Elementary integrals; a short table

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http://www.archive.org/details/cu31924001533334

ELEMENTARY INTEGRALS

A SHORT TABLE

COMPILED BY ,( ,

T. J. I'a. BROMWIGH, Sc.D., F.R.S.

Fellow and Lecturer of St. John's College, Cambridge ;
University Lecturer in Mathematics.

BOWES AND BOWES

1911

s

fN.Z-b^S^i

INTRODUCTION.

'PHE followiug table of integrals has been drawn up with
the hope of lessening the labour often involved in the
integration of elementary functions. The list includes all
the ordinary standard integrals and formulae of reduction,
arranged compactly, so as to allow of ensy reference.
Although the tables occupy less space than some which are
in existence, yet it is hoped that the present set may prove
no less useful in practical work ; for example, it proved
possible in many instances to condense three or more separate
formulfe into a single reduction formula ; and it is hardly
necessary to point out that a given result can be found more
quickly in a short than in a long table.

A few other useful forms, such as (57), (58), have
been added, which do not appear in the ordinary text-
books : and in selecting definite integrals, preference has
been given to examples which occur in potential-theory and
other branches of Applied Mathematics. Some numerical
examples have been added to illustrate the general formulae,
but it is not intended that these should be regarded in any
way as replacing the sets of examples provided by books
on the Calculus. Methods of approximate integration —
by Simpson's formulae and the planimeter— are briefly
formulated in ±he last two sections.

TABLE OF CONTENTS.

PAGE

I. 1 — 9. Fundamental Algebraic Integrals . 7

II. 10—15. Trigonometrical Integrals . . 9

III. 16—19. Integration by Parts ... 10

IV. 20 — 27. Integration by Snbstitntion . 11
V. 28 — 34. Rational Algebraic Integrals . . 13

VI. 35 — 48. Irrational Algebraic Integrals . 15

VII. 49 — 55. Trigonometrical Integrals . . 22

VIII. 56—79. Definite Integrals ... 26

IX. 80—85. Integration of Series ... 32

X. 86—88. Differentiation of Integrals . 34

XI. 89 — 95. Simpson's Formulfe ... 35

XII. 96—100. Planimetric Formula . . 37

ELEMENTARY INTEGRALS.

I. Fundamental Algebraic Integuals.
■ 1. \x''dx = ; if w + 1 is not zero.

2. \~ dx = \ogx or log(-a;),
according to the sign of w.

3. [e'^dz = -€'"'.
J a

dx

4. f^, = itan-p).
]d'-\-x^ a \aj

i\ f dx _], /x — a\ 1, fa—x

= --cotlrM'-) or -itanlr'f-)
a \al a \aj

or

-; — = — =sin'M-) if a>0,

f— = tan-'(-j if y=a^-^^

"• -77-3 ^ = sinlr'(-) if a>0.

8- -77-5 5^ = cosh"'(-) if x>a>0.

Both 7 and 8 are included in the one general formula

9. j — = log (a? + «/) if y^=x^+c,

which follows at once from the equations

, , dx dy dx-\-dy
X dx=y dy, or — = -^ = ~ ;

but the result can also be derived from (7) and (8) by the
method given in the Notes below.

Notes.
2 and 5. It is often felt to be strange than we can write
f dx

according as a; — a is positive or negative. A full explanation
involves a little acquaintance with the theory of logarithms
of complex numbers, from which it appears that

log (a; — a) - log {a — x)

is independent of x, and is an odd multiple of ttl ; but the
beginner should convince himself by differentiation that the
integral is really +Iog(a — :r) and not —log (a — a?).

1, 4, and 6. The inverse circular functions sin"'a7, tan"'ar
are to be understood as angles between — ^tt and + ^tt ;
with this convention they are uniquely determined by x.

The square-roots are to be understood as positive num-
bers, here and elsewhere, so that P must be positive if we
use such an equation as P >^/Q = i\/{P''Q).

5, 7, and 8. The inverse hyperbolic function sinh"'a: is
uniquely determined by x; and so is the function tanh~'a;,
so long as x is between —1 and +1; while coth~'a; is
uniquely determined so long as x does not lie between
— 1, +1. But cosh"' a? has two values (x being greater
than 1), and of these it is meant that the positive one
is to be taken. The beginner may find it instructive to
draw rough graphs of the inverse functions.

5 and 9. It is to be remembered that the inverse hyper-
bolic functions are in reality logarithms ; and that we can
always avoid their use by simple transformations. Thus,
if we write s = sinh u, c — cosh u,

we find s + c = e" or u = log (s + c).

Thus sinh"'5 = log(5 + e) = log|s + V(l +s')l.

and cosh"' c = log (s + c) = log \c + '^{c^— 1) j.

9
Similarly, if ^ = tauliM, we find

(1 + 0/(1-0=^'"

or taiilr'i! = ^log|(l +0/(1-01) -1<^< + 1;
and, if 7 = cotliM, we get

(7 + l)/(7-l)=e=»,
and so coth''7 = -|logj(7H- 1)/(7- 1)1,

where either 7 > 1 or 7<— 1.

II. TllIGONOMETRICAL INTEGRALS.

10. Jsin a; afar = — COS it;, Jcos a: ofa; = + sin a;,

11. Jtana;rfa; = — logcosa;, |cota;rfa; = + logsinar,

12. Jsec.'cofa; = log(seca; + tana;) = siuh~'(tana;)

Jcosec xdx = log ( tan \x^ = — log (cosec x + cot a;),

13. Jsin"a:a?a; = ^(a; — sina?cosa;),
|cos'''a; fl?a; = ^ (a; + siu a; cos x),

14. Jtan'a; dx — tan x — x, jeofa; dx = — x — cot x,

15. Jsec'^ar afa; = tan a:, Jcosec'''ara?j; = — cota?.

Note.

The integrals above are suggested at once from the
ordinary rules of the differential calculus, with the ex-
ception of (12) ; in this case we may introduce the new
variable ^ = tan^a; (in agreement with Vll. below). Then
we find

u = \6&cxdx=\,^--^, = \og[~-\ ,

and v = l cosec:r dx= rkr = ^^a^-

The integrals can also be fonnd by using tanx as a new
variable: and the reader may find it instructive to obtain
the same results by this method.

The former integral [u) has some interesting applications,
and it may be convenient to note a few of its transform-

10

ations here ; for simplicity, we suppose the angle x to lie
between — \ir and + ^tt, so that t lies between — 1 and + 1 .
Then (5) gives at once

M = 2 tanlr' t or tanh \u = tan \x ;
and so

sinhM = tan:r, cosh m = sec x, tanhM = sinx,

leading to

e''=sec« + tan.r, e"^ = sec x — tana-,

as given in the table above.

Cayley proposed to write x = gd m, naming the function
after Gndermann, who made tables for its use : but this
notation is now seldom adopted. However, the function
u is in constant practical use by navigators and others who
work with Mercator's charts : for this reason its value
(multiplied by 10800 /ir, the number of minutes in the
radian) is given in all sets of nautical tables and some others
(such as Chambers's) under the title. " Meridional Parts."

III. iNTJiGEATION BY PAETS.

16. The fundamental formula is

\u dv = uv — \v du.

17. If X = le'^cosbx dx, Y= le'^sinbx dx, prove that

aX-bY=e'^cosbx, bX + aY=e''^smbx.
Deduce the values of X, Y, and so show that
X+i,Y = e'^lc if c = a + tb.
This result indicates that formula (3) remains true for
complex values of the index a.

18. Prove that

jar''-Mog^rfx = "^ [\Qgx-^,

Ixe'dx = (x-l) e% JxVrf.c = {x'— 2x + 2) e"',
^xVdx=[x''-7ix''-'+n(n-l)x"-'-...{-iynl]e''.
19. Find ^(s[o:^x)dx, ^x(sin'^x)dx, ^xsiaxdx,
J(tan"'x) dx, Ja- (tan"' a.-) dx.

11

IV. Integration by substitution.

Integrals contaiaiug the general quadratic

^ = az'+2bx-\- c

are often simplified by using the substitution v = ax + b, for
then

aX = v' + I), if D=-ac-b\

20. Thus, if i* is positive, we find «?a;/Z=(^«j/(t)' 4- D), or
. - rdx 1 , _Jax-\-b\

which leads to a convenient formula (57) for definite integrals.
When D is negative, the roots of X=Q are real, say a, /3
and we get the formula

which reduces (for a definite integral) to a form similar
to the first integral.

21. Similarly, if y^=a.r'+ 2bx-\-c, we find

d.r^ dv

y ~ s/a ^/(v' + U)

when a is positive. Hence, from (9),

{a) \^ = -j-log{y \/a + ax + b).

When a is negative and equal to — oSp we write

v^ = a^x — b, a^y''=D^ — 'o^,

where D= — D is now necessarily positive.
Then, from (6), we find

The apparent difference between the two types disappears
when definite integrals are considered as in (58).

12

22. It is instrnctive to note that in (21) x, y are simple
functions of the integral u = \dxly\ in fact, we find the
following results at once : —

(1) a>0, i'>0,

ax + 5 = i^D . sinh (u »/a), y \Ja = \/D . cosh {u \/a) ;

(2) a>0, Z><0, !> = -!>„

a.v + b = a/-D, • cosh (u ^/a), y \la = V-O, • sinh (m \ja) ;

(3) a<0, i><0, a = -a„ D = -D^,

a^x — b = \/Dj. sin (u^/a^), y '\/a,= \/-D|.cos(M V«,)-
From these formulae we deduce, if :;:„ y^, m, are corresponding
values of a.-, y, u.

>Ja. ^- = iaM\\[\\Ja(u — u^\ ia cases (1), (2),

or '\Ja^.'- ^ = tan{^\/a, (m — M,)j in case (3).

23. With the same notation as in (20)— (22), we have

the formulae

Cxdx 1 , „ b Cdx

Cx dx y h rdx
] y ~ a a] y '

34. By writing x- p^ljv, and then using (21), we can
prove that

C dx _ 1 , \yq + apx + b {p -\- x') -\- c\
]{x-p)y~ q ^1 x-p y

if ap'' + 25jo + c is positive and equal to q^. But if ajo' + 25/> + c
is equal to - q', we reduce the integral to the form

r dx _1 _i {apx + b {p + x) + c')
]{x-p)y~ q, ^° 1 q,y j"

And if ap^-\-2bp + c = 0, so that x—p is a factor of y', we

find

C dx \ y

](jio—p)y ap + x — p'

13

25. Apply (21) and (24) to the integrals

Cdx r dx

where (1) /=a,-+4x + 2, (2) y=-a;'+4A- + 6.

26. (a) If /= (x - a) (.« - /3), where :r > a > /3, prove that
Jy = 2 log [V(x - a) + ^{x - )3)!, cf. 21 (a),

by using the substitution x = o. cosh" - j3 sinli" ^.
When a>j3>x, use the substitution

a; = /3 cosh" — a sinh'' ^.
(6) Ify'=(a — ;r) (« — j3), where a>^>^, prove that

Jf = 2tan-'{y{'^)}, cf. 21 (i) and 22.

by means of the substitution ar = asin''0 + /3cos'^.

[c] If y^= {x — a) [x - /3), then (24) gives
C_dx____ 2 y _ 2 l/x-l3\

][x-oi)y (3-ax — a~(3 — a.y\x — aj'
which should be verified by differentiation.

27. If ^ = tan^.r, then prove that

r ^^' —of '^^

Ja + b cos a- J (a + 6 + (a - d) i" '

r ^-"^ — 9 / ^'^

Ja + 6cosx + csina;~' j(a + 6) + 2c^+ (a — 6j t^'

Complete the integration in the cases

(1) a = 5, 5 = 4, (2) a = 3, f) = 5,

(3) a=13, 5 = 4, c = 3, (4) a = 5, 5 = 7, c = l.

V. Eatioxal Algebraic Intkgrals.

When a rational fraction has been resolved into partial
fractions by the ordinary rules of algebra, the integration
can be carried out at once, except for fractions of the type

14

where ^ is a quadratic in x with complex factors. Such
fractions are reduced to (20) by nsing the formulge of (28)
below.

If the fraction originally proposed for integration is of
the form i2/Z""', where R, Z are polynomials in x, it is
usually simpler not to divide the expression into partial
fractions, but to assume

Z"*' dx \Z"I'^ Z'

where Y, Y, are polynomials of degrees 7ik—l, k — 1 respec-
tively (k being the degree of Z), with coefficients to be
determined. Thus we have the identity

E = Z^-nY'^+Y,Z'',
ax ax

from which the (n + l)k coefficients in Y, Y^ are found. This
method is illustrated by (33), (34) below : in each of these
we can foresee that even powers of x cannot occur in Y
nor odd powers in F,. The same method could be used
also for (32), but it is probably quicker to use (28) in
this case.

28. Write Z= ax' + 2bx + c, D = ac-b%fixid v=^ dx / X"*\

then prove, by differentiating [Ax + B)IX", and adjusting

the coefficients A, B, that we have the following reduction

formnlse

1 {ax-\-b . ,, ")

^»=2^i^--+('"-'^«M'

Deduce that

[x dx 1 \hx-\-c , , \

at

29. Prove that, if ar + c/3 - 2^5' = P,
f , , „ ,dx vx-\-q P [ax + b Cdx\

15

30. By taking w = i in (28) obtain the two standard
forms of (35)

f da; _ax + l> He dx _ bx -\- c

31. Showthatif«'-a: + l=Z

fx dx , /x — 2 \

and find »„.

32. Prove that

[ dx 3a;'+5x _ _.

/

33. Prove that
do- _^ X — x'

..■1_1_ ^2 . , ,a '6 TS"

+ i tanh- f^) + -^ tan- f^.V

34. Prove that, if Z = x*— 2x''cos2a4 1,

/ = ril±^- ^ _J_ tan- f^) ,
J A 2sma \l-a;V'

J ^ 2cosa V l+.fV

and show that

rrfx _ X (.rVos 2a — cos 4a) 1+4 sin^'a 1 f 4 cos'a
JX " 4jrsin''2a "^ 16sin'a "^ 16 cos'a '

VI. Irrational Algebraic Integrals.

As a general rule we cannot reduce to elementary functions
any integral in which the irrational element is more compli-
cated than the square-root of a linear or quadratic expression
in X. We consider, therefore, integrals of the type

where y^= ax'+ 2bx + c.

16

Any algebraic function f{_x, y) is reducible to the form

(-p Ti \

J, + -o- ) by the ordinary rules of algebra, where P, Q,

li, S are polynomials in x. Thus we need only consider
integrals of the form

{R dx_ {(rpU\ dx

where T is the quotient, U the remainder when R is divided
by /S ; we may regard U\ S as expressed in terms of partial
fractions, and then the new integrals to be discussed are
of the three forms

/■ ,^dx C I dx cAx + B dx

where -Z is a quadratic with complex roots. It will appear
from the reduction formulas (36)-(41) below that these
integrals can be reduced to the case n = 0. But the labour
involved in the third is almost prohibitive except in the
special case considered in (39) ; however, this covers the
practical applications which are of principal importance.

On account of the fact that we have often to deal with
the integrals ly'dx, ^dx/y", it is worth while to give a special
formula of reduction (35) for them, although this may be
regarded as a special case of (36) or (38).

35. Let y^=ax^+1bx + c, and D — ac — b'\ then verify
by differentiation that

{n + \)a ly"dx = {ax + 5) y" 4 nD Jy''"W«,
' dx ax + b C dx

-^ Cdx ax + b , , , f dx
Deduce that if m = [dxjy, then

Utandard forms.
tdx ax + o Cxdx bx + ci

17

lu the special case b = 0, we note the results
(?z + 1 ) J fd.'c = xf +ncj f dx,
C dx X , . [dx

{d.e

jydx = ^{xy + cu), l7=.7'

j/dx = }xi/ {2ax'+5c) + ^c'-u, J-^ = ^^ (2ax'+ 3c).

30. If we write

u^= \x — , with u^ = u,
prove that

(w + 1) aM„^, + (2?2 4- ] ) bu^ + '^^z'n-i = '^'''y-
Hence show that

au^ = y— bu, 2a'^u^ = y (ax — 3b) — (ac — 3b^) u.

Thus u^ can be reduced to the form yP^_^{x) + ku, where
P,., is a polynomial of degree (?e — l) in ,i', and k is a
constant.

37. Similarly, if we write
d.

,=\^r and .„=.,

V

■• jx-y
we have

(71 + 1) c«„^,+ (271 + 1) bv„+ 7iav,^_^=- y/*"*',

and so, in analogy with (36),

cv^ = -(yjx + bv), 2c\ = y (3bx - c)la-' +(3b''- ac) v.

In the special case c — we can use the reduction-formula
to evaluate v, and we find

bv = -yjx, (27i + l)bv^ + '2*Vi = "" yl^"*^^
so that », can then be found without using any logarithmic
function or inverse tangent.

38. To deal with the integral
r dx

18

we write x- p = .-Cj, aud tlieu

y'= ax^-^ 2 {ap + h) x^ + {ap^-\- 2bp + c).
We write, therefore, a„ b^, c^, x, for a, S, c, :» in the formulEe
of (37), where a^=a, b^ = ap + b, c^ = ap^+2bp + c.
The reduction formula is

{n + I) c^v,_^^ + (2?i + 1) b^v,^ + na^v^_, ^-yfic;"''.
The simple case c, = occurs when x—p is a factor of
y^, and then v^ can be found without using transcendental
functions.

39. li y''=ax^-{-c, X=ra:'-\-s, A = as - re, and the integral
to be evaluated is

_ r dx

the reduction formula is

2(re + 1) (5A)»„^,- (2?2+ l)(a6' + A)»„ + 2a?z2J^_, = -rxy/Z"*'.

In this way i\^ is reduced to depend on y,,, which we integrate
by the substitution t = yjx ; for then
r (/x r dt

which is of the type (4) or (5).
If the integral is of the type

r X dx

we write axdx=y dy, aX=ry' + ^, and the integral

becomes

dy

•'/(

l(r/+ A)"""
which is reduced by (28).

40. Consider now the general case of the integral ^dxl{Xy)
where X is a quadratic with complex roots.* We can then

* It is at present usual to handle this integral by the substitution v = y/X*;
but although the same variable can be used for the two integrals, yet the algebra
involved is no less than in the method suggested here, in any practical case.

This method has the further theoreutical Objection of introducing a second
square-root jX in the result. And according to a general theorem of Abel's, the
only root needed is y, whenever the integral can be expressed by elementary
functions.

19

always find a, ^ snch that

and if we write ^,=y/(x-^), t.^ = yj{x-a), we iiud
Hx - g) dx _ ^L_ r dt^ r (x - 0) d.if _ 1 r dt,
J Xij - a-0Jrt,'+^' j X^ "ie^ajs^T^^'
where A=ps -qr.

To find a, j3 when X, y' are given quadratics, we first
determine the two values of X for which A'- \y^ is a perfect
square in x: these values of X are known to be real and
unequal if the factors of X are complex (see Cambridge
Mathematical Tracts, No. 3, Oh. I.). Suppose that we have

X-xy=A{x-oi)\ X-\f=B(x-l3y,

then {\-X^)y'=-A(x-ay-^B{x-i3y,

and (X,, - \J Z = - A\ (x - a)" + 5\ (x - j3)'.

f dx
41. The general integral ^^^ can be reduced to (39)

by writing S, = (x- «)/(.).■ - /3) in the notation of (40). We
find then {iix>(3)

, J^ , =(oL-(5) ^•'""^^'^^" , where v' = J^f + ?,

and

(i-g)^_r«-/3y

so that we have

[ L{x-a.)^M{x-^) _ 1 r (Lg + iy)(l-g r

J A'"> '^-(a-^)-'] (rr + ^r''? ^"

42. Apply the formulee of (37) to prove tliat
{dx 3a' — 2 f fl?a;

iS

^"*j^' '^ y'=*''+^ + i.

'.r'y 4. 1- -^ '^ } xy'

and evaluate the last integral by changing the variable
to 1/a', or by (22).

20
By means of (38) show that

and prove similarly that

r dx _^y[2x — b) ^r dx

and evaluate the last integral by (22).

43. Apply the method of (40) to evaluate the integrals
''[x — \)dx [{x — i)dx

nx-\)dx n

Xy ' ] Xy ■■
where

(1) Z=3«'-10a- + 9, j/'=5;r'- 16^ + 14,

(2) jr=3.i;'-10;i-+9, y'= x'-%x -\-lO.

4:4:. Prove that, if a >j3,

C dx _ 2 _j //«+^\

''~J(x + aV(^-+/3)~V{a-^) VVa-jS/'

, r^___dx 1 f V(A- + /3) , 1

''"'^ J(x + a}V(^H-/i^) a-/3 t a- + a ^H

J(a'+a)(^+;«)f~~^^ tvi^Tpy^^j ■

45. Integrals of the types considered in this section can
also be reduced to integrals of rational fractions by sub-
stitution ; the general method is the following : —

Let [p, q) be a point on the conic y^= ax''+ Ihx + c, and
consider the other intersection of the conic with a variable
line through (/», — q). Thus write t = {x —p)l{y + q), then
X, y are rational functions of t, given by

U [q + rt) ~ q[\ + af) + 2rt ~ l-at" r-ap-^b,

dx . 2dt r. ,„„,

and 7 = 13^- «M22).

If q=0, the formulae are very simple : and the method has

01

usually been restricted to this special case, but the restriction
is not necessary.

46. In the case of a hyperbola (a>0), we can take
[p, q) as a point at infinity and use the lines parallel to an
asymptote, instead of lines through [p', q) : thus, if we write

th

en ax + b = ^(v j, t/ ^/a = ^(v + — ] ,

and — = -; , cf. 21 (a).

y sja V ^ '

ExAMPLltS OF SIMPLE PSEUDO-ELLIPTIC CASES.

Although an integral which contains the square-root of
a cubic (or bi-quadratic) must lead, in general, to elliptic
functions, yet it is sometimes possible to complete the
integration by means of elementary functions ; such integrals
are called pseudo-eLliptic, and a few simple examples are
given below. More complicated cases have been worked
out by Greenhill with a view to physical applications.

47. 1^ y^ = X [a [x"" + I) + bx] and «J=y/a-, verify that

Cx - 1 dx _ r 'idv rx + 1 dx _ r 2dv

]x+l 'J ~ ]v'-b-\-2a' ]x-\ "y ~ ]v'-b-ia'

and so evaluate the integrals

Cx'—l dx Cx^—\ dx Cx^+l dx

Similar methods apply to any integral of the type - — - dx,
where /(a:) is a fraction such that /(.?;) +/(!/«) =0.

48. Prove that if v^= x^ + x, then

[x-^\ dx .^ , , . rx \/2

= - V2 tanh ' ( — ^

}x-l y \ y

rii:i^-=v2tan-'m.

}xA-\ y \x^/2)

22

VII. Teigonometrical Integrals.

The cases of most common occnrrence are given in (49)-
(51) ; these formulEe show that the integrals can always be
reduced to (10)-(15) and (27).

When the integrand is more complicated, but is expressed
as a rational fraction in sin a,- and cos .7;, the integral can
always be transformed to an algebraic fraction by means
of the substitution

, , . 2t 1 - f , 2dt

t = tan -^x, sin x = ^ ^ ^^ , cos x = ^ ^ , ax = -,

l+f' 1+t" 1 + f

But it is often possible to bring the integral to simpler forms
by taking as the new variable

tauA-, or sin a;, or cos*-,

according to the form of the integrand ; for instance, when
the integrand is a rational fraction in tanj;, we should use
tana; as a new variable rather than tan^a:.

In other cases we can use trigonometrical transformations
to bring the integrand into a form similar to that of partial
fractions (for algebraic fractions). A common type is the
fraction

P^(sina:, cos a;)

nsin(x — a) '

where P^ is homogeneous of degree r in sina; and cosa;, and
there are n different factors in the denominator. Here there
are two types of partial fractions, when ji > ?-,

(1) ^—. — 7 r, when w - r is odd,

sm(a; - a)

(2) S-; ; r, wheu w — r is cvcu.

tan (a;— a)

The value of the coefficient A is found by multiplying by
sin(a; — a) and then writing x = a; this gives

A= P^(sina, cosa)/n'sin(a — j3),

the accented n' containing (?z — 1) factors, in each of which
a comes first.

23

The proof that such resolutions are possible, and the
necessary additional terms when n^i\ are ionnd most
quickly by using the complex variable
^ — Qix — COS g; ^ J sin g;_^

and then applying the ordinary rules of algebra. For
instance, if r = n, we find the equation

P„(sinas cos«)_^^^ A

n sin (a' — a) tan (,« — a) '

where the value of A is given by the same formula as
before if C is the real part of e"'P^(l, t) and o-=Sa. In
particular, if

P^(sinx, cosx) = n sin(x— ^),

where there are n factors in the product (not necessarily all
different), we find from this formula

C = cos (S6I - 2a).
When the values of a are complex, as in (53) below, this
method is not satisfactory, and the use of a new variable
is simpler.

49. If s = sin.r, c = cos.t', verify that

(m + n) [s'^d'dx = s"'"'c"-'+ (w - 1) ^s'V-'clx
= - .s'^''c"*' + im - 1) js"'-'c"dx,

(»« - 1) j^ «?^ = - ^1 - (?« - 1) Jp;Pi '

50. Useful special cases of (49) are given by taking
»2 = 0, M = 0, or m = n; then, with i! = tana-, we find the
following six cases : —

s in" X 71 Is" dx = - s"-'c +(n-l) ^s'"' dx,

cos" X n Jc" d.L- = + sc"-' + (w - l ) Jc""' dx,

dx

24

■' n-l '

sec x (?2-i)J— = _ + (re-2)j^-^,
cosec ^ (re_l) __ = __j + (^_2) U^.

] & S J s

51. If v^ = '\dxjX"*^, where X=a + 5cosa;, verify that
[n+ 1) {a'-F) w„^,- (2?2+l)a»„+?z»„_,= -5sina,-/Z".
More generally, if ^ = a + 5 cos « + c sin :r, we have

(n + 1) (a=- b'- c') »„„- (2w + 1) a»„ + wVi

= (— 5 sin ,r + c cos a;)IX''.

Alternative methods of transformation are suggested for
these (and other similar) cases in (55).

52. Reduce to partial fractions and so integrate the
fractions

P/sin(:c — a) sin(x — j8) and Q /tan (a; — a) tan (a; -j3)

where (1) P=l, or sin«, or sin^x,

(2) Q=l, or tanx, or cot^r.

If P is sin^^•, prove that the first fraction is equal to

. , „ , 1 f sin'a sin'jS ]

sm(a — /y) [sm(A- — a) sm(a; — /S)j

Integrate also the second fraction by using the variable
t = ta,Txx, and consider further the case with Q = tan'x.

53. By taking i; = tan-|x and applying (34), or » = cosa;
and using (40), prove that

(2cosa; — l)(^a; 1 > / 2i! a/2 '

r (2cosx — l)dx _ 1 /2ts/2\

J 2-4cos«+3cos'a' " V2 ^^ \1 + 3f)

i-tanh-( ^^^^°n ,

V2 \2-cos«;'

25

r (2 - cos a:) dx _, / 2^ \_ _j / sin.r \

J2 — 4cosa- + 3cos"a;~ ^^ \1 — 3tV " ^^ V2cos:j. — ij '

Similarly use the variable » = sinh.r to evaluate

f (1 +2sinh;i-)f/x , r (2 — sinh a;) ^a^

J 5 — 4 siuha; + 2 sinlfa; J 5 - 4 sinha? + 2 siuh'j: '

54, Fiud formulEe of reduction for

^cosnxcos'"aid.v, ^cosna:sin"'x-dx, etc.,
and verify that they can be brought to the forms
(m + n) Je'"^ cos"' a: dx= — le"'" cos™.r + m Je'("-i)^ cos"*-! x dx;
{m + n) Je'"* sin™ xdx = — te'"^ sin™ x-\- im Je'(''-i)« sin'"-! x dx.
So also prove that

f e'«^ , fe'(«-l)« fe'(n-2)a;

ax = 2 ;— dx — dx,

Jcos'^x Jcos™-!^ Jcos™.t'

f e'"" f e'(''-i)»^ fe'C»-2)a:

^ a« = 2t ^ ; — a*' + —. dx,

J sm'" X J sm™-! a; J sm™ x

55. If (a + b cos 6)(a — b cos (f) = d'— b'\ where a > and
a'>b'', and ^, ^ both lie between and v, verify that

dd d(h . sin sin
^ and -

« + 6cos^ '^(a^—b''), a + b cos 6 'J{a^—b'')'

If (a + b cosh u) (a — b cosh v) = d'—b'', where a>0 and
a' > 6\ and u, v are both positive, verify that

du dv , sinliM sinh»

and

a + bcoshu >J{d'—l)') a-\-bcoshu '^{d'—b')'

If (a + bsmhu)(a — bsmhv) = a'+b', verify that, when
a + b sinb u is positive,

du dv , coshw cosh»

and

a + 6sinhM s/{d'-^ b'') a + bsi\ihu -^{a^+h')'

Consider the various cases which arise when V > a\ and
either

(a + ^cos^) {a — bcQshv) =a'—b',

or (a + bco&hu){a — bcos^) = a^—b''.

26

VIII. Definite Integrals.

The followiug integrals can all be evaluated by the
fundamental foi-mula

j f{x)dx = F{b)-F{a), if \f{x)dx = F{x);

but in (58) the preliminary transformations of p. 12 may
be useful.

In the numerical examples (60) it should be remembered
that logarithms are taken to base e, and that angles are
expressed in circular measure. It is sometimes easier not
to use the tables, but to calculate by means of series such as

tan"'^- =x-\x^+\x^—..., i'A.n\r^x = x+^gx' + j^x^+...

1 x' 1.3 x^ 1.3.5 x'
sm-. = . + --+— -+^-^g- + ...

. ,_, 1 x^ 1.3 x' 1.3.5 x'
smh x = X — I- — — I- . . .

2 3 2.4 5 2.4.6 7

which are obtained (whenx'<l) by integrating the series
for 1/(1 ±.«'), (1 ±x'yK (See p. 32 below.)

When the indefinite integral is found by a change of
variable » = <^ {x), reducing it to the form J^ (») dv, then we
shall have as a rule

f{x) dx=^ \ g (v) dv, if a = (^ (a), j3 = ^ (b).

But it sometimes happens that ^' (a;) changes sign between
a, b ; and if so, it is usually necessary to subdivide the
integral into two or more parts, tlie poiuts of division being
the values of x at which these changes take place.
For instance, we have from (10)

sin :!; fl?:r = cos — cos 7r = 2.

J

But if we try to calculate the integral by changing the
variable to 2) = sinx, we should have to put it in the form

r dv f" —dv_ /■' vdv _

VV(r^^)^J/V(T^^)-'i„V(T^^)-''
the point of division being at v = l, corresponding to a- = ^7r,
where (j)' (x) changes sign.

27

A different point arises in connexion with the trans-
formation ^ = taua:, or t = ia,n^x; for instance, consider the
former and suppose that x varies from to ir. The limits
for t are then both : and this appears to be absurd,
until we note that as x approaches ^tt from the left (that
is, through values less than ^tt), t tends to + oo , while
t tends to — 00 , when a: approaches ^tt from the right.
Consequently t must vary first from to + oo , and secondly
from — 00 to ; and this explains the apparent absurdity.

56. Prove that

f" 'sinarf.i'

J _j 1 — 2,K cos a + x"
is equal to + -^tt if < a < tt, and is — ^tt if w < a < 27r.

57. Prove from (20) that if ac - b''=p',

p. dx ^ 1 ^^^_, [ P(x,-^,) ]

where the angle lies between and tt, if x^p^x^ and a > 0.

But if ac-V——p^, we get a similar formula with
tanh"' in place of tan"' and ^, in place of p ; only care
must be taken to see that both roots of the quadratic fall
outside the range of integration.

58. If y'=az"+ 1hx-vc, prove from (22) or (45) that

r^ = Atanh- p--"°-^^i , if a>0;

when a is negative, we replace sja by s/^—a) and tanh"'
by tan"'.

Also r'_^ = ltanh-I ^1^^ \

if ap'+2bp-\-c is positive and equal to q' ; when this
expression is negative, we use V(- {ap'+ '^bp + c)\ and tan"'
in place of tanh"'. In applying this formula it is important
to note that p must not fall between x^ and .f,.

28

59. If a, b both lie betweea and 1, prove from (58)
that

J _, Vi(l - 2aA' + a') (1 - 2bx + b'\ 'J[ab)
and evaluate the integral if a > 1 > ^ > 0.

60. As exercises on numerical work prove that

[ V(-i-' + 4)(^:r = 8-96, [ V(^'- 4) rfx = 6-89,

i ^/(4:-x'')dx=l■n.

■'

61. Prove that

riir rijr

sin^« dx = ^TT = cos':i- dx.

Jo J a

This result is of constant application in physical problems :
it is conveniently stated in the form : —

The mean value of sin'iu or cos'"'*- (integrated over any
multiple of ^tt) is \.

62. If m, n are unequal integers, prove that

smmxamnxdx = (i, j cosmx cosnxdx = 0.

■' ■'o

r*" • 2„ 7 1.3.5...(2w-l) TT

63. sin'"xdx = ^ '

J '^

■'

2.4.6...2re 2

r*' • »„.. J 2.4.6...2W

sm'" 'a- dx=--—- — J- -T ,

J„ 3.5.7...(2n+l) '

where n is an integer ; these follow at once from (50).
64. Prove that the integral

riir

f{m, 71)= sin'"a;cos''a;£^a;

J

satisfies the relationy(»2, 7i) —J(n, m) ; and that (49)
m + n)/'(m, n) = {m— '^)f{m, — 2,n) = (n — l)f{'m^ n - 2).

29
Deduce that when m, n are both even integers

but that when m is odd

f(m, n) = i .

■^ ^ ' ^ {n + l){n + 3)...im + n)

"When n is odd, there is a similar formula which can
be found by interchanging m, n.

65. Prove from (54) that

C08"'a: COS w A- dx = cos"' 'x cos (w — i ) jj: c^x,

Jo m + n}^

and so find the integral when vi'>n. In particular
cos"a.- cos nx dx = ■

/:

2"*'
adx

67

I -^ — ^ = ±i's-, according to the sign of a.

I ^ — ^7 = - tan"' \\\ , if »' = ac - V,

J„ ax'+25.r + c /» \hl ^

where a, ac — V are positive and the angle lies between
and IT. Hence also

r dx -n

J -00 «*•'+ '^hx + c p '

68. If Z= a;r' + 25a; + c, and «„ = f dxjX"'', it follows
from (28) that

(2?2-l)a r"a;«?.r_ (2??- 1)6

l)a C xdx _ I

2?2/ "-" J_„JL"'' Inp' ""'

Thus we find from (29) and (67)

J -00 JL ip

1.3...(2w-l) Tra"
^""^ ^"^ 2.4... 2« 7^'-

30

69. (a) If X=X+x*-2x^ cos 2oL, where 0<a<7r, prove
by using Ijx as a new variable that

"dx rx'dx f'l+x\ IT

rdx _ rx'dx _ r
J„z-J„^~-J„

the final result being found from (34).

{b) Again by using 1 /a; as a new variable

70. By writing x = f, and then applying the method of
partial fractions, we can prove that

r x'-' _ r 2nf""dt _ -77
J„ i^"^^-]^ 1 + r -sin(a,7)'
where a is a proper fraction of the form \{2m + l)/2wj, ?w and
n being integers and m < ?z. The final result is true when
a has any value between and 1 ; but the proof depends on
more advanced methods, except in cases such as 69 (b).

71. (a) Prove from (30) that if a, c are positive, and
b + \J[ac)'>0, we have

f" dx — I C ^^^ 1

^^ J „ [ax' + 2bx + c)tf ~ ^*J „ (a:r' + 26a7 + c)l ~ s/(ac) + i "

(5) Prove from (58) that, if 0<a<7r,

r" (^a; _ a

J, (a; + cosa)\/(^" — 1) sina'

Verify by writing x= coshw, t = e'", and using (57).

72. As easy applications of (58) we note the following :
(a) If y- = {x -a){b — x) (a<x< h),

then — = TT,

(6) If j/' = (a; — a) (ic — §), and a<6, then

£1— -V(H)-»'j:f-'»""Vc^:).

according as x is less than a or greater than b.

31

dx

J„a + ocosx V(a -^) ^ '^ ' ^ -"

■'

siix'xdx

a + dcosx a + \/{a' — b')'

„ - [^^ dx^ i^ / -^ n^

],p'cos'x^q'5iu'x~M ■^^
by taking tan x = t as & new variable.

75. If we write {p, q>0),

„_ r^" cos''xdx c_[*' sin'xofx

J „ p^ cos'' a- + g'' sin'jc ' ] o P' cos^r + q' sin'' a:

prove that C + S = ^Trl(pq), p'C +q''S=^iT^
and deduce that pC=qS=^Trl{p + q).
Show that, if c>a>0,

['' ^/(a'-x')dx IT

6. If a, (5 are positive and r' = a' + b' — 2ab cos:!:, prove

a —b cos.r)

that

r d

{b) sin^^ -

•'0

, , r . dx

W J^ sin^ —

[d] {a — b cos ^) sir

where in each case the first value is to be taken when a>b,
the second when a < §.

r" a r b

77. e''^cosbxdx = ~^^rj-j^, J e''^ sinbx dx = ■^^-^, ,

where a is positive. These two are contained in the single
equation

[ e'^^dx^- (if c = a + tb).

Jo ^

ax IT

or 0,

2a'

°^ 26' '

2
a

2
or ^,

dx 2
r a

or 0,

■'

32

71 ^

'dx=-:^^ if a>0.
a

This result is valid when a is replaced by c=a + t5; but
a simple proof requires more advanced methods than
for (77).

_„ r°° dx TT .„

79. — =-— =— if a>0.

J„ coshaa- 2a

IX. Integration of Series.

It is impossible to give any detailed explanation here of
the conditions under which term-by-term integration of an
infinite series is permissible : some of the more useful tests
will be found in my book on Infinite Series (Arts. 44, 45,
175, 176). But the majority of ordinary cases can be tested
by the simple rule : —

If for all values of x from a to b, J„{x) is numerically
less than i/„, where M^^ is a positive constant, and if the
series Slf is convergent, then the equation

fjV.{^)]da: = 2j''/Jx)dx,

will be correct.

Thus for instance the two series

1 — f
(a) — --, = l + 2t cos X + 2f cos 2x + 2;;' cos 3.!; + . . . ,

t SIQ X

(B) - — 5= tsmx+ <'sin2x+ f sm3x + ...,

^' ' 1 -2tcosx + t

in which 0<i<l, can be compared with the series of
positive constants if ; and so term-by-term integration is
allowable. This same test can still be applied when the
series are multiplied by co&px, where p is any integer.
Again, if x is numerically less than c (where c < 1) the

series

H[l-x'') = \ + x' + x' + ...,

l/(l + x') = l-x'' + x*-...,
l/V(l-.^•^) = l + ^x' + i^^^ + ...,

33

can be compared with 2c", and so term-by-term iategration
can be applied to obtaia the series quoted on p. 26. Clearly
a similar test can be applied to any convergent power-series.
It is usually possible to establish the differentiation of a
series, by applying the comparison test to the differentiated
series.

80. Apply the series (a), (|8) above to deduce (76) (a), [b).

81. Deduce from (a) that (/>, q being positive)
^'^ = 1 - 2icos2j:-f 2i;*cos4A'-... ,

p cos x-^ q sin x
where t=[p — q)/[p-{-q). Hence obtain (74) and (75).

82. Deduce from (/3) that if n is an integer

r • / tsmx \ „

sin nx — -, ax = -kirt .

J„ \l-2(tcosa; + i;7

Hence prove that if a', 6' are both less than 1,

r sixi'wdx _ ""■

J„ (1 +a'-2acos«)(l+6'-2(5cos:i-) "" 2{l-ab)'

83. Deduce by differentiating (a) that (if <^<l),

(l-QsiUA- ^ gj^^ ^ 2if sin 2a' + U' sin3« + ... .
{\-2tcosx + ty

Hence prove that

r sin^x cosxdx _ t<

j„ (1-2;; ~c,osx + ff ~ 2(1-0 ■

84. Dednce by integrating (a) or (|3) that

log ( 1 - 2;; cos a- + O = - 2 (« cos a: + !«' cos 2a' + -t (' cos 3.1; + . . . ),

/•TT

and so log(l - 2i cos«+ f)c?ar = 0,

r log ( 1 - 2< cos X + f) coanxdx = - Trf /w.

85. The results obtained in (82)-(84) can be extended

34

to cases where the integrand depends on a + b cos .v, (a>0,
a' >5*J by writing a = c{l +f), b = - 2ct, so that

^/{a'-F) = c{\-f), aud 2e=a^ ^J{a.' -h').
Thns we find for instance

r

log {a + b cos x) dx = IT logc,

— irb

, f sinxcosa; , — tto

J „ (a + i cos x)' icW{a' - 6')

X. DlFFBKENTIATION AND INTEGRATION OF 1nTE!GRALS.

From any integral containing a parameter we can obtain
others by differentiation or integration with respect to the
parameter ; but difficulties sometimes arise in the case of
infitiite integrals (that is, integrals which extend over an
infinite range, or which have an infinite integrand). For
some of the more useful working rules in such cases, see
my Infinite Series (Arts. 171, 172, 177] ; but even without
a knowledge of these tests we can still regard these methods
as useful for suggesting results. For example, (78) can
be derived from (77) by differentiation with respect to a ;
but this could not at present be regarded as a complete
proof, because the range extends to infinity. In like manner
the results of (68) are suggested from (67) by differentiation
with respect to a, b, c. It has sometimes been suggested
that the method of differentiation should be applied instead
of formula? of reduction to cases such as (28), (29), (38),
(41) : but in general this process is tedious and in the case
of (41) it is hardly practicable.

86. By differentiating (73) with respect to a and b,

prove that

r dx Tra

J „ [a+bcosxf ^ [a:' - b')i '
r cosxdx %'nab

{a + bcoBxY {a' -by

sin' a: cos .r , —Trb

— dx =

/„ {a + b cosa:/' ' ^c'^J[d' — b'') '
where 2c = a + \J [a" — b') ; compare (85).

35

87. Verify the accuracy of the equation (85)

I log (a + 5 COS .«) = TT log c,

by differeatiation with respect to a and b.
Deduce also (76) [d) from (76) (c).

if i^(x) is a polynomial of degree not greater than n, and
c>6>a.

XL Simpson's Formula.

89. If y is a cubic function of «, and y,, y^, yj are the
values of y at the beginniug, middle and end of an interval
of length I, then the integral of y over this interval is
equal to

y(y> + 4y, + y3)-

90. If y is any function f[x), which does not vary
rapidly in the interval I, the last formula can be applied as
an approximation to the integral ; the error involved can
be put in the form

2880'^ ^^''

where 5 is some point within the interval. Thus if f" {x)
is positive throughout the interval, the approximation will
be too large.

91. Apply (90) to calculate the values of the integrals
in (60), and estimate the order and sign of the error in each.

92. The approximation can be improved by dividing the
range of integration into n parts and applying (90) to each
of them. In this way the interval I is divided into 2m parts,
and the values of y must be found at each point of division ;
we denote these values by y,, y,, y,, ..., y,„^„ and then the
ap^proximation to the integral is

36

The error is less than Ml^l(28S0n*), if M is the greatest
numerical value oi f"{x) in the interval.

93. An important application of (90) is to find the area
of a segment of a circle of large radius. If h is the height
of the segment, and I the length of the chord, we can write
y, = 0, y, = h, y, = : and the rule then gives as the
approximate area

A = llh.

The exact formula is J = \r'Q — ^{r — h) l\, where

'2rh = lV-^h' and tanJ0 = 2A/^.

We can estimate the accuracy of the approximate
formula by expanding A in powers of t=2hjl, which gives

^ = ^ (^ + rXI - 37577 + 57779 —•)•
Thus if k does not exceed ^l, the approximation

will generally suffice, for practical calculations. For such
segments the original form of A is not very convenient
in numerical work ; because A is given by the difference of
two nearly equal areas.

94. Another practically important application of (90) is
to the calculation of volumes and centroide. If a solid is
obtained by the revolution of a straight line, or of an arc of
a conic (about one of its axes), the area of a section at
distance x along the axis of revolution is expressible in the
form

S^ax' ■\-2bx+ c,

and so the exact volume is given by

V = ISdx = \l (S, + S,+ AS,),

where S^, S, are the areas of the ends, and S, of the middle
section. When the value of the coefficient a is known, we
may use the equivalent formula

V=ll(S, + S,)-ial\

derived from 3^ = I (S^ + S,) — \al\

The centroid is the same as that of 3 particles propor-
tional to -Sj : 4*Sj : S^ placed at the corresponding points
on the axis of revolution.

For a cone (of angle a), a=7r tan'a ; for a sphej'e, a=—ir\
for a paraboloid^ a = 0.

95. The formnlaa of (94) apply also to a solid bounded
by planes, two of which (the ends) are parallel ; it is then
necessary to interpret I as the distance between the ends.

Examples are given by a mound, with horizontal top and
bottom, and plane faces ; or a straight embankment on a
hill-side, in which the ends are vertical planes perpendicular
to the length of the bank. Another case is a tetrahedron
in which two opposite edges are regarded as the ends.

XII. Planimeteic Formulae.

96. When a rod of length I receives a differential
displacement, prove the equation

dS,-dS=l{.d<i -\dp\

where dS^, dS, are the polar elements of area (about a fixed
origin 0) traced by the ends of the rod, d<T is the displace-
ment of the centre of the rod perpendicular to itself, and
p is the perpendicular from to the rod.

97. In the planimetric applications, the end 1 of tlie
rod is made to oscillate to and fro along an arc of a curve
(generally a circle or straight line) and the displacement z
perpendicular to the rod is registered at some point K on
the rod, while the end 2 traces out a closed curve, the rod
being brought back to its initial position without making
a complete turn. Thus

dff = dz + cdO,
if Q is the inclination of the rod to a fixed line, and c is
the distance of K from the centre of the rod : and so
dS= dS, + lidz + cde - \dp).

38

But wlien the rod returns to its original position, the
total change in S^, 6, and p is zero in each case ; thus

S, = Iz,

which gives a mechanical means of finding /S„ by
observing z.

98. If in a planimeter the end 1 describes a complete
circle of radius a, the rod at the same time making one
complete torn, then

S^= w (a' + 2lc) + Iz ;

but if the end 1 oscillates along a straight line, while the
rod makes a complete turn

5, = •2Trlc + Iz.

99. Deduce from (96) that if A, B, C are three points
in order on a rod, and Ǥ,, S^, S^ are the areas of three
closed curves described by A, B, C respectively, in a single
turn of the rod, then (Holditch's Theorem)

aS^ + cS.^ —[a + c)S^ = irac {a + c),

where a = BG, c = AB.

In particular if the ends A, C move on the same curve

/S, — (Sj, = "Trac.

100. Dednce from (99) that if A, B, C, the vertices of
a triangle, trace closed curves of areas S^, S^, S^, while a
fourth point P in the plane ABC describes a curve of
area S^, then the expression

IS^ + mS, + nS^ -{l + 'm + 7i) <S,,

is independent of the curves traced out, P being the
centroid of masses I, m, n placed at A, B, C.

By taking the points A, B, C, P as fixed in turn, the
value of the expression can be found to be

TT {a^mn + ¥nl + c^lm) j [l + m + n) = — irf (I + m -\- n),

if t is the tangent from P to the circle ABC.

BY THE SAME AUTHOR.

AN mTROBUCtrON TO THE THEORY OF
' INEINITE SEfilES.

8vo. 15j; net,

Mathemdtical GascWi;.-^" Mr. Bromwich has rendered an
important service to mathetftatical students and teachers by
writing an elaborate and scholarly treatise divided explicitly to
series and to allied portions of the theory of limits. . ,;. Of
the importance and viaiue of Mr. Bromwich's book as a store-
house of accurance information oil the subjects withwhick it
deals there can hardly be two opinions. . . Mr. Bfomwich's
book, is decidedly more complete, and in other ways more
satisfactory, than any of the continentar treatises with wlrich it-
is at all comparable."

Athenaeum. — "We bdieye that Professor Bromwich's book
contains the most complete account which has been, published
in English o^ the theory p^ infinite series. The style qt the
work is exceedingly clear, so thatTve can recommend it without
reservation as a most useful addition to any mathematical
library."

Educational TMnes. — By its publication the student's library
acquires another yery important accession to its mathematical
section. . . There is every reason to'believe that Professor
Bromwich's treatise will be invaiuable to those for whom it is
intended." . '

JJwflr^jaB.-T-*' We welcome the publication dt the present
volumcf and are confident that it vwill meet a very real need.
The reputation of the author is a sufficient guaranty that the
methods of proof employed are lucidj scholarly, and rigorous."

Honion : Macmillan & Go. Ltd.

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