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INTEGRAL CALCULUS

FOR BEGINNERS



INTEGRAL CALCULUS

FOR BEGINNERS

WITH AN INTRODUCTION TO THE STUDY OF

DIFFERENTIAL EQUATIONS



BY

JOSEPH EDWAEDS, M.A.

FORMERLY FELLOW OF SIDNEY SUSSEX COLLEGE, CAMBRIDGE



MACMILLAN AND CO.

AND NEW YORK

1896
All rights reserved



First Edition, 1890.

Reprinted 1891, 1892, 1893.

With additions and corrections, 1894 ; reprinted 1896.



GLASGOW : PRINTED AT THE UNIVERSITY PRESS
BY ROBERT MACLEHOSE AND CO.



PREFACE.

THE present volume is intended to form a sound
introduction to a study of the Integral Calculus,
suitable for a student beginning the subject. Like its
companion, the 'Differential! Calculus for Beginners,
it does not therefore aim at completeness, but rather
at the omission of all portions of the subject which
are usually regarded as best left for a later reading.

It will be found, however, that the ordinary pro-
cesses of integration are fully treated, as also the
principal methods of Rectification and Quadrature,
and the calculation of the volumes and surfaces
of solids of revolution. Some indication is also
afforded to the student of other useful applications
of the Integral Calculus, such as the general method
to be employed in obtaining the position of a
Centroid, or the value of a Moment of Inertia.

As it seems undesirable that the path of a student
in Applied Mathematics should be blocked by a
want of acquaintance with the methods of solving

M298720



vi PREFACE.

elementary Differential Equations, and at the same
time that his course should be stopped for a sys-
tematic study of the subject in some complete
and exhaustive treatise, a brief account has been
added of the ordinary methods of solution of the
more elementary forms occurring, leading up to and
including all such kinds as the student is likely to
meet with in his reading of Analytical Statics,
Dynamics of a Particle, and the elementary parts of
Rigid Dynamics. Up to the solution of the general
Linear Differential Equation with Constant Coeffi-
cients, the subject has been treated as fully as is
consistent with the scope of the present work.

The examples scattered throughout the text have
been carefully made or selected to illustrate the
articles which they immediately follow. A consider-
able number of these examples should be worked
by the student so that the several methods explained
in the " book-work " may be firmly fixed in the
mind before attacking the somewhat harder sets at
the ends of the chapters. These are generally of a
more miscellaneous character, and call for greater
originality and ingenuity, though few present any
considerable difficulty. A large proportion of these
examples have been actually set in examinations, and
the sources to which I am indebted for them are
usually indicated.



PREFACE. vii

My acknowledgments are due in some degree to
the works of many of the modern writers on the
subjects treated of, but more especially to the
Treatises of Bertrand and Todhunter, and to Pro-
fessor Greenhill's interesting Chapter on the Integral
Calculus, which the more advanced student may
consult with great advantage.

My thanks are due to several friends who have
kindly sent me valuable suggestions with regard to
the desirable scope and plan of the work.

JOSEPH EDWARDS.

October, 1894.



CONTENTS.



INTEGRAL CALCULUS.

CHAPTER I.

NOTATION, SUMMATION, APPLICATIONS.

PAGES

Determination of an Area, ...... 1 3

Integration from the Definition, 4 9

Volume of Revolution, 10 13

CHAPTER II.
GENERAL METHOD. STANDARD FORMS.

Fundamental Theorem, ....... 14 19

Nomenclature and Notation, ...... 20 21

General Laws obeyed by the Integrating Symbol, . . 22

Integration of x n , x~ l , 2326

Table of Results, 2628

CHAPTER III.
METHOD OF SUBSTITUTION.

Method of Changing the Variable, 29 32

The Hyperbolic Functions, ...... 33 36

Additional Standard Results, 3741



CONTENTS.



CHAPTER IV.
INTEGRATION BY PARTS.

PAGES

Integration "by Parts" of a Product, .... 4347

Geometrical Proof, 4849

Extension of the Rule, 5052



CHAPTER V.
PARTIAL FRACTIONS.

Standard Cases, ........ 55 57

General Fraction with Rational Numerator and De-
nominator, 5861



CHAPTER VI.
SUNDRY STANDARD METHODS.

Integration of f^L ... . 6568

J \/K

Powers and Products of Sines and Cosines, . . . 6974

Powers of Secants or Cosecants, ..... 75 76

Powers of Tangents or Cotangents, ..... 77 78

/rfv

, etc., 7983
a + o cos x



CHAPTER VII.
REDUCTION FORMULAE.

Integration of x m - l XP, where X = a + bx n , . . . 8789
Reduction Formulae for / x m ~ l X p dx, .... 90 93

Reduction Formulae for / sin^a; cos^a: dx, . . . 94 95

7T IT

j"z r -i

Evaluation of / sii\ n xdx, I ain^x cos^x dx, . . . 96 102



CONTENTS. xi

CHAPTER VIII.

MISCELLANEOUS METHODS.

PAGES

Integration of / ^).f x . 9 ....... 109117

J X. i\f Y



Integration of some Special Fractional Forms, . . 118 119

General Propositions and Geometrical Illustrations, . 120 124

Some Elementary Definite Integrals, .... 125 127

Differentiation under an Integral Sign, .... 128 129



CHAPTER IX.
RECTIFICATION.

Rules for Curve-Tracing, ....... 135137

Formulae for Rectification and Illustrative Examples, . 13S 139

Modification for a Closed Curve, ..... 140

Arc of an Evolute, ........ 143

Intrinsic Equation, ........ 144 149

Arc of Pedal Curve, ........ 150



CHAPTER X.
QUADRATURE.

Cartesian Formula, ........ 153 157

Sectorial Areas. Polars, ....... 158160

Area of a Closed Curve, ....... 161163

Other Expressions, ..... '.''-. . 164165

Area between a Curve, two Radii of Curvature and the

Evolute, ......... 166167

Areas of Pedals, ........ 168175

Corresponding Areas, ....... 176 177



CHAPTER XI.
SURFACES AND VOLUMES OF SOLIDS OF REVOLUTION.

Volumes of Revolution, ....... 183184

Surfaces of Revolution, ....... 185187



xii CONTENTS.



PAGES

Theorems of Pappus, ....... 188 191

Revolution of a Sectorial Area, ...... 192



CHAPTER XII.

SECOND-ORDER ELEMENTS OF AREA. MISCELLANEOUS
APPLICATIONS.

Surface Integrals, Cartesian Element, . 195 198

Centroids ; Moments of Inertia, ..... 199 201

Surface Integrals, Polar Element, 202203

Centroids, etc., Polar Formulae, ..... 204 207



DIFFERENTIAL EQUATIONS.

CHAPTER XIII.
EQUATIONS OF THE FIRST ORDER.

Genesis of a Differential Equation, ..... 211214

Variables Separable, 215

Linear Equations, 216 219

CHAPTER XIV.

EQUATIONS OF THE FIRST ORDER (Continued}.

Homogeneous Equations, 221 226

One Letter Absent, 227229

Clairaut's Form, 230233

CHAPTER XV.

EQUATIONS OF THE SECOND ORDER. EXACT DIFFERENTIAL
EQUATIONS.

Linear Equations, 235 236

One Letter Absent, 237238

General Linear Equation. Removal of a Term, . . 239 240

Exact Differential Equations, . . . . . . 241242



CONTENTS. xiii



CHAPTER XVI.

LINEAR DIFFERENTIAL EQUATION WITH CONSTANT
COEFFICIENTS.

PAGES

General Form of Solution, 243244

The Complementary Function, ..... 245 251

The Particular Integral, 252263

An Equation Reducible to Linear Form with Constant

Coefficients, 264265



CHAPTER XVII.
ORTHOGONAL TRAJECTORIES. MISCELLANEOUS EQUATIONS.

Orthogonal Trajectories, 266269

Some Important Dynamical Equations, .... 270 271
Further Illustrative Examples, 272277

Answers, 278308



ABBREVIATION.

To indicate the sources from which many of the examples are
derived, in cases where a group of colleges have held an examination
in common, the references are abbreviated as follows :

(a) = St. Peter's, Pembroke, Corpus Christi, Queen's, and St.

Catharine's.

(j8) = Clare, Caius, Trinity Hall, and King's.
(7) = Jesus, Christ's, Magdalen, Emanuel, and Sidney Sussex.

(d) = Jesus, Christ's, Emanuel, and Sidney Sussex.

(e) = Clare, Caius, and King's.



INTEGRAL CALCULUS

CHAPTER I.
NOTATION, SUMMATION, APPLICATIONS.

1. Use and Aim of the Integral Calculus.

The Integral Calculus is the outcome of an en-
deavour to obtain some general method of finding the
area of the plane space bounded by given curved
lines.

In the problem of the determination of such an
area it is necessary to suppose this space divided up
into a very large number of very small elements.
We then have to form some method of obtaining
the limit of the sum of all these elements when
each is ultimately infinitesimally small and their
number infinitely increased.

It will be found that when once such a method of
summation is discovered, it may be applied to other
problems such as the finding of the length of a curved
line, the areas of surfaces of given shape and the
volumes bounded by them, the determination of
moments of inertia, the positions of Centroids, etc.
E. i. c. A <E



2 INTEGRAL CALCULUS.

Throughout the book all coordinate axes will be
supposed rectangular, all angles will be supposed
measured in circular measure, and all logarithms
supposed Napierian, except when otherwise stated.

2. Determination of an Area. Form of Series to
be Summed. Notation.

Suppose it is required to find the area of the portion
of space bounded by a given curve AB, defined by
its Cartesian equation, the ordinates AL and BM of
A and B, and the cc-axis.




L 0,0,0,0,

Fig. 1.



Let LM be divided into n equal small parts, LQ V
QiQz, Q^Qv > eacn f length A, and let a and 6 be
the abscissae of A and . Then b a = ?i/L Also if
y = (f)(x) be the equation of the curve, the ordinates
LA, QiPp $2^2* e ^ c -' through the several points L,
Q v Q 2 , etc., are of lengths <j>(a), ^(a+K), ^(a+2A), etc.
Let their extremities be respectively A, P 1? P 2 , etc.,
and complete the rectangles AQ V PjQg, P 2 Q 3 , etc.

Now the sum of these n rectangles falls short of
the area sought by the sum of the n small figures,
1 , P 1 J2 2 P 2 , etc. Let each of these be supposed



NOTATION, SUMMATION, APPLICATIONS. 3

.4

to slide parallel to the o>axis into a corresponding
position upon the longest strip, say P n _^Q n _iMB.
Their sum is then less than the area of this strip,
i.e. in the limit less than an infinitesimal of the first
order, for the breadth Q n .iM is h and is ultimately
an infinitesimal of the first order, and the length
MB is supposed finite.

Hence the area required is the limit when h is
zero (and therefore n infinite) of the sum of the
series of n terms



The sum may be denoted by

a+rh = b-h a+rh=b-h

S <>(a + rJi.h or



where S or 2 denotes the sum between the limits
indicated.

Regarding a+rh as a variable x, the infinitesimal
increment h may be written as to or dx. It is
customary also upon taking the limit to replace the

symbol S by the more convenient sign I , and the

limit of the above summation when h is diminished
indefinitely is then written

f 6

I <f)(x)dx,

a

and read as "the integral of <j)(x) with regard to x
[or of (f>(x)dx] between the limits x = a and x = b"
or more shortly " from a to b."

b is called the " upper " or " superior limit."
a is called the " lower " or " inferior limit,"
The sum of the n + l terms,



differs from the above series merely in the addition of



INTEGRAL CALCULUS.



the term h(f>(a+nh) or A0(6) which vanishes when
the limit is taken. Hence the limit of this series
may also be written

f 6

I <t>(x)dx.

a

3. Integration from the Definition.

This summation may sometimes be effected by
elementary means, as we now proceed to illustrate :



Ex. 1. Calculate



Cb

/ e*dx.



Here we have to evaluate

Lt h==Q h[e a + e a+h + e a+ + . . .
where b = a + nh.

This =Lt h ^h^p\e a =Lt h ^(e b - e a )-^-=e* - e\
& 1 & X

[By Diff. Calc. for Beginners, Art. 15.]

/b r=n-l

xdx we have to find Lt 2 (+rA)A, where

r=



Now 2(a + rh)h =



and in the limit becomes



2 22'



/ 6 1
$x we have to obtain the limit when h is

a

indefinitely diminished of



NOTATION, SUMMATION, APPLICATIONS. 5



>



a b + h



a-h &'

and when h diminishes without limit, each of these becomes

II
a b'

Thus f*JL /&==*.*

J .r 2 a b

a

Ex. 4. Prove ab initio that

/&
sin # ofo? = cos a - cos 6.

We now are to find the limit of

[sin a + sin(a + k) + sin(a + 2A) + . . . to n terms]A,

sinf a+n l- Jsin n-
\ 2/ 2,

sin |

*

This expression = cosf a - J cos < a + (2n - 1)- j- -

2JJ sin-

2



sm-



which when A is indefinitely small ultimately takes the form
cos a cos b.



INTEGRAL CALCULUS.

EXAMPLES.

Prove by summation that



/ sir



2. / sinh xdx cosh b cosh a.
3.



/b
cos OdO = sin 6 sin a.



4. Integration of x m .

As a further example we next propose to consider
the limit of the sum of the series

h[a m + (a + h} m + (a + 2h) m +.



i 7 6 a

where h = - ,

n

and n is made indefinitely large, m + 1 not being zero.

fy I I\m+l _ yin + 1

[Lemma. The Limit of v>/ % - 2 is m + 1 when A is

Ay m

indefinitely diminished, whatever y may be, provided it be of
finite magnitude.

For the expression may be written



- 1

y

and since h is to be ultimately zero we may consider - to be

y
less than unity, and we may therefore apply the Binomial

/ ^\7?l + l

Theorem to expand ( 1 -J - J , whatever be the value of m+l.



NOTATION, SUMMATION, APPLICATIONS. 7

(See Dif. Gale, for Beginners, Art. 13.) Thus the expression
"becomes



-x(a convergent series)

y

m + I when A is indefinitely diminished.]
In the result



put i/ success! vely = a, a+h, a+2h,etc....a + (n l)h,
and we get

l -a m + l _ T (
~

_ r ,



_ 1 - (a + n^



h(a+n-Ui) m

or adding numerators for a new numerator and de-
nominators for a new denominator,



fe[a w + (a + /t) w + (a + 2h) m + . . . + (a + n^l
or
Lt h=Q h[a m + (a + A) m + (a



m+1 '

In accordance with the notation of Art. 2, this
may be written



' 6 7 b
x m dx=



m+1



8 INTEGRAL CALCULUS.

The letters a and b may represent any finite quantities what-
ever, provided x m does not become infinite between x=a and

When a is taken as exceedingly small and ultimately zero, it
is necessary in the proof to suppose h an infinitesimal of higher

order, for it has been assumed that in the limit - is zero for
all the values given to y. V

When 6 = 1 and a = 0, ultimately the theorem be-
comes



x m dx= 7 if m + 1 be positive,
o
or = oo if m + 1 be negative.

This theorem may be written also



r



according as m+1 is positive or negative. The limit



or, which is the same thing,

-Lst n= oo M4-'



differs from the former by , i.e. by in the limit,

1 n
and is therefore also -?, or oo according as m+1 is

positive or negative. The case when m + l=0 will
be discussed later.

Ex. 1. Find the area of the portion of the parabola 7/ 2 =4a#
bounded by the curve, the #-axis, and the ordinate xc.



NOTATION, SUMMATION, APPLICATIONS.



Let us divide the length c into n equal portions of which
NM is the (r+l) th , and erect ordinates NP, MQ. Then if




PR be drawn parallel to NM, the area required is the limit
when n is infinite of the sum of such rectangles as PM (Art. 2),



i.e. Lt^PN.NM or

where nh = c.

Now




[By Art. 4.]



Area =f



=f of the rectangle of which the extreme ordinate arid abscissa
of the area are adjacent sides.

Ex. 2. Find the mass of a rod whose density varies as the
with power of the distance from one end.

Let a be the length of the rod, o> its sectional area supposed
uniform. Divide the rod into n elementary portions each of

length -. The volume of the (r+l)th element from the end

of zero densitv is w-, and its density varies from ( | to

(7+la\ m n * *"

1 - ) . Its mass is therefore intermediate between
\ n )

coa** 1 - and **



10 INTEGRAL CALCULUS.

Thus the mass of the whole rod lies between



and



and in the limit, when n increases indefinitely, becomes



ra+1

5. Determination of a Volume of Revolution.

Let it be required to find the volume formed by
the revolution of a given curve AB about an axis
in its own plane which it does not cut.

Taking the axis of revolution as the cc-axis, the
figure may be described exactly as in Art. 2. The




Fig. 3.

elementary rectangles AQ V P-fy^ P 2 Qz> etc., trace in
their revolution circular discs of equal thickness, and
of volumes <jrA L 2 . LQ 19 nrP^ . Q&, etc. The several
annular portions formed by the revolution of the
portions AR^^ P^R^P^ P 2 E 3 P 3 , etc., may be con-



NOTATION, SUMMATION, APPLICATIONS. 11



sidered made to slide parallel to the #-axis into a
corresponding position upon the disc of greatest radius,
say that formed by the revolution of the figure
Pn-iQn-iNB. Their sum is therefore less than this
disc, i.e. in the limit less than an infinitesimal of the
first order, for the breadth Q n -\N is h, and is ulti-
mately an infinitesimal of the first order, and the
length NB is supposed finite.

Hence the volume required is the limit, when h is
zero (and therefore n infinite), of the sum of the series



or as it may be written



Cb

7r\



or



Cb

T y 2 dx.



Ex. 1. The portion of the parabola y 2 = 4a,r bounded by the
line # = c revolves about the axis. Find the volume generated.

P




Let the portion required be that formed by the revolution of
the area APN^ bounded by the parabola and an ordinate PN.



12



INTEGRAL CALCULUS.



Then dividing as before into elementary circular laminae, we
have






/c re

y^dx 4a:r / xdx



[Art. 4.]



2 AN



=J cylinder of radius PA 7 ' and height AN.
[Or if expressed as a series

[c

Volume = 4a?r I x dx
o



r
. = 2a?rc 2 .]

2



[Art. 4.]



Ex. 2. Find the volume of the prolate spheroid formed by the

revolution of the ellipse ~+^- = l about the #-axis.
2 *




Fig. 5.

Dividing as before into elementary circular laminae whose
axes coincide with the #-axis, the volume is twice



/ Try^dx.



-a 2 - x*)dx

a



Now

which, according to Article 4, is equal to
5[a*.(-0)-^] or
and the whole volume is



NOTATION, SUMMATION, APPLICATIONS. 13

[or if desirable we may obtain the same result without using
the sign of integration, as



EXAMPLES.

1. Find the area bounded by the curve y ^^ the #-axis, and
the ordinates #=a, #=&.

2. If the area in Question 1 revolve round the ,#-axis find the
volume of the solid formed.

3. Find by the method of Art. 2, the area of the triangle
formed by the line y=x tan 0, the #-axis and the line x = a.

Find also the volume of the cone formed when this triangle
revolves about the #-axis.

4. Find the volume of the reel-shaped solid formed by the
revolution about the y-axis of that part of the parabola y^^ax
cut off by the latus-rectum.

5. Find the volume of the sphere formed by the revolution of
the circle x 2 +y 2 = a 2 about the .r-axis.

6. Find the areas of the figures bounded by each of the
following curves, the #-axis, and the ordinate x = h ; also the
volume formed by the revolution of each area about the #-axis :

(a) 7/3 = a *a



(8) aty

7. Find the mass of a circular disc of which the density at
each point varies as the distance from the centre.

8. Find the mass of the prolate spheroid formed by the
revolution of the ellipse ^ 2 /a 2 -f^/ 2 /6 2 = l about the #-axis, sup-
posing the density at each point to be //x,



CHAPTEE II
GENEEAL METHOD. STANDAED FOEMS.

6. Before proceeding further with applications of
the Integral Calculus, we shall establish a general
theorem which will in many cases enable us to infer
the result of the operation indicated by

n

I <p(x)dx

a

without having recourse to the usually tedious, and
often difficult, process of Algebraic or Trigonometrical
Summation.

7. PROP. Let </)(x) be any function of x which is
finite and continuous between given finite values a
and b of the variable x ; let a be < 6, and suppose the
difference b a to be divided into n portions each
equal h, so that b a = nh. It is required to find the
limit of ike sum of the series

ft[0O) + <p(a + h) + 4>(a + 2h)+... + 0(6 - h) + 0(6)],
when h is diminished indefinitely, and therefore n
increased without limit.

[It may at once be seen that this limit is finite, for if <$>(a+rh)
be the greatest term the sum is



- a)<t>(a + rh) + h<$>(a +



GENERAL METHOD. STANDARD FORMS. 15

which is finite, since by hypothesis <(#) is finite four all values of
x intermediate between b and a.]

Let \fs(x) be another function of x such that <j)(x) is
its differential coefficient, i.e. such that



We shall then prove that
Lt h ^[<fa)+^a+h)+^

By definition ^a)*
and therefore a =



where a : is a quantity whose limit is zero when h
diminishes indefinitely ; thus

h(j)(a) =\/s(a+ 7i) t/r(a) +ha lt

Similarly
h<f>(a

etc.,



Ih) = \[s(a + nil) \

where the quantities a 2 , a 3 , ..., a n are all, like a v
quantities whose limits are zero when h diminishes
indefinitely.
By addition,

h[<f>(a) + 0(a + h) + <f>(a



Let a be the greatest of the quantities a v a 2 , . . . , a n ,
then

Afoi+ag+^.+On] is <nha, i.e. <(& a)a,
and therefore vanishes in the limit. Thus



16 INTEGRAL CALCULUS.



The term fc</>(6) is in the limit zero; hence if we
desire, it may be added to the left-hand member of
this result, and it may then be stated that



.e.



1 </)(x)dx = \ls(b) \ls(a).



This result \[s(b) \fs(a) is frequently denoted by the
notation p\/r(a3) J .

From this result it appears that when the form of
the function ^fs(x) (of which </>(x) is the differential
coefficient) is obtained, the process of algebraic or

rb
trigonometric summation to obtain I <j)(x)dx may be

avoided. a

The letters b and a are supposed in the above work
to denote finite quantities. We shall now extend our

notation so as to let I <f>(x)dx express the limit when

a

b becomes infinitely large of ^(6) ^js(a), i.e.
I (j)(x)dx = Lt b=x I <j)(x)dx.

a a

fb
(j)(x)dx we shall be understood to

fb
-\I,(a)] or Lt a=00 \ <f>(x)dx.






Ex. 1. The differential coefficient of ^ - is plainly x m .



Hence if <$>(x)=x m we have

df(x):L- and / x
^ ' m+l J



m+1 m + \ m+l



GENERAL METHOD. STANDARD FORMS. 17

Ex. 2. The quantity whose differential coefficient is cos a? is
known to be sin x. Hence






6

cos x dx = sin b sin a.



Ex. 3. The quantity whose differential coefficient is e* is
itself e x . Hence



Ex. 4.

EXAMPLES.

Write down the values of
1. /Vdr, 2. /Vcfo?, 3. cfo, 4.



/b rl ,-2

X CiX) 2i, I X Cf/JCm o. I X d/X^

a 1

it ir

/2 rA r4

cos x dx, 6. / sec 2 ^; dx^ 7. / \



o

ia



8. Geometrical Illustration of Proof.

The proof of the above theorem may be interpreted geo-
metrically thus :

Let AB be a portion of a curve of which the ordinate is finite
and continuous at all points between A and B, as also the
tangent of the angle which the tangent to the curve makes
with the a?-axis.

Let the abscissae of A and B be a and b respectively. Draw
ordinates A N, BM.

Let the portion NM be divided into n equal portions each
of length h. Erect ordinates at each of these points of division
cutting the curve in P, Q, R, ..., etc. Draw the successive
tangents AP^ PQi, QRi, etc., and the lines

AP 2 ,PQ 2J QR 2 ,...,

parallel to the ,r-axis, and let the equation of the curve be
y = ^r(x\ and let V^') = <M>

then <f>(a\ <$>(a + h\ <$>(a + Zh\ etc., are respectively
tanP.JPj, taii^Pft, etc.,

E. I. C. B



18
and



INTEGRAL CALCULUS.

-h), ..., are respectively the lengths



Now it is clear that the algebraic sum of

P 2 P, 2 , R 2 R, ..., is MB-NA, i.e.
Hence



u



s,



L
Fig. 6.



M x



Now the portion within square brackets may be shewn to
diminish indefinitely with h. For if R^ for instance be the
greatest of the several quantities PjP, Q^ etc., the sum

[P 1 P+Q 1 Q+...] is <nR 1 R, i.e. <(b-a)-}~.
But if the abscissa of Q be called #, then



and



so that



(x) + -^"(x + Qh\
[Diff. Calc. for Beginners, Art. 185.]
R^R = "(x 4- <9A) = (x + Oh),



and (6 - a)



which is an infinitesimal in general of the first order.



GENERAL METHOD. STANDARD FORMS. 19



Thus

Lt h=0
or Z
Also since Z^= A^(6) = 0, we have, by addition,



9. Interrogative Character of the Integral Cal-
culus.

In the differential calculus the student has learnt
how to differentiate a function of any assigned char-
acter with regard to the independent variable con-
tained. In other words, having given y = \f^(x) )
methods have been there explained of obtaining the
form of the function \[s'(x) in the equation



The proposition of Art. 7 shews that if we can reverse
this operation and obtain the form of ^(x) when \l/(x)


1 3 4 5 6 7 8 9 10 11

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