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8/10/2019 Tables and Formulas by WR Longley 1915 50s
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TABLES AND
FORMULAS
FOE SOLVING NUMERICAL
PROBLEMS
IN
ANALYTIC
GEOMETRY,
CALCULUS
AND APPLIED
MATHEMATICS
ARRANGED
BY
WILLIAM
RAYMOND
LONGLEY,
PH.D.
ASSISTANT PROFESSOR OF
MATHEMATICS
IN
THE SHEFFIELD
SCIENTIFIC
SCHOOL,
YALE
UNIVERSITY
REVISED
EDITION
GINN
AND
COMPANY
BOSTON
NEW
YORK
CHICAGO
LONDON
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COPYRIGHT,
1913,
1915,
BY
WILLIAM RAYMOND
LONGLEY
ALL
BIGHTS
RESERVED
615.2
Cftc
GINN
AND
COMPANY
PRO-
PRIETORS BOSTON U.S.A.
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PREFACE
This
collection
of
tables
and
formulas is
intended
for
use as
a
handbook
for
solving
numerical
problems
in
connection
with
the
courses
in
mathematics
in
technical schools and
colleges.
The
numerical
tables
are
those
which
have been
found
by
experience
to
be
most
useful
in
performing
the
calculations
necessary
for
careful
plotting
of
various
algebraic
and
tran-
scendental
curves
in
rectangular
and
polar
coordinates
(includ-
ing
parametric
representation),
for
the
approximate
solution
of
equations,
particularly
transcendental
equations,
and
for
many
types
of
problems
in
calculus and mechanics. Much
emphasis
is
being given
to
numerical
problem
work
illustrating
the
mathematical
theory
in
engineering
schools,
and
it
is
believed
the
student
should
have
every
time-saving
facility
for
obtain-
ing
numerical results
of
a
given
degree
of
accuracy
commen-
surate
with that
required
in
practical
work. For this
purpose
the
more
extended tables are
usually
too
cumbersome.
The
formulas
are
selected
from
algebra,
trigonometry,
geom-
etry,
and calculus
(including
a
short
table
of
integrals),
and
the
collection
includes
those
that
have been
found
useful
in
the
courses
in
mathematics
in
the
Sheffield
Scientific
School.
It is
not
intended
to
provide
the
student
with a
compendium
of
all
formulas that
he
may
require,
so that
problem
work
becomes
mere
substitution of
numbers
in
equations,
but
to
give
those
which
save
time
and relieve
the
burden
on
the
memory. Following
this
idea certain
formulas
have
been
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iv
PREFACE
omitted
;
for
example,
the mass
of
a
body
of
variable
density,
center
of
gravity,
and moment
of
inertia.
It
is
impossible
to
apply
these
formulas
to
a
practical
problem
without
a
thorough
understanding
of the fundamental theorem
of
the
Integral
Calculus.
If this theorem
has
been
mastered
the
formulas
are
unnecessary.
W. R.
LONGLEY
NEW
HAVEN,
CONNECTICUT
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CONTENTS
TABLES
PAGE
NUMERICAL
CONSTANTS
1
COMMON
LOGARITHMS.
BASE
10 2
NATURAL VALUES OF TRIGONOMETRIC
FUNCTIONS
AND
THEIR
LOGARITHMS
TO
THE
BASE
10
6
RADIAN
EQUIVALENTS
OF
DEGREE
MEASURE
... ...
7
NATURAL
VALUES OF
TRIGONOMETRIC
FUNCTIONS
FOR
ANGLES
IN
RADIAN
MEASURE
8
SQUARES
AND
CUBES;
SQUARE
ROOTS AND
CUBE ROOTS
. .
9
RECIPROCALS
14
NAPERIAN LOGARITHMS
14
CONVERSION OF
LOGARITHMS
14
EXPONENTIAL
AND
HYPERBOLIC
FUNCTIONS
15
FORMULAS
ALGEBRA AND GEOMETRY
16
TRIGONOMETRY
17
ANALYTIC GEOMETRY 18
CALCULUS
19
FORMULAS
FOR DIFFERENTIATION
21
TABLE
OF
INTEGRALS
23
DIFFERENTIAL
EQUATIONS
36
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TABLES
AND
FORMULAS
NUMERICAL
CONSTANTS
TT
=
3.14159.
7T
2
=
9.86960.
=
1.77245.
-
=
0.31831.
7T
=
=
0.10132.
7T
=
0.56419.
1
radian
=
0.1 radian
0.01 radian
1
degree
1 minute
e
(Naperian
base)
log
e
10
=
M
1
Iog
10
TT
=
0.49715.
log
e
TT
=
1.14473.
57.296
degrees.
3437.75 minutes.
206264.8
seconds.
5
43' 46.5 .
34' 22.6 .
0.01745
radian.
0.00029 radian.
2.71828.
2.30259.
Angle
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COMMON LOGARITHMS.
BASE 10
N.
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COMMON
LOGARITHMS.
BASE
10
N.
50
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COMMON
LOGARITHMS.
BASE
10
N.
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COMMON
LOGARITHMS.
BASE
10
N.
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TRIGONOMETRIC
FUNCTIONS
NATURAL
VALUES
OF
TRIGONOMETRIC
FUNCTIONS AND
THEIR
LOGARITHMS TO
THE
BASE
10
De-
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RADIAX
EQUIVALENTS
OF
DEGREE
MEASURE
Deg.
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NATURAL
VALUES
OF
TRIGONOMETRIC
FUNCTIONS FOR
ANGLES
IN RADIAN
MEASURE
Rad.
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SQUARES
AND
CUBES
9
SQUARES
AND
CUBES;
SQUARE
ROOTS
AND
CUBE
ROOTS
No.
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10
SQUARE
ROOTS
N.
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SQUARE
ROOTS
11
N.
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12
CUBE
HOOTS
N.
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CUBE
ROOTS
13
N.
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14
RECIPROCALS,
NAPERIAN
LOGARITHMS
Table of
Reciprocals
of
Numbers
from
1
to
10,
at
Intervals of .1
N.
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EXPONENTIAL
AND
HYPERBOLIC
FUNCTIONS
15
Hyperbolic
sine
of
x
=
sinh
x
=
Hyperbolic
cosine of x cosh
x
=
Hyperbolic tangent
of
x
=
tanh
x
=
2
e* e-
x
&
-f
e-f
X
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COLLECTION
OF
FORMULAS
FORMULAS
FROM
ALGEBRA
AND
GEOMETRY
'1.
Binomial
Theorem
(n
being
a
positive
integer):
[2
[3
n(n-ln-2...n-r
r
l
2.
n =[n=1.2.3.4...(n-l)n.
3. The solution
of the
quadratic equation
ax
2
+
bx
+
c
is
b
V6
2
4
ac
2a
4. When
a
quadratic
equation
is reduced
to
the
form x
2
+
P%+
q
=
0,
p
=
sum
of
roots
with
sign
changed,
and
q
product
of
roots.
5. In
an arithmetical
series,
I
=
a
+
(n
-
l)d
s
=
-
(a
+
1)
=
-
[2
a
+
(n
-
l)d].
6.
In a
geometrical series,
rZ a
a
(r
n
1)
Z
=
ar
n
-
1
;
s
=
-
=
-
--
-
r
_l
r
i
7. Circumference
of
circle
=
2
TIT.*
8. Area
of
circle
=
-m*
2
.
Area of
sector
=
l
r
2
o:
(radians)
.
9.
Volume
of
prism
=
Bh.
10. Volume
of
pyramid
=
^
Bh.
Frustum
=
-
(J5
+
B'
+
VBB').
11. Volume
of
right
circular
cylinder
=
Trr
2
h.
12.
Lateral
surface
of
right
circular
cylinder
=
27irh.
13. Total surface of
right
circular
cylinder
=
2
irr(r
+
h).
14.
Volume
of
right
circular
cone
=
^7rr
2
h.
15.
Lateral
surface of
right
circular
cone
=
irrs.
*
In
formulas
7-20,
r
denotes
radius,
h
altitude,
B area
of
base,
s slant
height,
and
a,
&,
c
the
semiaxes
of
an
ellipse
or
ellipsoid.
16
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FORMULAS
FROM
TRIGONOMETRY
17
16.
Total
surface of
right
circular cone
=
7rr(r
+
s).
17.
Volume of
sphere
=
f
Trr
3
.
18.
Surface
of
sphere
=
4irr
2
.
19.
Area of
ellipse
=
Trab.
20.
Volume of
ellipsoid
=
FORMULAS
FROM TRIGONOMETRY
21
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18
TABLES AND
FORMULAS
36. sin*
=
J
1
-
008
*;
\
2
tan*=
A
E^.
x
_
/I
+
cos
x
2
\
1
+
cos
x
37. sin
x
+
sin
y
=
2 sin
|(x
+
y)
cos
1
(x
y}.
38. sin
x
sin
y
=
2
cos
^
(x
+
y)
sin
^
(x y).
39.
cosx
+
cos?/
=
2
cos
i
(x
+
y)
cos
l
(x
y).
40.
cosx
cos?/
=
2
sin
1
(x
+
2/)
sin
^(x
?/).
41.
=
-
=
;
law
of
sines.
Area
=
\bc
sin^4.
sin
A
sin
B
sin
C
42.
a
2
=
6
2
+
c
2
2
6c
cos
J.
;
law
of
cosines.
FORMULAS FROM
ANALYTIC
GEOMETRY
43.
d
=
V
(Xj
x
2
)
2
+
(y
1
y
2
)
2
;
distance -between
points
(x
1?
y)
and
2 2/2)-
44.
d
=
AXl
+
^
+
C
-
;
distance
from
line
Ax
+
By
+
C
=
to
(x
1?
y.).
;
(x,
y)
is the
point
dividing
the
line
p
A
,
jPa
in
the ratio
\.
46.
x
=
x
+
x
x
,
y
=
y
+
y
7
; transforming
to new
origin
(x
,
y
).
47.
x
=
x
x
cos
^
2/
x
sin
0,
y
=
x' sin
+
y'
cos
6
;
transforming
to new
axes,
making
the
angle
9
with
old.
48.
x
=
p
cos
#,
y
=
/o
sin
#
;
transforming
from
rectangular
to
polar
coordinates.
49.
p
Vx
2
+
y'
2
,
=
arctan-
;
transforming
from
polar
to rectan-
gular
coordinates.
50.
Equation
of a
straight
line
:
(a)
Ax
+
By
+
C
=
0,
general
form
;
(&)
y
y
l
=
m(x
x
x
),
slope-point
form
;
(c)
x
cos
a
+
y
sin a =
p,
normal form.
51.
tan
6
=
;
angle
between
two
lines
whose
slopes
are
m^
andm
2
.
l
+
m
*
m
*
m
l
=
m
2
when
lines
are
parallel
;
m
l
=
when
lines
are
perpendicular.
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FORMULAS FROM
CALCULUS
19
52.
(x a)
2
+
(y
/3)
2
=
r
2
;
equation
of
circle
with
center
(a,
/3)
and
radius
r.
53.
Area
of
the
triangle
P
1
(x
1
,
y^,
P
2
(x
2
,
y
2
),
P
3
(x
3
,
y
8
)
is
54.
Equation
of a
parabola
with
vertex at the
origin
and
focus on
the
x-axis
is
y
2
=
2px.
55.
Equation
of
an
ellipse
with
center at the
origin
and
foci
on the
x-axis
is
b
2
x
2
+
a
2
y
2
=
a
2
b
2
.
56.
Equation
of
a
hyperbola
with
center
at the
origin
and
foci on the
x-axis
is 6
2
x
2
-
a
2
y
2
=
a
2
6
2
.
57.
Locus
of
^.x
2
+
Cy
2
+
-Dx
+
Ey
+
F
=
is
parabola
if
A
=
or
C
; ellipse
if
A and
C
have
same
sign
;
hyperbola
if
A
and
C
have
opposite signs.
FORMULAS FROM
CALCULUS
58.
Radius of curvature. Center of
curvature,
(a)
Rectangular
coordinates.
,
y
dx
2
rTj-2 r-
/j
x
P,
,
(dy
(b)
Polar
coordinates.
+
^j~
(c)
Parametric form.
dx
d
2
y
dy
d
2
x
eft
dt
2
~~
~dt dt
2
59.
Plane
area.
(a)
Rectangular
coordinates.
(b)
Polar
coordinates.
A
=
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20
TABLES AND
FORMULAS
60.
Length
of
arc.
(a)
Rectangular
coordinates.
(6)
Polar
coordinates.
61. Volume
of
solid
of
revolution about
the
JT-axis.
V
=
62. Area of
surface
of revolution about the
X-axis.
63.
Taylor's
Series.
64.
Maclaurin's
Series.
/(*) =/(0)
+
L
(a)
+
-
12
[8
65.
Standard
Series.
z'
2
z
3
z
4
z
3
,
z
5
z?
=z
-
+
-
z
2
z
4
z
6
COSZ
=
1
-^-^
+
' '
z
2
z
3
z
4
108(1
+
*)
=
*-?-
+
?--^...,
(1
+
)
=
+
-z
+
1.2
-Z
2
+
z
8
z
5
z
7
slnh2= ,
+
_
+
_
+
_
Z
2
Z
4
2
6
r*-
1+
i
+
.g*g
Convergent
for
all values of
z,
all
values of
z,
all values
of
z,
-Kz
1,
all values of
z,
all
values of
z.
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FORMULAS
FOR
DIFFERENTIATION
21
FORMULAS
FOR DIFFERENTIATION
In
these
formulas
w,
u,
and
w
denote
variable
quantities
which
are
functions of
x.
I
*=0.
dx
.
n
*
=
i.
dx
d
.
du
dv
die
HI
_(
+
_,)
=
_
+
___.
IV
<
CT
)
=
r-
dx
dx
d
.
du du
V
(uv)
=
u
--
}-
v
dx
dx
dx
VII
/^iy-mpi-i,.
dx dx
Vila
(a;)
=
nx-
1
.
dx
du
du
dx\u
VIII
a
d /u\
_
dx
dx\c/
c
dv
viii6
du
d
..
dx
IX
(log
a
)
=
log
a
6
---
dx
u
du
lo^-fi.
X
dx dx
d
.
,
dw
du
XI
M|J
=
VU
V
~
l
--
1-
lOf U
U
r
--
dx
dx
dx
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22
TABLES
AND
FORMULAS
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TABLE
OF
INTEGRALS
23
TABLE
OF
INTEGRALS
SOME
ELEMENTARY
FORMS
1. C
(du
dv
dw
-
)
=
C
du C dv
C
dw
.
2.
Cadv
=
aCdv.
4.
C
x
n
dx
=
?
+
C,nj*-l.
J
J
J
n
+
1
3.
f
df(x)
=
ff(x)
dx
=
/(x)
+
C.
5.
f
|?
=
log
x
+
C.
FORMS
CONTAINING
INTEGRAL
POWERS OF
a
-f-
bx
7. a
8.
iF(x,a
+
bx)dx.
Try
one of the
substitutions,
z
=
a
+
&x,
xz
=
a
+
bx.
9.
10.
=
ti(
a
-t
te)
2
-
2a(a
+
6x)
+
a
2
log
(a
+
to)]
+
C.
11.
x
(a
+
bx)
a x
12.
r
J
x
2
(a
+
bx)
ax a*
x
(a
15
C
dx
__
L_
1
\o~
a
+
bx
,
c
J
x
(a
+
to)
2
a
(a
+
to)
a
2
x
is.
r
^
=
ir--J_
+
__
__
:l
+
c
.
J
(a
+
6x)
3
6
2
L
a
+
to 2
(a
+
te)*J
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24
TABLES AND
FORMULAS
FORMS CONTAINING
a
2
+
x
2
,
a
2
x
2
,
a
+
6x
2
,
a
+
bx
n
17.
.
a
2
+
x
2
a
a
1
+
x
2
dx 1
.
a
+
,
~ /*
dx i
x
a
19.
f
-
-
=
tan-
1
x
\
-
+
C,
when
a
>
and
&
>
0.
Ja
+
&x
2
Va&
20.
dx
i
z
ow
2ab
a
ox
21. I
x
m
(a
+
bx
n
)Pdx
,
L I
x
m
~
n
(a
+
bx
n
)pdx.
b(np
+
m
+
1)
b(np
+
m
-f
1)
J
s*
22.
23.
bx
n
)P (m
_(mn+np
l)br
dx
24 .
r
^^
/
x
m
(a
+
6x
w
)p
1
m
n
+
nr> 1
/
x
m
(a
an(p
l)x
m
-
1
(a
+
frx ^-
1
an(p
1)
/
x
w
(a
+
bx
n
)p
/(a
+
bx
n
)Pdx
_
(a
+
xw
'f
c
m
a
(m
1)
x
w
-
:
6
(m
n
np
1)
/*
(<x
+
bx
n
)Pdx
a
(m 1)
/
x
m
-
n
(a
+
bx
n
)Pdx
(a
+
6x
n
)^
anp
/
(a
+
bx
n
)P
-
27.
C-?*
J
(a
+
a
+
6x
M
)p
6
(m
np
+
1)
(a
+
a(m
w
+
1)
r x
m
~
n
dx
a(m
w
+
1)
r x
~~
b(m
np
+
l)
J
(a
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TABLE
OF
INTEGRALS
25
x dx
V.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
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26
TABLES
AND
FORMULAS
_
2(8
a
2
-
4
a6x
+
36
2
x
2
)
^
3.
f
J
Va
15
b
3
dx
1
.
Va
+
bx
Va
.
=
=
log
=
+
C,
for
a
>
0.
r
dx
,
44.
/
=
log
-
-
*
x
Va
+
bx Va
Va
+
bx
+
Va
^
x
Va
+
6x V a
tan-
46
f
dx
_
Va
+
bx b
ax
b
r
dx
2a
J
x
-y/
a
_j_
^x
2
Va
+
bx
/Va
+
tedx
I r
dx
-
=
2Va
+
6x
+
a
/
.
*
J
xVa
+
6x
FORMS CONTAINING
Vx
2
+
a
2
48.
49.
50.
51.
52.
53.
55.
56.
=
-
Vx
2
+
a
2
+
22
+
Vx
2
+
a
2
)
+
C.
f(x
2
J
f
(x
2
+
a
2
)t
dx
=
-
(2x
2
+
5a
2
)Vx
2
+
a
2
+
log(x
+
Vx
2
+a
2
)
+
C.
J
8
8
C
(x*
J
n
+
1
m
+
2
f
x
2
(x
2
+
a
2
)*dx
=
-
(2
x
2
+
a
2
)
Vx
2
+a
2
-
log(x
+
Vx
2
+
a
2
)
+
C.
f
^ -
=
log(x
+
Vx^+^)
+
C'.
J
(x
2
+
a
2
)*
/
dx
(x
2
+
a
2
)
dx
xdx
(x
2
+
a
2
0.
a
2
x
2
(x
2
+
a
2
)*
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TABLE
OF INTEGRALS
27
=
X
+
log
(x
+
Vx
2
+
a
2
)
+
C.
(x
2
+
a
2
)f
VxM-a
2
/ dx
1
x
58. I-
-
=
-log
=
x(x
2
+
a
2
)*'
a
a
+
Vx
2
+
a
/dx
-2//>.2 i
59
+
a
2
x
2
(x
2
+
a
2
)*
a
'
x
60./
61.
x
3
(x
2
(x
2
+
2a
2
x
2
2 a
3
.(x
2
+
a
t/
X
2
FORMS CONTAINING
Vx
2
a
2
a
2
63.
f
(x
2
-
a
2
)^dx
=
-
Vx
2
-
a
2
-
log
(x
+
Vx
2
-
a
2
)
+
C.
J
2
2
3
a
4
64.
/
(x
2
-
a
2
)idx
=
-
(2x
2
-
5a
2
)Vx
2
-a
2
+
log(x
+
Vx
2
-a
2
)+C.
/
8
65.
dx.
n
+
2
_
(x
2
-
a
2
)
2
n+2
,,
66. /
x(x
2
a
2
)
2
dx
67.
fx
2
(x
2
-
a
2
)^dx
=
-
(2
x
2
-
a
2
)
Vx
2
-a
2
-
log (x
+
Vx
2
-a
2
)
+
C.
Jo
8
es.
r_*
/
/^.o
=
log (x
+
Vx
2
-
a
2
)
+
C.
69.
/
70.
/
dx
(
X
2
_
a
2
)t
a
2
Vx
2
a
2
xdx
(x
2
-
a
2
)*
=
Vx
2
-
a
2
+
C.
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28
TABLES AND
FORMULAS
-
-
Vx
2
a
2
-f
log
(x
+
Vx
2
a
2
)
+
C.
+
log
(x
+
Vx
2
a
2
)
+
C.
=
sec-ix
+
C.
72.
C
J
(x
2
-
a
2
)f
Vx
2
-
a
2
ftn r
dx
1
,
x
r
73. I
=
-sec~
1
-
+
<7;
/
J
~/~2
~2\4-
a
a
J
-1
/
75.
/-
dx
x
2
(x
2
a
2
/~5
-
5 1
=
Vx
2
a
2
a cos-
*
-
+
C.
a
2
.
/
y-r
-
TN.
-
+
log
(x
+
Vx
2
a
2
)
+
C.
76 -
/-
^
77.
{=*
X
2
FORMS CONTAINING
Va
2
x
2
78.
f
(a
2
-
Z
2
)*dx
=
-
Va
2
-
x
2
+
-
sin-i-
+
C.
J
2
2
a
79.
f
(a
2
-
x
2
)tdx
=
-
(5a
2
-
2x
2
)
Va
2
-
x
2
+
sin-i-
+
C.
J
CL
80.
. .
n
+
1
n
+
1
J
n
+
2
_(a
2
-x
2
)
2
81.
Tx(a
2
-x
2
)
2
dx
=
-^
^~
/
n+2
82. f
x
2
(a
2
-
x
2
)^
dx
=
-
(2
x
2
-
a
2
)
Va
2
-
x
2
+
sin-
1
-
+
C.
J
8
8
a
83.
f
=
sin-i-+<7;
f
^
=sin-ix
+
C.
J
/g2
x
2
^
ft
V^l
iC
2
dx
(a
2
x
2
)
t
a
2
Va
2
x
2
85.
/
xcZx
(a
2
-
=
_
Va
2
-
x
2
+
C.
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TABLE
OF
INTEGRALS
29
86.
f
X2dx
(
2
-x
2
)*
x
2
dx
x
(
a
2
_
X
2)f
Va
2
-
x
2
88.
f_J^_
=
_^l Va^r
/
/
a
2
_
^axi
m
89.
/
dx
1.
=
-log
x
(a
2
x
2
)^
a
a
+
Va
2
x
2
C7.
90.
/
91
r
J
x
2
(a
2
-
-x
2
///72
e
93.
_
r-z
5
a
+
Va
2
x
2
dx
=
Va
2
x
2
a
log
+
C.
2
-x
2
.
,x ^
sm-
1
-
+
C.
x
a
FORMS
CONTAINING
V2
ax x
2
,
V2
ax
-f
x
2
94.
/V2ax
x
2
dx
=
V2 ax x
2
+
vers-
1
-
+
C.
2
2
a
95.
f
^
-=vera-ig4.(7;
f
^
v2
ax x
2
a
J
/
2
x x
2
96.
97.
/-
<
dx
m+2
V2
ax x
2
m
-l)aJ T
-i
dx
x
J
V2ax
x
2
(2m
l)ax
m
'
(2m
l)a
x
m-i
2ax
x
2
98
x
dx
_
x
t
-
1
V2ax x
2
(2m
1)
a
/
x
m
~
l
dx
ax
x
2
w m
J
V2
ax
x
2
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30
99
100
101.
102.
103.
104.
105.
106.
107.
108.
109.
110.
111.
112.
113.
TABLES
AND
FORMULAS
/
x
m
dx
_
(2
ax
x
2
)?
m 3 r
V
2
ox
-
(2
m
3)
ox
(2
m
3)
a
J
x-
1
x
V2 ax
x
2
.
|
x
V2
ax
x
2
dx
=
/
/
/
(2m-3)ox^
(2m-
3)
3
a
2
+
ox
2
x
2
'2ax-x
2
+
2
a
dx.
x
'2
ax
-
x
2
xdx
V2
ax
-
x
2
=
V2
ax x
2
+
a
vers-
*
-
+
C.
a
V2ax-x
2
2
r 2x
x
^
_
^/
2
ax
x
2
+
a
vers-
1
-
+
C.
J
x
a
2
ax
x
2
,
2
V2 ax
x
2
/
-
vers-
l
-+
C.
a
3
ax
3
.
(2
ax
x
2
)f
a
2
V2
ax
x
2
f
xdx
=
_
x
=
+
C.
J
(2ax
x
2
)i
a V2
ax x
2
fF(x,
V2
ax
-
x
2
)
dx
=
F(z
+
a,
Va
2
-
z
2
)
dz,
where
=
a
-
/*
dx
=log(x
+
a4-
V2ax4-x
2
)4-C'.
J
V2ax
+
x
2
fF(x,
V2
ax
+
x
2
)
dx
=
^(2
-
a,
Vz
2
-
a
2
)
dz,
where
z
=
x
+
a.
FORMS CONTAINING
a
+
6x ex
2
a
+
bx
+
ex
2
V4
ac
6
2
V4
ac
f
_
*5__
=
_
1
Ja
+
to
+
cx
2
V6
2
-
2
-4ac
2cx
+
6
+
fi
2
-
4ac
when
6
2
>
4
ac.
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114.
116.
117.
118.
119.
120.
r
^
J
a
+
bx
ex
2
y&
2
+
TABLE
OF
INTEGRALS
log
31
4
ac
V6
2
+
4ac
2
ex
+
6
^
=
-L
log
(2
ex
+
t>
+
2
Vc
Va
+
bx
+
ex
2
)
+
C.
/
a
+
bx
+
cx
2
Vc
O
nrf
I
7)
Va
+
fee
+
cx
2
dx
=
^
Va
+
bx
+
ex
2
4c
6
2
-
4
ac
2cx+_
c
log(2cx
+
6
+ 2VcVa+6x+cx
2
)+
C.
/Va
+
to
-
cx
2
dx
=
2CX
~
Va
+
bx
-
ex
2
J
4c
/
Va
+
6x
+
ex
2
V
a
+
te
+
ex
2
.log (2
ex
+
6
+
2VcVa
+
6x
+
ex
2
)
+
(7.
2cf
/-
xdx
V
a
+
6x
ex
2
6
.
,
2
ex
:
--
1
--
sin- C.
OTHER
ALGEBRAIC
FORMS
121.
122.
123.
124.
+
(a
6)
log
(Va
+
x
+
Vb
+
x)
+
C.
+
C.
V(a
_
x) (6
+
x)
+
(a
+
6)
sin-
=
_
V(a
+
x)(6-
x)-
(a
+
6)sin-i
C.
125.
f-
V(x
/3
a
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32
TABLES
AND
FORMULAS
EXPONENTIAL
AND
TRIGONOMETRIC FORMS
126.
Ca
x
dx
=
-^
+
C.
129.
f
sin
xdx
-
cos
x
+
C.
J
log
a
J
127.
|
e
x
dx
=
e?
+
C.
130.
J
cos xdx
=
sin
x
+
(7.
128.
f
e*dx
=
+
C.
131.
ftan
xdx
=
log
sec
x
=
log
cosx
+
C.
132.
foot xdx
=
log
sin
x
+
C.
133.
f
sec
xdx
=
C-^L
=
log
(sec
x
+
tan
x)
=
log
tan
(-
+
-\
+
C.
/
/
cosx
\4
2/
//*
dx ijC
cosec xdx
=
/
=
log
(cosec
x cot
x)
=
log
tan
-
+
C.
t/
sin
x
2
135.
/ sec
2
xdx
-
tan x
+
C.
138.
rcosecxcotxdx=
cosecx+
C.
136
.
fcosec
2
xdx
=
cot x
+
C.
139.
Tsin
2
xdx
=
-
--
sin
2
x
+
C
J
J
2 4
137.
fsec
x tan xdx
=
sec x
+
C.
140.
fcos
2
xdx
=
-
+
-
sin
2
x
+
C.
J
J
2 4
2
xdx.
-..
/
.
,
smw-ixcosx
nl/*.
141.
/ sm
M
xdx
=
1
/
sm -
J
n
n
J
142.
fcos-xda
=
*-
1
+
gnl
rcos^-^xdx.
J
n
n
/
11^1 r
^
^
cosx
n2r
dx
J
&m
n
x~ n
1
sin -
1
*
n
1
J
sin -
2
x
1
.
. r
dx 1
sin
x
n
2
r
dx
t/
cos
w
x
n
1 cos
n i
x
n 1
J
cos
n
~~
2
x
145
. Tcos
1
x sin
n
xdx
=
^-^
1
^
|cos
m
-
2
x sin
w
xdx.
J
m+n
m
+
n/
146.
|cos
m
x sin
w
xdx
=
'
'
cos
m
x
sin
n
~
2
xdx.
m
+
n
147.
f
ta
=-J
___
J
J
sin
m
xcos
n
x
w 1
sin^-ix
?n
+
n
2
/
dx
nl
J
sin
m
xcos
w
-
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148.
149.
150.
151.
152
153.
154.
155.
156.
157.
158.
159.
160.
161.
f-
J
si
dx
sin
m
xcos x
TABLE OF
INTEGRALS
1 1
m
1
sin-ix cos ~
1
x
ra
+
n
2
33
n
2 /
-1
J
I
dx,
2
x
sin
m
-
2
x
cos
x
/cos
m
xdx
_
cos
m
+
1
x
m
n
+
2
r
cos
m
x
~
sin x
(n l)sin-
1
x
n 1
J
sin-
;
/COS
OT
xdx
_
cos
m
-
*
x
m
I
r
sin
x
(m n)
sin
-
1
x m
nJ
/cos
+
*
x
sin
x cos xox
=
h
C.
siu
n
x
n
+
l
.
f
J
rin-x
cos
arete
=
tan - x
n-1
1
X
cot
-
2
xdx.
fcot
xcZx
=
-
COtH
X
-
f
J
n
lJ
r
.
sin
(m
+
n)
x
sin
(m n)
x
/
sin rax sm
nxdx
=
--
J
H
--
}
--
'
+
C.
J
2
(m
+
n)
2
(m
-
n)
r
sin(m
+
n)x sin(m
n)x
/
cos
mx
cos
nxcfcc
=
-
5
_
-|
--
5
--
>
+
c.
J
2
(7ft
+.
n)
2
(m
-
n)
/
.
cos(?n
+
n)x cos(m ri)x
I
sm
mx cos nxdx
=
--
s
--
-
-
'-
\-C.
J
2
(m
+
n)
2
(m
n)
f
^
=
/
a
+
6
cos
x
_
52
f
J
a
a
+
a
tan
-
+
V&
+
a
+
6cosx
/dx
+
6 sin
x
a
2
log
+
C,whena<6.
/
6
a
tan
--
V
6
+
a
2
atan|
+
6
tan-
1
^^^^- +
C,
when
a
>
6.
6smx
-b
2
a
tan
-
+
b
V&
2
a
2
log
h
C,
when
a<ft.
~
a2
a
tan-
+
&
+
V^^
2
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34
TABLES AND
FORMULAS
162.
f-
*
-
=
tan-i^Uo.
a
2
cos
2
x
+
^sin
2
x ab
\
a
/
,
r
e
flKC
(asinnx
ncosnx)
~
163.
/
e*sm Tixdx
=
* *
+
0.
J
a
2
+
n
2
164. -eoBKrto
=
+
g
t/
a
2
+
w
165.
fxe^dx
=
(ax
-
1)
+
0.
/
a
2
166.
167.
m
log
a
m
log
a
168
r
aa *B
__
__
og
r
a
x
z
'
J
x
m
(m
l)x
w
-
1
m It/ x
7
- 1
- _
rt
/
e
5
cos
n
-
l
x
(a cos x
+
n sin
x)
169.
{
e
cu
cosxax
=
-
\
-
-
'-
*
J
a
2
n
2
a
2
+n
2
1
70.
Cx
m
cos axdx
=
2
(ax
sin
ax
+
m cos
ax)
a
2
/x
m
~
^
x
m
sin
ax
=
: (m
sin
ax
ax
cos
ax)
_m(m-l)
r
a
2
/
LOGARITHMIC
FORMS
171.
flog
xdx
=
x
log
x
x
+
C.
172.
f-^L
=
log
(log
x)
+
logx
+
ilog
2
x
+
....
J
logx
2
J
173.
r-^-log(logx)
+
C.
J
xlogx
174.
f
J
n
+
1
(n
+
1)
2
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TABLE
OF INTEGRALS
35
175.
a
a
x
176.
Cx
m
\og
n
xdx
=
-
logx
--
-
fx
nt
log
n
-
1
xdx.
J
m+1
m
+
1
*
rx
m
dx_
x
m
+
l
m
+
1
r
x
m
dx
J
log
n
x
(n
l)log
n
~
1
x
n I
J
log
n
~
l
x
r
b
APPROXIMATE
EVALUATION OF
I
=
I
f(x)dx.
J
a
The
interval
(db)
is divided
into
n
parts,
each
equal
to
Ax.
Let the
abscissas
of
the
points
of
division be
x
=
a,
x
x
,
x
2
,
x
=
6.
The
corresponding
values of
/(x)
are
Trapezoidal
rule
(any
number of
parts)
Simpson's
(parabolic)
rule
(even
number
of
parts,
n
even)
I
=
(y
Q
+
4
Vl
+
2y
2
+
4y
8
+
2y
4
+
-
.
.
+
y,)
o
Note.
The
integrals
containing
Vx
2
+
a
2
(48
62)
can
be
expressed
in
terms
of
the
inverse
hyperbolic
sine
by
the relation
log(?
The
integrals
containing
Vx
2
a
2
(63
77)
can be
expressed
in
terms
of
the
inverse
hyperbolic
cosine
by
the
relation
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36
TABLES
AND
FORMULAS
DIFFERENTIAL
EQUATIONS
1. The
differential
equation
of
HARMONIC
MOTION.
x
=
ct
+
c
2
e-
x
=
Acoskt
+
Bsinkt.
The
general
solution
may
be
written in
the
following
forms
:
(a)
(&)
We
give
to
(6)
another
form,
thus
:
Draw
a
right
triangle
with sides
A
and
B.
Since
A
and
B
are
arbitrary
constants,
this
right
triangle
is
arbitrary,
and
hence also the
hypotenuse
C
and
the
angle
/3.
Now,
A
=
C
sin
/3,
B
-
C cos
/S,
and
substitution
in
(b)
gives
x
=
C
(sin
j8
cos
kt
+
cos
j8
sin
fc),
or
(c)
x
=
C
sin(kt
+
j8).
If
in
(c)
we write
for
/3,
ft'
+
,
we obtain
x
=
(d)
35
=
In
these formulas c
1?
e
a
, -4, .B,
C, /3,
/3'
denote
arbitrary
constants.
2.
~
k
2
x
=
Q.
The
general
solution
is
3.
The
differential
equation
of DAMPED VIBRATION..
The
general
solution
is
x
=
e~^(A
cos
Vfc
2
-/i
2
+
-B
sin
VA:
2
-/*
2
),
or
x
=
Ce-
***
cos
(VA;
2
tft
+
j8)
.
4.
The
differential
equation
of
harmonic motion
with a
constant
dis-
turbing
force.
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DIFFERENTIAL
EQUATIONS
37
The
general
solution
is
x
=
A
cos
kt
+
B
sin
kt
H
--
,
or
fc
5.
The differential
equation
of FORCED VIBRATION.
(a)
-
+
k
z
x
=
L
cos nt
+
M
sin
n,
where
n
96
k.
The
general
solution
is
x
=
A
cos
kt
+
B
sin to
H
--
cos nt
-\
--
sin
ni,
fC
-~
Tl
/C
71
where
-4
and
B
are
arbitrary
constants.
tf
2
rr
(6)
(
L
+
]
C
2
X
=
L
cos to
+
Jf
sin to.
The
general
solution is
L M
x
=
A
cos
to
+
-B
sin
to
+
sin
to
--
cos to.
2A;
2fc
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a
*>
a
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