article, change m into m + n, and p intoj? — ^ •* ^Q thus
hare

\x'^-^Ho,-\-bx''y-^dx^ - — J^— i — — -^

and this value of Jx"''^" (a -f- hx'')^-^ dx, substituted in Eq. 1,
giveS; after reduction,

/ x^'ia + hx'^ydx = ^-^L^ J^

^ ^ ' ' np 4- m -f- 1

-I j^^^-P— ./a?^4a + 6x")''-^da? (B).

np -\-m -\-\'^ ^ ^ ' ^ ^

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REDUCTION FORMULA, 363

By the repeated application of tbi» ibrmnla, the exponent p
will be diminished by all the units it contains. This formula
will not admit of application when «p -f- ^ + 1 = ^ > ^^^ ih^n
the integral Jx^{a-\'hx^Ydx can be found at once (Art. 202).
By means of formulae A and B, the integral Jx^{a + hx^ydx^
whea m and p are positive, may be made to depend on the
more simple integral Jx^~'^{a-\-hx^y~^dx; t/i being the
greatest multiple of n less than m, and q the entire part of ^.

216. Formula for the reduction of tlie exponent m wtwn
m is negative.

From Eq. A, Art» 214, by transposition and division, we find

a(m — n+I) -^ ' '

Changing m — n into- — m, this becomes

Jx-"*(a + hx'^ydx =z ; ^ ^, ^

■^ ^ ^ a(«» — 1)

' a(m — 1 j . •^ ^ ^ ^ ^

If in denote the greatest multiple of n contained in m, then,
by i -f- 1 applications of this formula, the integratioai of

will depend on that of a;-"»+^'+^^"(a + 5x'»)''rfar; and, if we

have — m + (* -f- 1)^ = ^* — 1,

we have /ic" Va -^ox'^)'ax =z ^~ ' — .A. ,— .

But under tliis supposition, since ^^ = — i, aiv entire

number, the original expression is immediately initegrable
(Art. 202).

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364 INTEGRAL CALCULUS.

217m Formula for the reduction of the exponent p when p
13 negative.

From Formula B (Art. 215), we finfl

fx"Ua + hx'^y-^dx ~ — ^ ^ ^

•^ ^ ' anp

np+m±l J. J^nydx;

and, if in this we replace p — 1 by — J?; it becomes

/ a?'"(a + hx^'Y^dx = — -^- ^4

•^ ^ ' ^ a7i(^ — 1)

_m + n + l-p nr ,^,^

an{p — 1) -^ ^ ' ^ ^ '

By the continued application of this formula, the exponent
of the binomial will finally be reduced to a positive proper
fraction. Whenjp = l,it cannot be applied; but then the
integration of the given expression may be brought to that of
a rational fraction.

218, The preceding formulsD facilitate the integration of
binomial differentials ; but it is to be observed that the exam-
ples to which they are applicable belong to cases of integra-
bility before established (Art. 202), and the results may there-
fore be obtained independently.

By the application of Formula A, we have

r x'^dx __ x'^-W^ — x^ m — \ rx"'-'^dx ^

and, by making m = 1.3.5... successively, this gives

/xdx ,

c x'dx _ X* 2 r xdx

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REDUCTION FORMULA. 365

rx^dx __ x^ . 4 r x\dx

whence

/xdx /-

x^dx

r x^dx __ /x^ 4:x^ 2.4 \

J vi -a;'^~ ~ U "'■3:5 +o:5]'^i -

ic-

When m is an odd number, we have the general formula

/x"^dx

^ m ^ (m — 2)m ^ ^ 1.3.. .m y^ ^'
and, when m is an even number,
/• x*^dx
J Vl- x^

m

. {rn^l)x^-' . 1.3.5.. .(m-1) ) ,-

+ (m^2)m +'"+ 2.4.6...n - ^j^l-^'

.1.3.5.. .(m-1) . ,
2.4.6...m

2- f — T = fx—^{a'' + xT^dx.

^ x^(a^ + xy ^

Comparing this with Formula C, and making
we find

/-

dx

(a2 _f. ^2)i

(m — l)a2a;"»-i

m — 2 /•

cfo;

(m— l)a2'^a;''*-

-2(a2 + aj2)*

366 INTEGRAL CALCULUS.

Without referring to the formula, this expression may be found
as follows : —

'^a;»(a» + x«)* *^ dx a^+i

= ^ - + ("» + !)/ ^ -,dx;

whence, by transposition and reduction,

(». + l)a'/ ^? , = <-^l±^* - m / ^_,..

(m — l)a*^ ir'»-2(a2 + a;^)**

;J19. By means of Formula D, the expression .^ .^ "^ ^ . >

which occurs in Art. 210, may be integrated by successive
reduction.

Let a + ^^ — 1, a — ^^ — 1, be the roots of the equation
x^ +|>x + J = ; then
{ Mx + N)dx _ (Mx + N)dx

M(x — a)dx M yitr , xT\ dx

Putting a: — a = 2, we find

if 1

■2(m— 1) (z'+^*)"-'

Jf 1

■2(n-l) j(a;_«)'+^«j-''

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REDUCTION FORMULAE. 3t57

Making a; — a = y, and Ma -|- 2f= M\ we have

and therefore, by combining these results, we have
r {Mx- \-N)dx _ M \ '

^ }(x-a)»+^*j"~~2(»-l) |(a;-a)»+fi'^^"-'

Formula D may now be applied to the term under the sign \

in the second member of this last equation ; and, by repeated
applications, the exponent — n will be reduced to — 1, when
the integral will be completely determined.

220. Reduction formulae may also be constructed to facih-
tate the integration of trigonometrical functions. Let the
integral of ^\n.^x^o^,^xdx be required.
Make sin. a; = 2 ; then

cos.a; = (1 - z'f, rfa; = (1 - z'Y^dz,

and sin.^a;Gos.^a:cii;= «^(1 — z^) ^ dz.

Now, if q be an odd number, whether positive or negative,

we may always effect the integration of z^(\ ^ z^) 2 dz^ what-
ever may be the value of jo. In like manner, by making
COS. a; = 2, we see that the integration can be effected whenjp
is an odd number, whether positive or negative, whatever be
the Talue -pf 5.

In any case in which i> > t), and - -^^ > 0, by applying

Si

Formulaa A and B, Ave get

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/.ni-.T^cf. = -^-^'(^ - ')""'

368 INTEGRAL CALCULUS.

If j[> < 0, Formula C gives

•/ 2>— 1

and when 2-_ — <^ o, by Formula D, we have

By the aid of the foregoing formulae, we are enabled to make
the integration of Bin.^xcoa.^xdx depend on that of expres-
sions in which the exponents j? and q are numerically less than
in the original function. From (A') we have

/. « „ y sin.^~^a;co8.^+^aj
sm.^a; cos. ^icaa; =

+ ^-^ — ; sin.^-'a?cos.^a;eZaj (1):
p + qJ ^ ^'

and from (B^),

/. „ „ , sin.^ + ^a;cos.'~^iC
sm. ''a; cos. ^a:aa?= — -

+ *-^ fsin.^rccos.^'-'icdaj. (2).
^jp + qJ ^ ^

When q is positive, by the application of (2).
fsm,^x cos,^xdx

will finally depend on jB\Ti.^xdx, or on f sin.'* x cos. ocdx^
according as q is even or odd.

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REDUCTION FORMULM, 369

By making g = 0, in (1), we have

8in.^a?aa? = 1- -^ I 9>m.^ ^xdx;

and thus, if j? be a positive integer, and even, Jsin.^xdx will
at last depend on fdx =.x; and if ^ be a positive integer, and
odd, the final integral to be found is J&m,xdx = — cos. a?. In
the second case, that is, when q is odd, we have

jBin.^xco9.ocdx=Jsin.^xdsin.x = — ' .. •

It is therefore always possible to find the integral

J'sin.^ a; COS. *a;da?

whenp and q are entire and positive.

Formulas 1 and 2 are inapplicable when i> = — g; but in
this case

fQiiL^x cos.^xdx =:fta,n.^xdx =J'tan.^""*a? tan.'axfo;
=^tan.^~*a;(sec.^a? — l)dx
=^ftsLn.^^^xdta,n.x ^ftan.^'-^Qcdx
= ^P^"^'^ _ fUin.^-^xdx. (3).

This formula serves for the reduction of the exponent of
tan. a;/ and the integration will at last depend on that of

Jdx = a;, or ^tan. ocdx == I cos. x,
according as p is even or odd,

221* In (1) of the preceding article, making q=^0, we
have

/. _ , sin .''■"^ a; COS. a; , Jt> — 1 /• . „ , ,
sm.^xdx = f - ^^ J &iu.P-^xdx;

47

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370 INTEGRAL CALCULUS.

and hence, when p is even,

/Bia.''xdx= '—J9in.'~^x 4-^ — oain-'~'a;

+ (i,-2)(p-4)«"'- * + •••
ip-l)(p-3)...iA

^ip-2){2>-i)...i.^
(p-l)(p-Z)...Z.l x]

"*"(i>-2Xl»-4)...4.2j7j ^'■^'
and, when^ is odd,

/, COS. x\ . , p — 1 . ,

8in.''xox = ■lBm.P~^x-{ - ^^—^Bin.''~'x-\- •••

( p-l)(^-3)...2 >

-2)(i,-4)...lJ (2)-

In like manner, by making p = in (2) of the preceding
article, we fibonld l»Te farmulas fer

fcos.^xdx.

EXAMPLES.

1 f ^"^^ — — ^^'^'H^^ — ^^)

(2aa; — x^) n

. 2n — 1 r x^'^^dx

+ — "j 1-

n '' {2ax — x*y

f

„ r x'dx

+ '{P-

dx • 1 «

— = ver. fiin.~* -•

(2ax + x^f

n n {2ax + «^)

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EXAMPLES, 371

and ultimately we should have (Ex. 7, p. 328)

r Y = \x + a + y/2a^ + x''),

^ {2ax + xY

1 q g

If n be a fraction belonging to the series - , ^r , -..., this

A M M

process of reduction would at last lead to

/dx . .X
Y = sm.-"^ -.

(a^ - 0?^)* ^

4. J(a» - x^)^dx = I (a^ - x'^f ^

, 3 „ / - „,} , 3a* . I it

13 5

When n is any one of the fractions -, -, o««»7 the integr€j

J 2 2

will finally depend on

/cLc _ 1 ic

(a"^ - x'^Y "" 2(71 — i)a' {a' — a?2)'»-i

2n— 3 1 /. rfa;

"•"2(71- 1) a^ J {a^^x'f-

)f the series of fracti
tegratioft will finally depend on that of

3 5 7
When n is one of the series of fractions ^, ~, !>•• •; tbe in-

^ t2 -2

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372 INTEGRAL CALCULUS,
r dx X

by Formula D.

7 r ^^ _ J: a? , ^ x

a:»(a2 — x^f {n — l)a2a;»-i

. n — 2 1 r rfg;

n— 1 a^ ic'— 2(a2 — a;2)*

When n is even, the application of the formula will lead to
r ^dx __ _ (a2_a;2)i
•^ a;2(a2 — a;2)* a^a;

and, when n is odd, the integral will depend on that of

C dx _ 1 ^ X ___ 1 ^ g — (gg — a?^)^

J i — i —

a;(g2 — flj^)^ g g + (g^ — a;^)* g a;

(Ex. 17, p. 332).

Q r ^**^^ _ 3x"(g + bx) 3ng r x^'-^dx

(g + bx)^ (3/1 + 2)6 3n + 2&*' (g + fta?)*

By the application of this formula, the exponent n will at
last be reduced to zero, and the integration will depend on
that of

•' (a + hxy 2b

10. /•_^!^ = (a + fe)M^-^ + !!^7.
ia + bxf (85 206' 406')

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EXAMPLES, - 373

11. /• x^dx _ x''-'^{a -\-hx-\- cx'^)

\a + h + cx^)^ nc

n— 1 a r x^^^dx

^ ^•^(a + 6a? + ca;2)*
2n— I h c x^~^dx

-I

2n c'' {a + bx + cx^)^
and, ultimately, we shall have to find

r xdx ___ (a + &a; + cx^) b r dx

{a + hx + cx'^)^ c ^2c (a + hx + cx^)^'

dx
but the integration of Y has been explained

(a + bx + cx^Y

(Ex. 7, p. 328).

12. / ^^—- = ^-±1^2^2x + x^)^

•^ (2 - 2a; + x'^y 2

+ lz|a:-l + (2-2a: + a;2)^j.

13. fx(2ax - x'')^dx = — J {2ax - x^)^

+ a / (2aa; — a:*) dx.
U. J^'^x{2ax'-x^)^dx=z~'

15. /*a;2(2aa? - a;^)*^^ = "" Z (2^ - ^')*

+ ^Ja:(2aa;-a;2)W

16. r'V(2aa;-.a;2)*rfx=— .
•/« 8

|H. /•sin.'ccrfx sin. a? j^ 1 7 1 ~" sin. a?

J cos.'a; 2008.^^0? '4 1+sin.a:

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SECTION IV.

GEOMETRICAL SIGNIFICATION AND PROPERTIES OP DEFINITE INTEJ-
GRALS. — ANOTHER DEMONSTRATION OP TAYLOR'S THEOREM.

DEFINITE Integrals in which one op the limits becomes

INFINITE. — DEFINITE INTEGRALS IN V.HICH THE FUNCTION UN-
DER THE SIGN J BECPMES INFINITE. — DEFINITE INTEGRALS THAT
BECOME INDETERMINATE. — INTEGRATION BY SERIES.

222* Assume CFD to be the curve of which the equation,

when referred to the rectangu-
lar axes OXf Oy, is y =/(^x).
It has been shown (Art. 164)
ih2Lt/{x)dx is the diflFerential
of the area of a segment of
the curve terminated by a va-
riable ordinate ; and therefore
Jf(x)dx is to be regarded as
the expression for the area bounded by the curve, the axis of
abscissae, and any two ordinates whatever. If this integral
be taken betweea assigned values, a and 5, for a;, the area will
be limited in the direction of the axis of x by the ordinates cor-
responding to these values of x. But the arbitrary constant
may be determined by the condition that the area shall bo
nothing for x = OA = a; that is, shall be limited on one side
by the fixed ordinate AOj while on the other side it is bounded
by a variable ordinate corresponding to the variable abscissa
0M= X.

874

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DF.FINITE INTEGRALS.

o7o

Jf f/{x)dz=z'tif{x) + C, thfco, by the above condition, we

should have

t^(a) + (7=0, C= - \p{a)]
and

f/{x)dx=i^p{x)-'xp{a)

is the expression for an indefinite area taking its origin from
the fixed ordinate AC. -

When the other value of a;, x = OW = 5, is assigned, the
integral becomes definite, and we have

i:

Definite integral, when applied in the determination of the
length of curves, the surfaces and volumes of solids, admit of
a like interpretation.

223* In Art. 192, it was shown that a definite integral was
to be regarded as the limit of the sum of an infinite number
of very small terms.

To illustrate this proposition geometrically, let y =i/(x) be
the equation of the curve CMD referred to rectangular co-
ordinate axes, and suppose that, between the assumed limits
x=:af x = bf y increases continuously.

Then /{x)ax measures the area
of one of the small rectangles
MP'.M'P^.,.] and if -S/(x)Aa? do-
notes the sum of all these rectan-
gles, the area ACDB included be-
tween the curve, the axis of x, and
the ordinates p=/{a), y=/(b),
will be expressed by

lim. ^/{x) AX = J^ /(x)dx =zyp(b) - xp (a),

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376 INTEGRAL CALCULUS.

if '^{x) be the function of which /(x) is the differential co-
efficient.

224:. The order in which the h'mits of a definite integral
are taken may be inverted, provided the sign of the result be
changed for

J'/{x)dx=xp(b)-^xp{a),
r/{x)dx = t/;(a) — t/;(6):

r'*/{x)dx = — C f{x)dx.

Also, if c be a value of x intermediate to the limits a and 6,
we have

J'/{x)dx = i/;(c) — xp{a),

j'/{x)dx=:xp{b)^'ip{c),

J'/{x)dx=zxp{b) ^xp{a)',

fAx)dx = jyix)dx +J'/{x)dx:

and generally, if there be any number of values c, c', c^'...,
between the values a and 6, it may be shown that

j''Ax)dx = jy{x)dx+jyix)dx ...+ f^^^/{x)dx.

225. Let /{x)j (p{x)j be two functions of a?, so related that
/{x) > q:{x) for all values of x from x = a to a; = 6/ then,
taking /(x) — 9 (x) for the differential co-efficient of another
function, we should have

£{/(a;)-q>(a;)}db>Oi

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DEFINITE INTEGRALS. 377

since the derivative /(a;) — g) (a?) is constantly positive be-
tween the limits a and 6, and the function (\f{x) — (p{x)\dx
is an increasing function of x : hence

I f{x)dx > r cp(x)dx.

Also if cpi{x) is another function of a?, such that cpi(x) ^f{x\
for all values of x between the limits a and ft, we should have

/f{x)dx<^ C (pi{x)dx:
a J a

tljerefore

/q>i(x)dx > r f{x)dx > T cp(x)dx.
a J a J a

When a given diflFerential cannot be integrated, it is desira-
ble, and sometimes possible, to find two other integrals be-
tween which the required integral, at assigned limits, will be
included.

Example, f . For values of x between and

J

1, we have

(1 - a;')*

1<— ^<

(1 - a;')* (1 - a;2)*

0.5 < r*

•/

J^ < sin.-i 1 = 0.5236.

j?^6» Demonstration of Taylor's Theorem dependent on
the properties of definite integrals.
The equation

fix + h) -f{x) = rV(x + A - <)d«

J
48

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£78 I y TEG JUL CALCULUS,

is identically true ; but successive integration by parts gives
f f'{x + h -'i)dt =zif (x + h - t) + r tf"{x + A — tyU,

Jo «/

f t/"ix + h- t)dt = ^J"{x + h - t)

Making t=ih in these equations, and then adding them
member to member, we have

Ax+h)=/ix)+h/'(x)

+ rx:^ />'"""(- +^-')'^

If the function to be expanded, and also its differential co-
eflScients up to the order denoted by n-^ly are finite, and
continuous between the limits x and x-\-h, the residual term

~-^ — r y^" +'^ (x + 7a — t)dt may be replaced by

1 . 2..71 •/

and the expansion then agrees with that of Art. 61.

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DEFINITE INTEGRALS. ' 379

227. In what precedes, it has been supposed that the limits
a and 6 of the definite integral \ f{x)dx were finite, and that

the function f{x) was also finite, and continuous between
these same limits. It may happen that one of the limits, 6,
becomes infinite while tho other is finite, the function re-
maining finite and continuous. Then the value of the inte-
gral is the limit of the value of / /(a:)ctr when ft is increased

J a

without limit. This value may be finite, infinite, or indeter-
minate.

Example 1. r*e~^da;.

For tiie indefinite integral, we have

Ce-'dx = — e-» + C:

f e-"dx=:l— - ^,
Ex. 2. f^e'dx.

J

The indefinite integral is

r*e*d!r = e* — 1 = oo .

'

Ex.3, r ^

J

We have

x^

+ a^-

h

dx

= — tan.
a

-.? + <7..

r_

dx

1^
= -tan.

-..=i:

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380 INTEGRAL CALCULUS.

Ex. 4. / cos.xdx.

J

In this case, ^cos.iTcZr = Biii.a;+ (7; and, taking the inte-
gral between and the finite limit 6, we have

/COS. a; do; = sin. 6 ;

but, when b becomes infinite, the value of sin. 6 will be inde-
terminate, though confined within the limits and 1.

The following investigation will sometimes enable us to

decide whether the definite integral I f{x)dx is finite or

J a

infinite for ft = cx> or 6 = — cx> .

First suppose that h is very great, but not infinite, and let
c be a number comprised between a and b; then (Art. 224)

f'/{x)dx = r/{x)dx + f'/ix)dx.

Since /(o;) is finite, C'^/(x)dx is also finite; and it remains

only to examine the value of j /{x)dx when b becomes in-
finite.

Put /(a;) = — ^, (p{x) being a function that remains finite

for all values of x greater than c. If A denotes the greatest
and B the least of the values of qp (x) for all values of x greater
than c, we shall have

or

;V,^<_4^(_L_±..).

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DEFINITE INTEGRALS, 381

Now, when n > 1, the second member of this last inequality

A \
for J = GO reduces to ^^ —-—. : hence, in this case, w^e

know that the integral f^/{x)dx has a finite value. When
n < 1, we have

• • • ^ ff{x)dx > j-^ (fti-" - c^-").

Now, when 1 — n > 0, the second member of this inequality
becomes infinite for 6 = oo : hence / /{x)dx, and therefore

j f{x)dx is infinite for 6 = oo .
If n = 1, then

but /f - J = 00 when 6 = oo : hence f/{x)dx = cx> .

Putting /(a?) under the form -~- , cp{x) being a function
that is finite for all values of x between — oo and some value
less than 6, it may be shown in like manner that j f{x)dx
is finite if n > 1, and infinite if n < 1 orn = 1.

Thus, if it be possible to put f{x) under the form ?-i~ , and

the condition imposed on (f{x) be satisfied, we can decide
whether the integral \ f(x)dx is finite or infinite when one

J a

of its limits becomes -\-qo or — cx> .

228. Definite integrals in which the function under the
sign of integration becomes infinite between or at the limits.

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382 INTEGRAL CALCULUS.

The function /{x) TDxy becooie infinite at one of the limits,
6, of the integral T f{x)dx; in which case the integral ia

defined as the limit of f ~ /{x)dx when a is decreased with-

•^ a

out limit. In like manner, if/{x) becomes infinite for a; =ia,

then f f(x)dx is the limit of / f{x)dx when a is indefi-

nitely decreased. FinaUy, if /(c) = <» ^ c being comprised
between a and 6, we should have

/ f{x)dx z^lixn. f"' /{x)dx + \im. f /{x)dx,

when a and ^ are decreased without limit. Should there be
more than one value of x for which f(x) becomes infinite be^
tween the limits a and 6, we learn from what precedes bow

to define the integral f /{x)dx,

229* It may sometimes be decided whether tha integral

f f(x)dx is finite or infinite when f(x) is infinite at one of

thfe limits. Suppose/(6) = cx> , and let f(x) = ,^ ^ ', g)(x)
being finite for x = 6 and for all values of a; < 6, and n being

>a.

If c be a naraber comprised betweea a snd h,. we have

r/ix)dx = r/(x)dx + ff{»)dx.

Now r*'/(a;)rfa; is finite; and hence i /(a;)da; will be finite
or infinite according as \ /{x)dx is finite or infinite.

Denote, by A the greatest and by B the least of the vakies
of (3p{x) &a Tahiea. of x included between o and b. If n^ < 1,

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DEFINITE INTEGRALS. 383

we shall have, for such values of x, fix) < ri v;. ; and

therefore

When a converges towards 0, the second member of this in-

A
equality converges towards tlie finite value ^j (6 — c)^"*:

hence, in this case, the value of lim. I f(x)dx^ and there-
fore of r f{x)dx^ is finite.

J a

But if 71. > 1, the proposed integral is infinite ; foi', since

/6-«-, w ^ ^^-tf Bdx __ -g j 1 1 )

and it is evident, that, when a becomes 0, the second member
of this inequality becomes infinite : hence, under the supposi-
tion, r f(x)dx is infinite.

J a

In like manner, when n = tj we have f(x) > j ; and

o —-' X

therefore

But Bl becomes infinite when a vanishes r hence

a

C f{x)dxy and therefore \ f{x)dx, is then infinite.

Example 1. T — ,

J ^ y/2h'' — bx—x'

P being a function of x that remains finite for all finite

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384

INTEGRAL CALCULUS.

values of Xj and a and b being two positive quantities, we

have

P P _ _\ (f{x)

X {b — x)h

by putting

= (p{x).

bx — x^ V26 + X Vb — ;
P

V^b-fx '

Since the exponent of 6 — a? is less than 1, it follows from
the rule just established that the proposed integral has a finite
value.

Ex. 2.

dx

dx

JqVI — a? J vl — X

s:

\-a

dx

dx

VT

= = 2-2Va; ,-. f _?f=
X J Q vl — x

dx

The expression . is the differential of the area includ-

X

ed between the axis of ar, and the curve having y = . —

for its equation.

This curve has two asymptotes ; the one the axis of x,
and the other a parallel to the axis of y, and at the distance

+ 1 from it. It is seen from

the figure the / , rep-

resents the area bounded by AC,
ABj the curve, and its asymptote
BD; and this area, although it
extends indefinitely in the direc-
tion of the asymptote BDj still
has the finite value 2.
230. A definite integral may become indeterminate, as is
the case for

r sin. xdx = cos. oo — cos. 0,

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INTEGRATION BY SERIES. 385

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