P
r
e
f
The pur-pose of this handbook is to supply a collection of mathematical formulas and tables which will prove...
75 downloads
921 Views
26MB Size
Report
This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form
P
r
e
f
The pur-pose of this handbook is to supply a collection of mathematical formulas and tables which will prove to be valuable to students and research workers in the fields of mathematics, physics, engineering and other sciences. TO accomplish this, tare has been taken to include those formulas and tables which are most likely to be needed in practice rather than highly specialized results which are rarely used. Every effort has been made to present results concisely as well as precisely SOthat they may be referred to with a maximum of ease as well as confidence. Topics covered range from elementary to advanced. Elementary topics include those from algebra, geometry, trigonometry, analytic geometry and calculus. Advanced topics include those from differential equations, vector analysis, Fourier series, gamma and beta functions, Bessel and Legendre functions, Fourier and Laplace transforms, elliptic functions and various other special functions of importance. This wide coverage of topics has been adopted SOas to provide within a single volume most of the important mathematical results needed by the student or research worker regardless of his particular field of interest or level of attainment. The book is divided into two main parts. Part 1 presents mathematical formulas together with other material, such as definitions, theorems, graphs, diagrams, etc., essential for proper understanding and application of the formulas. Included in this first part are extensive tables of integrals and Laplace transforms which should be extremely useful to the student and research worker. Part II presents numerical tables such as the values of elementary functions (trigonometric, logarithmic, exponential, hyperbolic, etc.) as well as advanced functions (Bessel, Legendre, elliptic, etc.). In order to eliminate confusion, especially to the beginner in mathematics, the numerical tables for each function are separated, Thus, for example, the sine and cosine functions for angles in degrees and minutes are given in separate tables rather than in one table SOthat there is no need to be concerned about the possibility of errer due to looking in the wrong column or row. 1 wish to thank the various authors and publishers who gave me permission to adapt data from their books for use in several tables of this handbook. Appropriate references to such sources are given next to the corresponding tables. In particular 1 am indebted to the Literary Executor of the late Sir Ronald A. Fisher, F.R.S., to Dr. Frank Yates, F.R.S., and to Oliver and Boyd Ltd., Edinburgh, for permission to use data from Table III of their book S T tf B a Aao i b a gtMy o R l n ir e l e e d si d 1 also wish to express my gratitude to Nicola Menti, Henry Hayden and Jack Margolin for their excellent editorial cooperation. M. R. SPIEGEL Rensselaer Polytechnic Institute September, 1968
o
s s
tc i
CONTENTS
Page 1.
Special
Constants..
.............................................................
1
2. Special Products and Factors ....................................................
2
3. The Binomial Formula and Binomial Coefficients .................................
3
4. Geometric Formulas ............................................................
5
5. Trigonometric Functions ........................................................
11
6. Complex Numbers ...............................................................
21
7. Exponential and Logarithmic Functions .........................................
23
8. Hyperbolic Functions ...........................................................
26
9. Solutions of Algebraic Equations ................................................
32
10. Formulas from Plane Analytic Geometry ........................................ ...................................................
34 40
11.
Special Plane Curves........~
12.
Formulas from Solid Analytic Geometry ........................................
46
13.
Derivatives .....................................................................
53
14.
Indefinite Integrals ..............................................................
57
15.
Definite Integrals ................................................................
94
16.
The Gamma
Function .........................................................
..10 1
17.
The Beta Function ............................................................
18.
Basic Differential Equations and Solutions .....................................
19.
Series of Constants..............................................................lO
20.
Taylor Series...................................................................ll
21.
Bernoulliand
22.
Formulas from Vector Analysis..
23.
Fourier Series ................................................................
..~3 1
24.
Bessel Functions..
..13 6
2s.
Legendre Functions.............................................................l4
26.
Associated Legendre Functions .................................................
.149
27. 28.
Hermite Polynomials............................................................l5 Laguerre Polynomials ..........................................................
1 .153
29.
Associated Laguerre Polynomials ................................................
30.
Chebyshev Polynomials..........................................................l5
Euler Numbers ................................................. .............................................
............................................................
..lO 3 .104
7 0 ..114 ..116
6
KG
7
Part
I
FORMULAS
THE
GREEK
Greek
name
G&W
ALPHABET
Greek name
Greek Lower case
tter Capital
Alpha
A
Nu
N
Beta
B
Xi
sz
Gamma
l?
Omicron
0
Delta
A
Pi
IT
Epsilon
E
Rho
P
Zeta
Z
Sigma
2
Eta
H
Tau
T
Theta
(3
Upsilon
k
Iota
1
Phi
@
Kappa
K
Chi
X
Lambda
A
Psi
*
MU
M
Omega
n
1.1 1.2
= natural
base of logarithms
1.3
fi
=
1.41421
35623 73095 04889..
1.4
fi
=
1.73205
08075 68877 2935.
1.5
fi
=
2.23606
79774
1.6
h
=
1.25992
1050..
.
1.7
&
=
1.44224
9570..
.
1.8
fi
=
1.14869
8355..
.
1.9
b
=
1.24573
0940..
.
1.10
eT = 23.14069
26327 79269 006..
.
1.11
re = 22.45915
77183 61045 47342
715..
1.12
ee =
22414
.
1.13
logI,, 2
=
0.30102
99956 63981 19521
37389.
..
1.14
logI,, 3
=
0.47712
12547
19662 43729
50279..
.
1.15
logIO e =
0.43429
44819
03251 82765..
1.16
logul ?r =
0.49714
98726
94133 85435 12683.
1.17
loge 10
In 10
1.18
loge 2 =
ln 2
=
0.69314
71805
59945 30941
1.19
loge 3 =
ln 3 =
1.09861
22886
68109
1.20
y =
1.21
ey =
1.22
fi
=
1.23
6
=
15.15426
=
0.57721
56649
1.78107
r(&)
=
79264
2.30258
190..
12707
6512.
9852..
00128 1468..
1.77245
2.67893
85347 07748..
.
1.25
r(i)
3.62560
99082 21908..
.
1-26
1 radian
1.27
1”
=
~/180
radians
.
= =
.. .
57.29577 0.01745
..
7232.
.
69139 5245..
.. = Eukr's co%stu~t
[see 1.201
.
38509 05516
II’(&) =
180°/7r
.
02729
~ZLYLC~~OTZ [sec pages
1.24
=
.
50929 94045 68401 7991..
01532 86060
F is the gummu
=
.
99789 6964..
24179 90197
1.64872
where
=
..
8167..
.O
95130 8232.. 32925
.
101-102).
19943 29576 92.
1
..
radians
THE
4
BINOMIAL
FORMULA
PROPERTIES
OF
AND
BINOMIAL
BINOMIAL
COElFI?ICIFJNTS
COEFFiClEblTS
3.6 This
leads
to Paseal’s
[sec page 2361.
triangk
3.7
(1)
+
(y)
+
(;)
+
...
3.8
(1)
-
(y)
+
(;)
-
..+-w(;)
3.10
(;)
+
(;)
+
(7)
+
.*.
=
2n-1
3.11
(y)
+
(;)
+
(i)
+
..*
=
2n-1
+
(1)
=
27l
=
0
3.9
3.12
3.13
-d
3.14
MUlTlNOMlAk
3.16
(zI+%~+...+zp)~ where
q+n2+
the
mm,
...
denoted
+np =
72..
by
2,
=
FORfvlUlA
~~~!~~~~~..~~!~~1~~2...~~~
is taken over
a11 nonnegative
integers
% %,
. . , np fox- whkh
1
4
GEUMElRlC
FORMULAS &
RECTANGLE
4.1
Area
4.2
Perimeter
OF LENGTH
b AND
WIDTH
a
= ab = 2a + 2b b
Fig. 4-1
PARAllELOGRAM
4.3
Area
=
4.4
Perimeter
bh =
OF ALTITUDE
h AND
BASE b
ab sin e
= 2a + 2b 1 Fig. 4-2
‘fRlAMf3i.E
Area
4.5
=
+bh
OF ALTITUDE
h AND
BASE b
= +ab sine
*
ZZZI/S(S - a)(s - b)(s - c) where s = &(a + b + c) = semiperimeter
b Perimeter
4.6
n_
L,“Z
.,
.,,
= u+ b+ c
Fig. 4-3
:
‘fRAPB%XD
4.7
Area
4.8
Perimeter
C?F At.TlTUDE
fz AND
PARAl.lEL
SlDES u AND
b
= 3h(a + b) = =
/c-
a + b + h
Y&+2 sin 4 C a + b + h(csc e + csc $)
1 Fig. 4-4
5 / -
GEOMETRIC
6
REGUkAR
4.9
Area
= $nb?- cet c
4.10
Perimeter
=
POLYGON
inbz-
FORMULAS
OF n SIDES EACH CJf 1ENGTH
b
COS(AL)
sin (~4%)
= nb
7,’ 0.’ 0 Fig. 4-5
CIRÇLE OF RADIUS
4.11
Area
4.12
Perimeter
r
= & =
277r
Fig. 4-6
SEClOR
4.13 4.14
Area
=
&r%
OF CIRCLE OF RAD+US Y
[e in radians]
T
Arc length s = ~6 A
8
0 T Fig. 4-7
RADIUS
4.15
OF C1RCJ.E INSCRWED
r=
where
&$.s-
tN A TRtANGlE *
OF SIDES a,b,c
U)(S Y b)(s -.q) s
s = +(u + b + c) = semiperimeter
Fig. 4-6
RADIUS- OF CtRClE
4.16
R=
where
CIRCUMSCRIBING
A TRIANGLE
OF SIDES a,b,c
abc 4ds(s - a)@ -
b)(s - c)
e = -&(a.+ b + c) = semiperimeter
Fig. 4-9
G
4
A
=.
4
P
.
&
sr s =
2e
s
1=
n +
1
=
FE
3 ise n
7
r n
OO
6
ni a
2 nr s i y 8
2r
RM
0
n
n ri i n
M7E
UT
°
2
r mn z
e
t
e
!
?
Fig. 4-10
4
A
=.
4
P
.
= 1 n r t a eL T n
t rZ n
n =
2e
2
t
9 r 2 a n a! 0
2 nr t a
=
2
n
n ri a n
T
!
I : e?
r m nk
T
t
e
0 F
SRdMMHW W
4
o .s
A
f=2 h +
pr
( -ae s
C%Ct&
e) 1 a r
e
OF RADWS
ra i
d2
4
i
-
g
1
T
tn
e T
e
d r
tz!? Fig. 4-12
4
A
=.
4
P
.
r
r
2
a
e
2
2 4 1 - kz rs
e c3
b
a
7r/2
=
e 5 4a
ii
m +
l
e
@
t
e
0 =
w
k = ~/=/a.h
4
A
4
A
l
[
27r@sTq See
p
e254 f
=.
$ab
r
2
.
ABC
r = e -&2dw
a
n a
e
r
to
4
c +n E5
p
u g
e
ar
p
m e
b F
r
4e
l
i
-r
o e g
a 4
gl 1
a )
tn
+
h
AOC
@
T
b Fig. 4-14
- f
1i
GEOMETRIC
8
RECTANGULAR
4.26
Volume
=
4.27
Surface
area
PARALLELEPIPED
FORMULAS
OF
LENGTH
u, HEIGHT
r?, WIDTH
c
ubc Z(ab + CLC + bc)
=
a Fig. 4-15
PARALLELEPIPED
4.28
Volume
=
Ah
=
OF CROSS-SECTIONAL
AREA
A AND
HEIGHT
h
abcsine
Fig. 4-16
SPHERE
4.29
Volume
=
OF RADIUS
,r
+
1 ---x
,-------
4.30
Surface
area
=
4wz
@ Fig. 4-17
RIGHT
4.31
Volume
4.32
Lateral
=
CIRCULAR
CYLINDER
OF RADIUS
T AND
HEIGHT
h
77&2
surface
area
=
h
25dz
Fig. 4-18
CIRCULAR
4.33
Volume
4.34
Lateral
=
m2h
surface
area
CYLINDER
=
OF RADIUS
r AND
SLANT
HEIGHT
2
~41 sine =
2777-1 =
2wh
z
=
2wh csc e Fig. 4-19
.
GEOMETRIC
CYLINDER
=
OF CROSS-SECTIONAL
4.35
Volume
4.36
Lateral surface area
Ah
FORMULAS
9
A AND
AREA
SLANT
HEIGHT
I
Alsine
=
=
pZ =
GPh
--
ph csc t
Note that formulas 4.31 to 4.34 are special cases. Fig. 4-20 RIGHT
=
CIRCULAR
4.37
Volume
4.38
Lateral surface area
CONE
OF RADIUS
,r AND
HEIGHT
h
jîw2/z =
77rd77-D
=
~-7-1
Fig. 4-21 PYRAMID
4.39
Volume
=
OF
BASE
AREA
A AND
HEIGHT
h
+Ah
Fig. 4-22 SPHERICAL
4.40
Volume (shaded in figure)
4.41
Surface area
=
CAP
=
OF RADIUS
,r AND
HEIGHT
h
&rIt2(3v - h)
2wh
Fig. 4-23 FRUSTRUM
=
OF RIGHT
4.42
Volume
4.43
Lateral surface area
+h(d
CIRCULAR
CONE
OF RADII
u,h
AND
HEIGHT
h
+ ab + b2) =
T(U + b) dF
=
n(a+b)l
+ (b - CL)~ Fig. 4-24
10
SPHEMCAt hiiWW
4.44
Area of triangle ABC
=
GEOMETRIC
FORMULAS
OF ANG%ES
A,&C
Ubl SPHERE OF RADIUS
(A + B + C - z-)+
Fig. 4-25
TOW$
&F
lNN8R
4.45
Volume
4.46
w Surface area = 7r2(b2- u2)
4.47
Volume
=
RADlU5 a
AND
OUTER RADIUS
b
&z-~(u+ b)(b - u)~
= $abc
Fig. 4-27
T.
4.4a
Volume
=
PARAWlO~D
aF REVOllJTlON
&bza
Fig. 4-28
Y
5
TRtGOhiOAMTRiC
D
OE T
FF R
WNCTIONS
F
l I FU
A R N G T ON
Triangle ABC bas a right angle (9Oo) at C and sides of length u, b, c. angle A are defined as follows. sintz . of A
5 5 5 5
5
sin A
1=
:
=
opposite hypotenuse
i
=
adjacent hypotenuse
cosine . of
A
=
~OSA
2=
. of
A
=
tanA
3= f = -~
. of
A
=
of A
tangent
c
5.5
=
secant
cosecant
. of
A
4=
k
=
adjacent t opposite
=
sec A
=
t
=
-~
=
csc A
6=
z
=
hypotenuse opposite
E
l O R RC
functions
G T
N I T
of
B
opposite adjacent
A
o cet
The trigonometric
I
TX A
c
z
A
n
g
hypotenuse adjacent
W OT
Fig. 5-1
N M
3 HG E
G A
TE I R N9L Y
H C E S0 E
A H A I ’
Consider an rg coordinate system [see Fig. 5-2 and 5-3 belowl. A point P in the ry plane has coordinates (%,y) where x is eonsidered as positive along OX and negative along OX’ while y is positive along OY and negative along OY’. The distance from origin 0 to point P is positive and denoted by r = dm. If it is described dockhse from The angle A described cozmtwcZockwLse from OX is considered pos&ve. OX it is considered negathe. We cal1 X’OX and Y’OY the x and y axis respectively. The various quadrants are denoted by 1, II, III and IV called the first, second, third and fourth quadrants respectively. In Fig. 5-2, for example, angle A is in the second quadrant while in Fig. 5-3 angle A is in the third quadrant.
Y
Y
II
1
II
1
III
IV
III
IV
Y’
Y’ Fig. 5-3
Fig. 5-2
11 f
TRIGONOMETRIC
12
FUNCTIONS
For an angle A in any quadrant the trigonometric
functions of A are defined as follows.
5.7
sin A
=
ylr
5.8
COSA
=
xl?.
5.9
tan A
=
ylx
5.10
cet A
=
xly
5.11
sec A
=
v-lx
5.12
csc A
=
riy
RELAT!ONSHiP BETWEEN DEGREES AN0
RAnIANS N
A radian is that angle e subtended at tenter 0 of a eircle by an arc MN equal to the radius r. Since 2~ radians = 360° we have 5.13
1 radian
= 180°/~
5.14
10 = ~/180 radians
=
1
r
e 0
57.29577 95130 8232. . . o
r
B
= 0.01745 32925 19943 29576 92.. .radians
Fig. 5-4
REkATlONSHlPS 5.15
tanA
= 5
5.16
&A
~II ~ 1
5.17
sec A
=
~
5.18
cscA
=
-
tan A
AMONG
COSA sin A
zz -
1
COS A
TRtGONOMETRK
5.19
sine A +
~OS~A
5.20
sec2A
-
tane
5.21
csceA
- cots A
II
III IV
1
A = 1 =
1
1 sin A
SIaNS AND VARIATIONS
1
=
FUNCTItB4S
+ 0 to 1
+ 1 to 0
+ 1 to 0
0 to -1
0 to -1 -1 to 0
OF TRl@ONOMETRK
+ 0 to m -mtoo + 0 to d
-1 to 0 + 0 to 1
+ CCto 0 oto-m + Ccto 0 -
--
too
oto-m
FUNCTIONS
+ 1 to uz
+ m to 1
-cc to -1
+ 1 to ca
-1to-m + uz to 1
--COto-1 -1 to --
M
TRIGONOMETRIC
E
Angle A in degrees
00
X
F
Angle A in radians
A T
A
O
RL
FC
R
1
IU
O UT
O S
sec A
csc A
0
1
0
w
1
cc
ii/6
1
+ti
450
zl4
J-fi
$fi
60°
VI3
Jti
750
5~112
900
z.12
105O
7~112
*(fi+&)
-&(&-Y%
-(2+fi)
-(2-&)
120°
2~13
*fi
-*
-fi
-$fi
1350
3714
+fi
-*fi
150°
5~16
4
-+ti
#-fi)
2-fi
&(&+fi)
fi
1
0
fi)
-&(G+
0
-*fi
-fi
-(2-fi)
-(2+fi)
180°
?r
-1
1950
13~112
210°
7716
225O
5z-14
-Jfi
240°
4%J3
-#
255O
17~112
270°
3712
-1
285O
19?rll2
-&(&+fi)
3000
5ïrl3
-*fi
2
315O
7?rl4
-4fi
*fi
-1
330°
117rl6
*fi
-+ti
345O
237112
360°
2r
-$(fi-fi)
-*(&+fi)
2-fi
-
1
4
-*fi
-i(fi-
2+fi 0
-(2+6)
&(&+
-ti
fi) 1
0
see pages
206-211
-(2
- fi) 0
++
-fi
\h
-+fi
2
-(fi-fi)
f
-(&-fi)
-2 -(&+?cz)
-@-fi)
&+fi
-(2+6) T-J
i
-36 -(fi-fi) -1 -(fi-fi)
2
-1
f
-fi
Tm
-*fi
-ti
-2
g
-fi
0
*ca -(&+fi)
i -
&fi 2-6
Vz+V-c? -1
3
1
km
*(&-fi)
6)
angles
ti
-&(&-fi)
1
l
1
-4
-&&+&Q
6
fi-fi
-2
2 + ti
&
1
-(&+fi)
Tm
0
fi-fi
km
-1
-1
TG
;G
&+fi 0
N
fi
2
2-&
*CU
fi)
fi
.+fi
2+&
R
2
$fi
1
C N
3
&+fi
fi-fi
fi
1
@-fi)
$(fi-
2+*
*fi
r1
i(fi+m
other
A
cet A
300
involving
FN A
tan A
rIIl2
tables
GE
COSA
0
llrll2
V
sin A
15O
165O
For
V
FUNCTIONS
fi $fi fi-fi
-$fi -fi -2 -(&+fi)
1
?m
and 212-215.
f
I
TRIGONOMETRIC
5.89
y = cet-1%
5.90
y
=
FUNCTIONS
19
sec-l%
5.91
_--/
y
=
csc-lx
Y
I T
---
,
/A--
/’
/ -77 -//
,
Fig. 5-14
Fig. 5-15
RElAilONSHfPS
BETWEEN
The following results hold for sides a, b, c and angles A, B, C.
5.92
ANGtGS
any plane triangle
ABC
OY A PkAtM
with
TRlAF4GlG
’
A
Law of Sines a -=Y=sin A
5.93
SIDES AND
Fig. 5-16
1
b
c
sin B
sin C C
Law of Cosines
/A
cs = a2 +
bz -
Zab
COS
f
C
with similar relations involving the other sides and angles. 5.94
Law of Tangents
tan $(A + B) a+b -a-b = tan i(A -B) with similar relations involving the other sides and angles.
5.95
sinA where
s = &a + b + c)
=
:ds(s
is the semiperimeter
- a)(s - b)(s - c) of the triangle.
B and C cari be obtained. See also formulas 4.5, page 5; 4.15 and 4.16, page 6.
Spherieal triangle ABC is on the surface of in Fig. 5-18. Sides a, b, c [which are arcs of measured by their angles subtended at tenter 0 of are the angles opposite sides a, b, c respectively. results hold. 5.96
Law of Sines sin a -z-x_ sin A
5.97
sin b sin B
a sphere as shown great circles] are the sphere. A, B, C Then the following
sin c sin C
Law of Cosines sinbsinccosA
cosa
=
cosbcosc
COSA
=
- COSB COSC +
+
Fig. 5-1’7
sinB sinccosa
with similar results involving other sides and angles.
Similar relations involving angles
2
T
0 L
5
o.
w
T
a
s
5
f 9
ri
s = &
S = +
f
e
E g
i
c
S(
8 n & + B
a
t(
t
&
a=
t(
4l
op
f r x F s ii
1e
o mn s
e
I
g
U
$ ) + n b )
sh a t ai v i
r)
G
e
aA
(
N
)
n
a A
n
O
n
C
T
t
u
n
u n h nl o d
N
s
i
l d e a g l e
t
r
O
(
rl v s
s
e i
u i hr
f e. o
.s
r)
A i rh
ra
0
FGR
RtGHT
o C it c na i b -va b i
m + ose
o sa t a i l i
r u n h n l dd
ld e g a e
t
r
lr s
0
( Se
RlJlES
a wn
- B
0
h+ B + C
NAPIER’S
a
t
1
w a
et
F
9
1+
h
.
S
w
9
w
5
a
i i
.
R
4o
f e. o
m g
.
meos
4
eu
ANGLED
rf , gh p e gor s i c gB e le, , u , . .
o sa t a i l i ,
l
SPHERICAL
ha e p t lef 9n A l
pv
Atr
r un h n l dd
ld e
ga
e
t
r
lr
a
TRIANGLES
t rwe , d
he i
Z aet
ih
ei f t r3s o a i
n r
h rC r nc
a
C
F S [
c
A a
t i
o p
5
uq ot
i h hn
o t n p n da t a t
-
g
a pu a
fi hce ri a m s ia ea d a e c to a om c r
of th y ac e rj n h w r p
T
s.
o
a
h m i1
fp
5.102
T
s
o
a
h m i
fp n ee i n
S
T
x
C
c
= 9i
o
ch
a-
n ee i n0 t
O C
s
a
s
( ba
=
t
n0= m 9A t =
o f
.
oe
F
9
ri p aar c s a en io nr Fi a n l p A a npi B
e m oc ayd
5
E
1
5
5e s n w s t a ir ndoc g . c
p rt ti tsl a ps r p ea Te
an cs laN
u d f oeh t oa a a l
a p y q d eo ht r rc
u d fo eht t ooo
O w c° h 0p,
B A
-
e e a °l n s
,
(
a
na
C o
C i
C a C
(
nO
Oo O C ~
uts
rl i rt 5 rhe o es p
a er fb
g
e 2p t
hd
ead
wl xi hr
pv le r l aao s
f p eh
B
dsp
et Cg.l h
eoc tn eu t
a -ei e p sr
.
0r t ri O ee e ni n s rl
da sl i n
ae ug j s
l e
ae
ve
r
uip s
r
frg di ie ce a n co
e:
i
AS =-r SC OaOs
2
ee
et
f eph dn d l e
a l
n = rOt
b it
-
w - a ehi ngc t dta l m p t
a p y q d eo1 ht r rt
--
i
O
-a B A
oa 1. s e n os a
n O -
mi 99 e
Bn
S i
) ug
a
)b
SB n
n .7
l e
e
A complex number is generally written as a + bi where a and b are real numbers and i, called the imaginaru unit, has the property that is = -1. The real numbers a and b are called the real and ima&am parts of a + bi respectively. The complex numbers a + bi and a - bi are called complex
6.1
a+bi
=
c+di
if and only if
conjugates
a=c
and b=cZ
6.2
(a + bi) + (c + o!i) =
(a + c) + (b + d)i
6.3
(a + bi) - (c + di) =
(a - c) + (b - d)i
6.4
(a+ bi)(c+
di) =
(ac- bd) + (ad+
of each other.
bc)i
Note that the above operations are obtained by using the ordinary rules of algebra and replacing 9 by -1 wherever it occurs.
21
22
COMPLEX
GRAPH
NUMBERS
OF A COMPLEX
NtJtWtER
A complex number a + bi cari be plotted as a point (a, b) on an xy plane called an Argand diagram or Gaussian plane. For example in Fig. 6-1 P represents the complex number -3 + 4i. A
eomplex
number
cari
also
be
interpreted
as
a
wector
p,----.
y
from
0 to P. -
0
X
* Fig. 6-1
POLAR
FORM
OF A COMPt.EX
NUMRER
In Fig. 6-2 point P with coordinates (x, y) represents the complex number x + iy. Point P cari also be represented by polar coordinates (r, e). Since x = r COS6, y = r sine we have
6.6
x + iy = ~(COS 0+
called
the poZar form
the mocklus
of the complex
and t the amplitude
i sin 0)
number.
L
We often
-
X
cal1 r = dm
of x + iy. Fig. 6-2
tWJLltFltCATt43N
[rl(cos
6.7
AND
DtVlStON
OF CWAPMX
el + i sin ei)] [re(cos ez + i sin es)] V-~(COSe1 + i sin el)
6.8
ZZZ 2
rs(cos ee + i sin ez)
If p is any real
number,
De Moivre’s [r(cos
rrrs[cos
1bJ POLAR
ilj 0”
FtMM
tel + e2) + i sin tel + e2)]
[COS(el - e._J + i sin (el - .9&]
DE f#OtVRtt’S
6.9
=
NUMBRRS
THEORRM
theorem
states
e + i sin e)]p
=
that rp(cos pe + i sin pe)
.
RCWTS
If
p = l/n
where
k=O,l,2
integer,
[r(cos e + i sin e)]l’n
6.10 where
n is any positive
OF CfMMWtX
k is any ,...,
integer. n-l.
From
this
the
=
n nth
NUtMB#RS
6.9 cari be written rl’n roots
L
e + 2k,, ~OSn of
a complex
+
e + 2kH
i sin ~
number
n cari
1 be
obtained
by
putting
”
In the following p, q are real numbers, CL,t are positive numbers and WL,~are positive integers.
7.1
cp*aq z aP+q
7.2
aP/aqE @-Q
7.3
(&y E rp4
7.4
u”=l,
7.5
a-p = l/ap
7.6
(ab)p = &‘bp
7.7
&
7.8
G
7.9
Gb
a#0
z aIIn
= pin
=%Iî/%
In ap, p is called the exponent, a is the base and ao is called the pth power of a. The function is called an exponentd function.
If a~ = N where a # 0 or 1, then p = loga N is called the loga&hm N = ap is called t,he antdogatithm of p to the base a, written arkilogap. Example:
Since
The fumAion
3s = 9 we have
y = ax
of N to the base a. The number
log3 9 = 2, antilog3 2 = 9.
v = loga x is called a logarithmic
jwzction.
7.10
logaMN
=
loga M + loga N
7.11
log,z ;
=
logG M -
7.12
loga Mp
=
p lO& M
loga N
Common logarithms and antilogarithms [also called Z?rigg.sian] are those in which the base a = 10. The common logarit,hm of N is denoted by logl,, N or briefly log N. For tables of common logarithms and antilogarithms, see pages 202-205. For illuskations using these tables see pages 194-196. 23
EXPONENTIAL
24
AND LOGARITHMIC
NATURAL LOGARITHMS
FUNCTIONS
AND ANTILOGARITHMS
Natural logarithms and antilogarithms [also called Napierian] are those in which the base a = e = 2.71828 18. . . [sec page 11. The natural logarithm of N is denoted by loge N or In N. For tables of natural logarithms see pages 224-225. For tables of natural antilogarithms [i.e. tables giving ex for values of z] see pages 226-227. For illustrations using these tables see pages 196 and 200.
CHANGE OF BASE OF lO@ARlTHMS
The relationship between logarithms of a number N to different bases a and b is given by
7.13
loga N
=
hb
iv
hb
a
-
In particular, = ln N
7.14
loge N
7.15
logIO N = logN
RElATlONSHlP
= 2.30258 50929 94.. . logio N =
0.43429
44819 03.. . h& N
BETWEEN EXPONBNTIAL ANO TRl@ONOMETRlC eie =
7.16 These are called Euler’s
COS 0 + i sin 8,
dent&es.
e-iO
=
COS 13 -
sin 6
Here i is the imaginary unit [see page 211.
7.17
sine
7.18
case =
=
eie- e-ie 2i
eie+ e-ie 2
7.19
7.20 2
7.21
sec 0
=
&O + e-ie
7.22
csc 6
=
eie
7.23
i
2i
eiCO+2k~l
From this it is seen that @ has period 2G.
-
e-if3
=
eie
k =
integer
FUNCT#ONS
;;
E
POiAR
T
p
XA
FORfvl OF COMPLEX
f
7
o h a co
o n
.
2
6
t
o
6
NUMBERS
.o hp r 2
(reiO)l/n E
LOGARITHM
7.29
COMPLEX
a.
l
OD
ym i a tm e
(
ffUMBERS
e7n ra m 2 t 1r t
(q-eio)Pzz q-P&mJ [
7.2B
OF
GU
EXPRESSE$3 AS AN
oxl + i r c u b w m a
WITH
7.27
PN
or rpe
N
AN 25
E
RC
N
EXPONENTNAL
n re b
[if lx 6
pi r e 2 st a ep .
a mr 2 et s x o6
g
4 6 + i sin 0) = 9-ei0 x + iy = ~(COS
OPERATIONS
F
fe
L
[~&O+Zk~~]l/n
q f og
M
t =
n
u
D
FORM
o 0eh uo ue
o
h
l
e
e
i
il
g
a
e
vl
v
h
s
o
NUMBER
k e=e i k
@n z
) t -
ao
r
rl/neiCO+Zkr)/n
OF A COMPLEX
= l r n + iT + 2
IN POLAR
e i
DEIWWOPI
OF HYPRRWLK
8.1
Hyperbolic
sine of x
=
sinh x
=
8.2
Hyperbolic
cosine
=
coshx
=
8.3
Hyperbolic
tangent
= tanhx
=
8.4
Hyperbolic
cotangent
8.5
Hyperbolic
secant
8.6
Hyperbolic
cosecant
RELATWNSHIPS
of x
of x
coth x
of x =
of x
AMONG
ez + e-=
2 ~~~~~~
2
ez + eëz
HYPERROLIC FUWTIONS
=
sinh x a
coth z
=
1 tanh x
sech x
=
1 cash x
8.10
cschx
=
1 sinh x
8.11
coshsx - sinhzx
=
1
8.12
sechzx + tanhzx
=
1
8.13
cothzx - cschzx
=
1
FUNCTIONS
2
= csch x = &
tanhx
8.7
# - e-z
ex + eCz = es _ e_~
= sech x =
of x
.:‘.C,
FUNCTIONS
cash x sinh x
=
OF NRGA’fWE
ARGUMENTS
8.14
sinh (-x)
=
- sinh x
8.15
cash (-x)
= cash x
8.16
tanh (-x)
= - tanhx
8.17
csch (-x)
=
-cschx
8.18
sech(-x)
=
8.19
coth (-x)
=
26
sechx
-~OUIS
HYPERBOLIC
AWMWM
FUNCTIONS
27
FORMWAS
0.2Q
sinh (x * y)
=
sinh x coshg
8.21
cash (x 2 g)
=
cash z cash y * sinh x sinh y
8.22
tanh(x*v)
=
tanhx f tanhg 12 tanhx tanhg
8.23
coth (x * y)
=
coth z coth y 2 1 coth y * coth x
8.24
sinh 2x
=
2 ainh x cash x
8.25
cash 2x
=
coshz x + sinht x
8.26
tanh2x
=
2 tanh x 1 + tanh2 x
=
* cash x sinh y
2 cosh2 x -
1
=
1 + 2 sinh2 z
HAkF ABJGLR FORMULAS
8.27
sinht
=
8.28
CoshE 2
=
8.29
tanh;
=
k
Z
sinh x cash x + 1
.4
[+ if x > 0, - if x < O] cash x + 1 -~ 2 cash x - 1 cash x + 1
’ MUlTWlE
[+ if x > 0, - if x < 0]
ZZ cash x - 1 sinh x
A!Wlfi WRMULAS
8.30
sinh 3x
=
3 sinh x + 4 sinh3 x
8.31
cosh3x
=
4 cosh3 x -
8.32
tanh3x
=
3 tanh x + tanh3 x 1 + 3 tanhzx
8.33
sinh 4x
=
8 sinh3 x cash x + 4 sinh x cash x
8.34
cash 4x
=
8 coshd x -
8.35
tanh4x
=
4 tanh x + 4 tanh3 x 1 + 6 tanh2 x + tanh4 x
3 cash x
8 cosh2 x -t- 1
2
H
8
YF
P
O
PU
HO
E N
FY& W
P
J
R C
E
E
B T
f
R
R
8
.
3
s
6=
&i c
2
-
4 na
8
.
3
c
7=
4 oc
2
+
$ sa
8
.
3
s
x
8=
&i s
3
-
8
.
3
c
x
9=
&o c
+
8
.
4
s
0=
8i -
4 c
2
n+
4 ca
4x
h
as
% 4
sh
x
h
8
.
4
c
1=
#o +
+ c
2
s+
& ca
4x
h
as
x 4
sh
x
h
S
D
8
U
.
AI
F
A
hs
zh
x
x
hs
zh
x
2 sn i
xx
ihn
nsh
2 cs o
x
ahs
ssh
K
NFO
x
W R
&
DFF F O P
Sl
h h3
E
x
UR D
R
s
4+
s
i
=
2 si2
& n
+ y
cn i
$ hx - y)
anh
(x
)
s hy
x
h
x
h
kR U
8
.
4s
-
s
3i
=
2 ci
n&
+ y
s an
$ hx - Y)
i sh
(x
)
n hy
8
.
4c
+
c
4o
=
2 co
is
+ y
c as
#(h
- Y)
a sh
xxx
)
s hy
8
.
4c
-
c
5o
=
2 so
$s
+ y
s is
$ (h - Y)
i nh
( xx
)
n hy
8
.
4s
x s
y 6i=
* i
n
{- n c
h
c ho
o
s
s
h
h
(
8
.
4c
x c
y 7 a=
+ a
s
{+ s c
h
c ho
o
s
s
h
h
(
s
x 4c
y
i=
+ a
n+ y
{- s s
x @ h- ) Y sl h i
) -i
n
} n
h
h
8
.
E
I
t
t
OX H
f
n
hw
.o
8 s
FP FY
x e>e 0 ls I
oa 1
x = u
i c
8(
= u
!R UPT
x < 0 u. l s t f
a
9
.
n o t
t s
x
i
n
h
c
x
a
s
h
t
x
a
n
h
c
x
o
t
h
s
x
e
c
h
c
x
s
c
h
= uh s a c
s ou h
s p
O
N ‘ E NEE
a e i wme
x = 1h n o s
i p b s fn i e
x =1 xu h t e c
h x
h
F OSC RR
g r 8y
o dn
x = xwh c s
T SB
n o .
rig
h c
HYPERBOLIC
GRAPHS
8.49
y = sinh x
OF HYPERBOkfC
8.50
29
FUNCltONS
8.51
y = coshx
Fig. S-l 8.52
FUNCTIONS
Fig. 8-2
y = coth x
8.53
/i
y
y = tanh x
Fig. 8-3
8.54
y = sech x
y = csch x Y \
X
1
7
10
X
0
-1
iNVERSE
HYPERROLIC
L
X
Fig. 8-6
Fig. 8-5
Fig. 8-4
0
FUNCTIONS
If x = sinh g, then y = sinh-1 x is called the inverse hyperbolic sine of x. Similarly we define the The inverse hyperbolic functions are multiple-valued and. as in the other inverse hyperbolic functions. case of inverse trigonometric functions [sec page 171 we restrict ourselves to principal values for which they ean be considered as single-valued. The following list shows the principal values [unless otherwise indicated] of the inverse hyperbolic functions expressed in terms of logarithmic functions which are taken as real valued.
8.55
sinh-1 x
=
ln (x + m
8.56
cash-lx
=
ln(x+&Z-ï)
8.57
tanh-ix
=
8.58
coth-ix
=
8.59
sech-1 x
8.60
csch-1 x
)
-m<x<m XZl -l<xl
O<xZl
=
[cash-r x > 0 is principal value]
ln(i+$$G.)
x+O
or xc-1
[sech-1 x > 0 is principal value]
HYPERBOLIC
30
FUNCTIONS
8.61
eseh-]
x
=
sinh-1
(l/x)
8.62
seeh-
x
=
coshkl
(l/x)
8.63
coth-lx
=
tanh-l(l/x)
8.64
sinhk1
(-x)
=
- sinh-l
x
8.65
tanhk1
(-x)
=
- tanh-1
x
8.66
coth-1
(-x)
=
- coth-1
x
8.67
eseh-
(-x)
=
- eseh-
x
GffAPHS
8.68
y =
OF fNVt!iffSft HYPfkfftfUfX
8.69
sinh-lx
FfJNCTfGNS
X
7 -ll
8.72
coth-lx Y
0
\
\
\
\
‘-. Fig. 8-9
L 11
x
y =
8.73
sech-lx
y =
Y
Il
0
I I’ Fig. 8-11
csch-lx Y
I
Fig. 8-10
\
Fig. 8-8
Fig. 8-7
l l l
x
-1 \
y =
tanhkl
l
Y
Y
8.71
y =
8.70
y = cash-lx
,
,
/
X
3
L 0
Fig. 8-12
-x
HYPERBOLIC
tan (ix) == i tanhx
sec (ix) = sechz
8.79
cet (ix)
8.81
cash (ix) = COSz
8.82
tanh (iz)
= i tan x
8.84
sech (ix) = sec%
8.85
coth (ix)
=
sin (ix) = i sinh x
8.75
COS(iz)
8.77
csc(ix)
8.78
8.80
sinh (ix) = i sin x
8.83
csch(ti)
-i
=
cschx
-icscx
In the following
31
8.76
8.74
=
FUNCTIONS
= cash x
== - 0n )
Solutions:
a a
EQUATION:
L
i e D =n 0 q i
e
I a a
Ef
lx
c i
x
o O A
o
Th 0 O e =S -RI&@
x s x e
S e
0
= - ,s ,
r z s
e
t
’
e
S
)
r
’ ax e
x r s
) ss
2 .
SOLUTIONS
QUARTK
Let y1 be a real root
9.7
Solutions:
ALGEBRAIC
EQUATION:
of the cubic
The 4 roots
OF
x* -f- ucx3 + ctg9 +
of ~2 +
xl, x2, x3, x4 are the four
u
3
+
a
3 =
0
4
3
$
equation
+{a1
2
a; -4uz+4yl}z
If a11 roots of 9.6 are real, computation is simplified a11 real coefficients in the quadratic equation 9.7.
where
EQUATIONS
by using
+ that
$&
* d-1
particular
= real
root
0
which
produces
roots.
-
FURMULAS Pt.ANE ANALYTIC
10
fram
GEOMETRY
DISTANCE d BETWEEN TWO POINTS F’&Q,~~) AND &(Q,~~) 10.1
d=
-
Fig. 10-1
10.2
mzz-z
EQUATION
10.3
OF tlNE JOlNlN@
Y -
Y1
x -
ccl
m
Y2 -
Y1
F2 -
Xl
TWO POINTS ~+%,y~)
Y2 - Y1 xz -
10.4
cjr
Xl
y = where
b = y1 - mxl =
XZYl xz -
EQUATION
XlYZ 51
tan 6
Y -
Y1 =
mb
ANiI
l%(cc2,1#2)
- Sl)
mx+b
is the intercept
on the y axis, i.e. the y intercept.
OF LINE IN ‘TEMAS OF x INTERCEPT a # 0 AN0 3 INTERCEPT b + 0 Y
b a Fig. 10-2
34
2
FORMULAS
FROM
ffQRMAL
10.6
ANALYTIC
FORA4 FOR EQUATION
+ Y sin a
x cosa
PLANE
=
where
p
=
perpendicular
and
a
1
angle of inclination positive z axis.
GEOMETRY
OF 1lNE
y
p
distance
35
from
origin
0 to line
of perpendicular
P/ ,
with
,
L LX
0 I Fig. 10-3
GENERAL
10.7
Ax+BY+C
KIlSTANCE
where
FROM
the sign is chosen
ANGLE
10.9
s/i BETWEEN
tan $ Lines are parallel Lines
POINT
SO that
=
(%~JI)
the distance
TWO
OF LINE
EQUATION
0
TO LINE
AZ -l- 23~ -l- c = Q
is nonnegative.
l.lNES
HAVlNG
SlOPES
wsx AN0
%a2
m2 - ml 1 + mima
=
or coincident
are perpendicular
if and only if mi = ms.
if and only
if ma = -Ilmr.
Fig. 10-4
AREA
10.10
Area
=
z=
where
*T
OF TRIANGLE
1
*;
the sign
If the area
Xl
Y1
1
~2
ya
1
x3
Y3
1
(Xl!~/2
+
?4lX3
is chosen
WiTH
VERTIGES
AT @I,z&
@%,y~), (%%)
(.% Yd +
Y3X2
SO that
is zero the points
-
!!2X3
-
the area
YlX2
-
%!43)
is nonnegative.
a11 lie on a line. Fig. 10-5
FORMULAS
36
TRANSFORMATION
1
10.11
FROM
PLANE
ANALYTIC
OF COORDINATES
x
=
x’ + xo
Y
=
Y’ + Y0
1 x’
or
y’
x
x
GEOMETRY
INVGisVlNG
x -
xo
Y -
Y0
PURE
TRANSlAliON
Y
l Y’ l
l
where (x, y) are old coordinates [i.e. coordinates relative to xy system], (~‘,y’) are new coordinates [relative to x’y’ system] and (xo, yo) are the coordinates of the new origin 0’ relative to the old xy coordinate system. Fig. 10-6
TRANSFORMATION
10.12
1
=
x’ cas L -
OF COORDIHATES
y’ sin L
or
-i y = x’ sin L + y’ cas L
x’ z
INVOLVING
PURE
x COSL + y sin a
ROTATION
\Y! \ \ \ \
yf z.z y COSa - x sin a
where the origins of the old [~y] and new [~‘y’] coordinate systems are the same but the z’ axis makes an angle a with the positive x axis. ,
,
,
,
Y
,
\o/ , ’ \
,
/
/
/
,x’
L
CL!
\ Fig. 10-7
TRANSFORMATION
OF COORDINATES
1 1
02 =
10.13
lNVGl.VlNG
TRANSLATION
x’ cas a - y’ sin L + x.
y = 3~’sin a + y’ COSL + y0
or
ANR
x’
ZZZ
(X - XO) cas L + (y - yo) sin L
y!
rz
(y - yo) cas a - (x - xo) sin a
1 \
ROTATION
/
,‘%02 \
where the new origin 0’ of x’y’ coordinate system has coordinates (xo,yo) relative to the old xy eoordinate system and the x’ axis makes an angle CYwith the positive x axis.
Fig. 10-8
POLAR
COORDINATES
(Y, 9)
A point P cari be located by rectangular coordinates (~,y) or polar eoordinates (y, e). The transformation between these coordinates is
10.14
x
=
1 COS 0
y = r sin e
or
T=$FTiF
6 = tan-l
(y/x)
Fig. 10-9
FORMULAS
RQUATIQN
10.15
FROM PLANE
OF’CIRCLE
(a-~~)~ + (g-vo)2
ANALYTIC
OF RADIUS
GEOMETRY
37
R, CENTER AT &O,YO)
= Re
Fig. 10-10
RQUATION
10.16
OF ClRClE
OF RADIUS
R PASSING
T = 2R COS(~-a)
THROUGH
ORIGIN
Y
where (r, 8) are polar coordinates of any point on the circle and (R, a) are polar coordinates of the center of the circle.
Fig. 10-11
CONICS
[ELLIPSE,
PARABOLA
OR HYPEREOLA]
If a point P moves SO that its distance from a fixed point [called the foc24 divided by its distance from a fixed line [called the &rectrkc] is a constant e [called the eccen&&ty], then the curve described by P is called a con& [so-called because such curves cari be obtained by intersecting a plane and a cane at different angles]. If the focus is chosen at origin 0 the equation of a conic in polar coordinates (r, e) is, if OQ = p and LM = D, [sec Fig. 10-121 10.17
T =
P 1-ecose
=
CD 1-ecose
The conic is (i)
an ellipse if e < 1
(ii)
a parabola if e = 1
(iii) a hyperbola if c > 1.
Fig. 10-12
38
FORMULAS
FROM PLANE
10.18
Length of major axis A’A
=
2u
10.19
Length of minor axis B’B
=
2b
10.20
Distance from tenter C to focus F or F’ is
ANALYTIC
GEOMETRY
C=d--
= c =
E__
10.21
Eccentricity
10.22
Equation in rectangular
a
-
~
0
a
coordinates:
(r - %J)Z + E b2 a2
Fig. 10-13
=
3
re zz
a2b2
10.23
Equation in polar coordinates if C is at 0:
10.24
Equation in polar coordinates if C is on x axis and F’ is at 0:
10.25
If P is any point on the ellipse, PF + PF’
=
a2 sine a + b2 COS~6
r =
a(1 - c2) l-~cose
2a
If the major axis is parallel to the g axis, interchange x and y in the above or replace e by &r - 8 [or 9o” - e].
PARAR0kA
WlTJ4 AX$S PARALLEL
TU 1 AXIS
If vertex is at A&,, y,,) and the distance from A to focus F is a > 0, the equation of the parabola is 10.26
(Y - Yc?
10.27
(Y - Yo)2 =
=
4u(x - xo)
if parabola opens to right [Fig. 10-141
-4a(x - xo)
if parabola opens to left [Fig. 10-151
If focus is at the origin [Fig. 10-161 the equation in polar coordinates is 10.28
T
=
2a 1 - COSe Y
Y
-x
0
Fig. 10-14
Fig. 10-15
x Fig. 10-16
In case the axis is parallel to the y axis, interchange x and y or replace t by 4~ - e [or 90” - e].
FORMULAS
FROM PLANE
ANALYTIC
GEOMETRY
39
Fig. 10-17 10.29
Length of major axis A’A
= 2u
10.30
Length of minor axis B’B
=
10.31
Distance from tenter C to focus F or F’
10.32
Eccentricity
10.33
Equation in rectangular
10.34
Slopes of asymptotes G’H and GH’
10.35
Equation in polar coordinates if C is at 0:
10.36
Equation in polar coordinates if C is on X axis and F’ is at 0:
10.37
If P is any point on the hyperbola,
e = ;
= -
2b =
c = dm
a coordinates:
=
(z - 2# os
(y - VlJ2 -7=
1
* a
”
PF - PF!
=
=
If the major axis is parallel to the y axis, interchange [or 90° - e].
a2b2 b2 COS~e - a2 sin2 0
22a
r =
Ia~~~~~O
[depending on branch]
5 and y in the above or replace 6 by &r - 8
11.1
E
i
p
qc
n r
E
1
1 i
1
A
b1
1
A
o 1o
r
l
+ y
An
o
r=
uo
= a c
2
. cn
u
q (
o
e
l 2
2 a
0
c
a
2 o
= C S - y* A e.
a
&f . n
B xga r
o
e
ao
a
o
’lx
o
B w
a
E
i
p q
fn [
C =
CE
L-
1y = a 1
A
1
A T
a r
o 1o l
a
1
2
o r ae
i a c dh a x ao
s l
.=
o
r 8f
(
F o o
E
1 i
r
q %
E
1
1
i
p q
A
11.11
A
u
b l
T i a c a i r o /t
brc o
e r
e
u
y
2 Z
a
=
a
s
li
dh s bu a p ei P o si t o o a n c4h n o rl f
d
A
o
’
s
\ l ’
eB,
/
xg
n
n
i 1
1
g-
l
m
i
2
,
tY
n-
nn
j
m i
O
:
e o
t n
n S
a
#
h
)
2
c g
h t
v i
ic f
a g
is
h er
nr n. F
1
d
c
ti g i
1
i
l
b g
-
u
ViflTH FOUR CUSf’S
/ Z
c 2
a
f
9o
o
3 Z
r
O
0
f= n6 c
a
1
o ss o r n n
8 o Z
l
u
a
r
a
a
t
m
3
ar ta
n gr
ya r c o ss o r n v i ai e s f al r.
i
d
m i
:
n
o
i
g
n
e o
t n
9
n
a
40
t 3
r
S
nu
t
3
i
o = & yeu ec
r
e
i
F
)
a
a
ya r c
. fn a u x = a C y
11.10
. cn +
c
7
HYPOCYCLOID
1
, e
o i
C
O
= 6e
nc rn
ei
p
a i
e
bu a p ei x l
r
a &
Y
- C
r = 3f . n
o
a u (s + +
r
i d\ ,
,
\
)!
5d
Y
r
t (
C
11.5
t
s)
t’=3 4 n
\
s
G
a 4e
tA r
z
2
v i
ic f a i sd
d
e
tv er c
r nr F d
1
e
d
he d i
e c l
ti i e
i 1 u
l e
b g
-
u s
.
SPECIAL
PLANE
CURVES
41
CARDIOID
11 .12
Equation:
11 .13
Area bounded by curve
11 .14
Arc length of curve
r = a(1 + COS0) = $XL~
= 8a
This is the curve described by a point P of a circle of radius a as it rolls on the outside of a fixed circle of radius a. The curve is also a special case of the limacon of Pascal [sec 11.321. Fig. 11-4
CATEIVARY
11.15
Equation:
Y z : (&/a + e-x/a)
= a coshs
This is the eurve in which a heavy uniform cham would hang if suspended vertically from fixed points A anda. B.
Fig. 11-5
THREEdEAVED
11.16
Equation:
ROSE \
r = a COS39
The equation T = a sin 3e is a similar curve obtained by rotating the curve of Fig. 11-6 counterclockwise through 30’ or ~-16 radians. In general n is odd.
v = a cas ne
or
r = a sinne
‘Y
\ \ \ \ \ , /
has n leaves if
/ +
,/
, Fig. 11-6
FOUR-LEAVED
11.17
Equation:
ROSE
r = a COS20
The equation r = a sin 26 is a similar curve obtained by rotating the curve of Fig. 11-7 counterclockwise through 45O or 7714radians. In general n is even.
y = a COSne
or
r = a sin ne has 2n leaves if
Fig. 11-7
a
X
42
SPECIAL
11.18
PLANE
CURVES
Parametric equations: X
=
(a + b) COSe -
b COS
Y
=
(a + b) sine -
b sin
This is the curve described by a point P on a circle of radius b as it rolls on the outside of a circle of radius a. The cardioid
[Fig. 11-41 is a special case of an epicycloid.
Fig. 11-8
GENERA&
11.19
HYPOCYCLOID
Parametric equations: z
=
(a - b) COS@ + b COS
Il
=
(a-
b) sin + -
b sin
This is the curve described by a point P on a circle of radius b as it rolls on the inside of a circle of radius a. If
b = a/4,
the curve is that of Fig. 11-3. Fig. 11-9
TROCHU#D
11.20
Parametric equations:
x =
a@ - 1 sin 4
v = a-bcos+
This is the curve described by a point P at distance b from the tenter of a circle of radius a as the circle rolls on the z axis. If
1 < a, the curve is as shown in Fig. 11-10 and is called a cz&ate c~cZOS.
If b > a, the curve is as shown in Fig. ll-ll If
and is called a proZate c&oti.
1 = a, the curve is the cycloid of Fig. 11-2.
Fig. 11-10
Fig. ll-ll
SPECIAL
PLANE
CURVES
43
TRACTRIX
11.21
PQ
x
Parametric equations:
u(ln cet +$ - COS#)
=
y = asin+
This is the curve described by endpoint P of a taut string of length a as the other end Q is moved along the x
axis.
Fig. 11-12
WITCH
11.22
Equation in rectangular
11.23
Parametric equations:
coordinates:
OF AGNES1
u =
8~x3
x2 + 4a2
x = 2a cet e y = a(1 - cos2e)
Andy
-q-+Jqx
In Fig. 11-13 the variable line OA intersects and the circle of radius a with center (0,~) at A respectively. Any point P on the “witch” is located oy constructing lines parallel to the x and y axes through B and A respectively and determining the point P of intersection.
y = 2a
FOLIUM 11.24
OF DESCARTRS Y
3axy
\
Parametric equations:
1
x=m
y =
11.26
Area of loop = $a2
11.27
Equation of asymptote:
3at
1
3at2 l+@ \
x+y+u
Z
Fig. 11-14
0
INVOLUTE il.28
Fig. 11-13
Equation in rectangular coordinates: x3 + y3 =
11.25
l
OF A CIRCLE
Parametric equations: x = ~(COS+ + @ sin $J)
I y = a(sin + - + cas +) This is the curve described by the endpoint P of a string as it unwinds from a circle of radius a while held taut. jY!/--+$$x . Fig. Il-15
I
44
S
11.29
E
i
r
q
(axy’3
+
P
11.30
e
x
(bvp3
1b = i t he u = 1e s z d
e
o
u tu3
-
1
P
of
6a
so h t i n r /i h lF 1a
+ qa4 .
a
s
z i
2
G 2
T I
i t 2 a
c
c d
hd s h i a c a p
i a ih F u 1 s t
b = u
cf
-
u b a p ie ib s o
t
u
A
C
a
m
r
R
N
I
V
E
A
a
i
d
t
e
n
o
i
g
2 u 3=
n
o he 2 s wg
lre
yr t sos t a 2s n
s[
a r
e1e
e lm l n. 1e F 1
F
o
i
rS
W
i oa 6d 1
p pi g
L t
i
m 1
v
i
r.
-k
o
1 7 eo
e
g
-
S
p u v h o i cih d r c e a f nrt e t i o hf trp t is ws d i .r t s t a t ] n a c
i g va1 - b s- a1 rc
r
t
s
e -h
A
aO e a 4
ba
Pe s
i
)
OF CASSINI V
so nFe i r1 1
i , a Zh
U
t
)
tf the s eov v a nb o i 1s
l~i
2
o
_--\ ++Y !---
T [
e 1
a
u C
O 1
E
by3
COS3 z 8
- b ys
L
ELLIPSE
c
r
- b
(
C
OF Aff
q = (
P
EVOWTE
=
a c
T c + y
cn
P
8
1c
oo b
n
sr
.
1a
P
X
a
F
1
i
1
g
LIMACON 11.32 t
P
L c T
c
O b i t c i a c
e
o r = qb
l
-
.
1
i
1
g
-
u+
a
aa
r
tc
y i gai p f a n s
s os nFe i 1r 1 a r i g a1v -b >c a og .b s< -e a1 r c r F r 1 v id e ig 4 o
ii h r. .
t
io
os
r in a0 t t
aTn h c nt s. s
2 I9 e o1 = a 1 i
t
0 f sr , . d
-
F
1
i
1
.
OF PASCAL
a l ej Q eo i 0to t a rp n Q ioo an c io eo dnn h l u o a s ph oe P rs f 1t oe Pc = vub 1h i Q u . ec i a ih F u 1 u [ s a1
17
F
g
-
.
1
F
1
9
i 1
g-
m hg r h
SPECIAL
PLANE
C
11.33
Equation
in rectangular y
11.34
Parametric
OF
CURVES
Ll
IS
x
2a -
2
3
x
equations:
i
=
2a sinz t
?4 =-
2a sin3 e COSe
This is the curve described by a point P such that the distance OP = distance RS. It is used in the problem of duplicution of a cube, i.e. finding the side of a cube which has twice the volume of a given cube.
SPfRAL
Polar
BS
coordinates: ZZZ
x
11.35
45
equation:
Y =
a6
Fig. 11-21
OF ARCHIMEDES
Y
Fig. 11-22
OO
C
FORMULAS APJALYTK
12
SCXJD GEOMETRY from
Fig. 12-1
RlRECTlON
12.2
COSINES OF LINE ,lOfNlNG
1 =
COS L
=
% - Xl
~
d
’
m
=
where a, ,8, y are the angles which line PlP2 d is given by 12.1 [sec Fig. 12-lj.
FO!NTS &(zI,~z,zI)
COS~
=
Y2 d,
Y1
n
=
AND &(ccz,gz,rzz)
c!o?, y
=
22 -
-
21
d
makes with the positive x, y, z axes respectively
and
RELATIONSHIP EETWEEN DIRECTION COSINES
12.3
or
cosza+ COS2 p + COS2 y = 1
lz + mz +
nz
=
1
DIRECTION NUMBERS
Numbers L,iVl, N which are proportional The relationship between them is given by
12.4
1 =
L dL2+Mz+
to the direction cosines 1,m, n are called direction
M
m= N2’
dL2+M2+Nz’
46
n=
N j/L2 + Ar2 i N2
numbws.
FORMULAS
OF LINE JOINING
EQUATIONS
12.5 These
FROM
x-
x,
% -
Xl
are also valid
Y-
~~~~ Y2 -
Y1
z -
Y1
752 -
ANGLE
are
also valid
+ BETWEEN
if 1, m, n are replaced
TWO
LINES WITH
12.7
12.8
x -
OF PLANE
AND
y
=
Y-
12.9
xz -
Xl
x3 -
Xl
2 -
Y1
m
FORM
Zl
n
IN PARAMETRIC
1 =
FORM
.zl + nt
by L, M, N respectively.
DIRECTION
mlm2
THROUGH
X
x -
Y =p=p
I’&z,y~,zz)
y1 + mt,
EQUATION
PASSING
Xl 1
COSINES
L,~I,YZI
AND
h
,
+ nln2
OF A PLANE
.4x + By + Cz + D
EQUATION
IN STANDARD
~&z,yz,zz)
or
47
by L, M, N respeetively.
COS $ = 1112 +
GENERAL
GEOMETRY
21
I’I(xI,~,,zI)
x = xI + lt, These
AND
.z,
if Z, m, n are replaced
12.6
ANALYTIC
~I(CXI,~I,ZI)
OF LINE JOINING
EQUATIONS
SOLID
=
[A, B, C, D are constants]
0
POINTS
Y1l
2 -
.zl
Y2 -
Y1
22 -
21
Y3 -
Y1
23 -
Zl
(XI, 31, ZI), (a,yz,zz),
=
(zs,ys, 2s)
cl
or
12.10
Y2 -
Y1
c! -
21
Y3 -
Y1
z3 -
21
~x _ glu
+
EQUATION
z+;+;
12.11 where a, b,c respectively.
are
the
z2 -
Zl
% -
Xl
23 -
21
x3 -
Xl
OF PLANE
z intercepts
~Y _
yl~
+
IN INTERCEPT
xz -
Xl
Y2 -
Y1
x3
Xl
Y3 -
Y1
-
(z-q)
FORM
1 on
the
x, y, z
axes
Fig. 12-2
=
0
48
FOkMULAS
FROM
E A
z
t
N
YB
X”
A
N t A B C
OQ
P
x -
-
Yn
P z
-
F I
2
P
T
T
”
R( S y
O
2 w
.
t
s
i hc
x
=
N
I
(
x,, + At,
r oe
T
O
+ B
F
A N
x
R
T E
+ C q+ D 3 d
EO
= yo + Bf, z =
y
R
y
O
I
N
.z(j
,
oees s
N
M
R
T
,
r e ir n
,NC ~
,
, B
n e t
FUM L
U
E
+ t
,
ct
+
A
OQ R P
O
nA b o +rhB c +l C + eDx =e p 0ey at a z
z
nas
o
E
I AZO + eM By L A ,+ Cz N + L ~ =A N0,
s teh d e Ogh i nhro i
F
H
ti mt et e pr
O xP T ,
1 k
h S it
GEOMETRY
R Ax O+ By L + C.z P + L =A 0
a al u ep
A 1
FU
PD
or
C
i ft
ANALYTIC
L
E
d o h n h , , .
D
SOLID
n na
A
A A
e
L
T N
1
1
2 x cas L + y COS,8 . i- z COSy w P a a
p = p C/ y a
h an x
de Xb3
=1
ef On d a e
r
4
p
0 i tr p r r a ,e,p e P xg y nz s
s op l to
eo t ,l , d .
t e a ws
m e
a n n ei
n d e et
s
Fig. 12-3
T
R
22
1
=
2y = z
w ( t t r t t x o t n s
=
x’ +
O
A
F
x’
x()
y’ + yo d
C
c
x -
y’ ZZZ 1Y -r
. o +
O IN
x
ON PS
(
T
RV UF
R
DO RO
A
l
J
5
Y0
z
(
a h o c% r e [l oc , e rird oy i (o y z y a v n a c z’ ’ s r e e[ ? o , ) t e s i oa ’ ( y vz a n yt q c s0 ee r d ’ h , o t, o f 0h r e r t t ’ e o e wq i c o h l l z go y s t
s
‘
J
e.e ro, rw , o ) e ze e da e
e o e io
el e
m
X
Fig. 12-4
d r~ r ’ t
m r m nr
.a l
d i) d i
.
] d ] d
FORMULAS
FROM
TRANSFORMATION
x
=
1
2
=
n
+ &
+
ANALYTIC
OF COORDINATES
+ 1
n
l+
y
1
y = WQX’+ wtzyf+
12.16
SOLID
3
r
INVOLVING
!
x
n
n
2
x
y
'
z
PURE ROTATION
*
% 1
p
3
49
GEOMETRY
? '
\
’
%
\
'
\
O i
=
Z
+' m
+I
y'
l=
1
+
x
=
z
+' m
m ?
T
1
X
z
y
l
+2
n
2
x
p
y
.
+z
?
a
x
%
y
g
\
Z
\ \
z
where the origins of the Xyz and x’y’z’ systems are the same and li, ' n 1 mm nl 1 2 m 2 l n 2 ; are 3, 3 the , , sdirection ; , , cosines of the x’, ,y’, z’ axes relative to the x, y, .z axes respectively.
3
1
, ,
,
\ X
’
\
, Y
, ?/‘ ’ ’ ~
’
Y
,,/ X Fig. 12-5
TRANSFORMATION
z
12.17
OF COORDINATES
Z
=
+ &
+ l&
+ I x.
INVOLVING
y
X
TRANSLATION
’
’
y = miX’ + mzy’ + ma%’ + yo 2
or i
=
n
+
n
l+
2+
zX
3
y
.'
y!
zz &z(X- Xo) + mz(y - yo) + n&
- 4
x’
=
- zO)
d-
y
x I+
F’
\
\
=
+t m
z
ROTATION
'
X
4
-' X
n
AND
n
-d z t
&(X - X0) + ms(y - Y& + 42
z
'
l
d y
COORDINATES
/
/
‘X’
(r, 0,~)
A point P cari be located by cylindrical coordinates (r, 6, z.) [sec Fig. 12-71 as well as rectangular coordinates (x, y, z). The
transformation x
12.18
=
between
these
coordinates
is
r COS0
y = r sin t
or
0 =
tan-i
r
(y/X)
z=z
Fig. 12-7
-
Y
1 '
$
l
Fig. 12-6
CYLINDRICAL
/
o
/
where the origin 0’ of the x’y’z’ system has coordinates (xo, y,,, zo) relative to the Xyz system and Zi,mi,rri; cosines of the la, mz, ‘nz; &,ms, ne are the direction X’, y’, z’ axes relative to the x, y, 4 axes respectively.
y
,
\ b
,
,
l
'
"
FORMULAS
50
FROM
SPHERICAL
[sec
SOLID
ANALYTIC
COORDINATES
GEOMETRY
(T, @,,#I)
A point P cari be located by spherical coordinates (y, e, #) Fig. 12-81 as well as rectangular coordinates (x,y,z). The
transformation
12.19
between
those
=
x sin .9 cas .$J
=
r sin 6 sin i$
=
r COSe
coordinates
is
x2 + y2 + 22 or
$I =
tan-l
(y/x)
e =
cosl(ddx2+y~+~~) Fig. 12-8
EQUATION
12.20
OF
SPHERE
(x - x~)~ + (y - y# where
the sphere
has tenter
IN
RECTANGULAR
+ (,z - zo)2 =
COORDINATES
R2
(x,,, yO, zO) and radius
R.
Fig. 12-9
EQUATION
12.21
OF
SPHERE
CYLINDRICAL
COORDINATES
rT - 2x0r COS(e - 8”) + x; + (z - zO)e where
the sphere
If the tenter
has tenter
(yo, tio, z,,) in cylindrical
is at the origin
the equation
12.22
7.2+ 9
EQUATION
12.23
OF
SPHERE
rz + rt where
the sphere
If the tenter
12.24
IN
has tenter
IN
and radius
= Re
SPHERICAL
COORDINATES
2ror sin 6 sin o,, COS(# - #,,)
the equation r=R
R’2
is
(r,,, 8,,, +0) in spherical
is at the origin
coordinates
=
is
coordinates
=
Rz
and radius
R.
R.
FORMULAS
E
FROM
OQ E
SOLID
ANALYTIC
C
tA (L
FW U L
51
GEOMETRY
E
A TTx I S
N
N HI ~P a Eb
T
D O, ,S M,
E
N y O dI
Fig. 12-10
E
1
C
2 w I
L
W Y
.
a I a
sh
b = a
i b
A I xL A X I
2
, f
A L
o re ee ac
c
ST
I X PI
H N I S T
D S I
6 fs e l
t e c
mr
r
e l
io r c y u
rf
ie
o
c i
a o l .
c
-
s
t p
d mi
u
a i
s
i t
en
l
x u
sd
Fig. 12-11
E
1
2
C
.
L
W
AO
2
L A I z A XN
J ST
X IE
P
H
I S
T
S
7
Fig. 12-12
H
1
2
$
.
Y O
z+
1
2
$
O
S
P F
8
_
N
H
E
E
$
Fig. 12-13
E
R
E
B
I
5
2
FORMULAS
FROM
SOLID
H
Note
orientation
of axes
ANALYTIC
YO
in Fig.
T
GEOMETRY
S
IF
W
H
’
O
E
E
E
12-14.
Fig. 12-14
E
1
2
P
.
L
3
A
L
R
I
A
P
0
Fig. 12-15
H
1
2 Note
xz --a2
orientation
y2 b2 of
axes
= .
PY
_z
3
AP
RE
AR
1
C
in Fig.
12-16.
/
-
Fig. 12-16
X
D
If y = f(z),
OE
A D
FF
E
t
R
N
t
~
lim f(X+ ‘) - f(X)
=
d
+h
hX
=
G
R
a
O
where h = AZ. The derivative is also denoted by y’, dfldx called di#e~eAiatiotz.
E
O
D
f
+ A
or f(x).
l
- fi
(
~
(r
~
)
~
F E
A
Ax
Ax-.O
The process of taking a derivative
N
F
t
t
E
F k
R
is
In the following, U, v, w are functions of x; a, b, c, n are constants [restricted if indicated]; e = 2.71828. . . is the natural base of logarithms; In IL is the natural logarithm of u [i.e. the logarithm to the base e] where it is assumed that u > 0 and a11 angles are in radians. 1
g(e) =3
1
&x)
0
=
3
c
.
2
.
3
1
3
.
4
1
3
.
5
c u
1
&
3
1
&
3
1
$-(uvw) 3 =
1 1 1 1
= =
du dx -H
v
3
du
_
ijii
-
du -=-
2
dv
3
du
du dx
1
dyidu
3
-
dxfdu
z n
c
6
.
u
7
dv + dx
vw-
u(dv/dx)
V
&
.
uw.
+
v(duldx)
dxfdu
=
v
-
3
dx
dy z
uv-
_
3z &
-
V
the derivative of y or f(x) with respect to z is defined as
13.1
1
l
$
-(Chai?
.
.
Z
du dx
gu gv g
8 9 1
0
. rule)
1
1
.
1
2
.
1
3
j
5
3
) )
+
E S
54
DERIVATIVES
AL”>. 1 _. .i
”
.,
13.14
-sinu
d dx
=
du cos YG
13.17
&cotu
=
-csck&
13.15
$cosu
=
-sinu$
13.18
&swu
=
secu tanus
13.16
&tanu
=
sec2u$
13.19
-&cscu
=
-cscucotug
13.20
-& sin-1
13.21
&OS-~,
13.22
u
-%
X
.3Lz2 &A?
-
x2)3/2
-
a2&z
=
2
sin-l-
dx
a
a+&GS
i31n
(
diFT1
dx x2(a2-x2)3/2
S
a2xF
8
dx= _~
x(a2-
+
5
dx @2ex2)3/2
s
x2)3/2
4
(a2 - x2)5/2
=
Wdx=
S S S S S
-x2)3/2
3
=
@=z -dx
S
-ta2
=
x3dmdx
s
14.248
14.251
sin-l:
s
14.246
14.247
$f
INTEGRALS
=
x
614x
dx
a4&iGz
-1
x3(a2-x2)3/2
=
3
+
x(a2
-
&51n
2a4&FG
2a2x2@T2
S($2 - x2)3/2 dx= Sx(&-43/2& = Sx2(& - &)3/2 ,&= S x2)3/2 dx=
-
+
3a2x&Ci3 8
x2)3/2
4
a+@? (
X
>
ia4 sin-l:
+
(a2-x2)5/2
s
(a2 -xx2)3'2
14.263
S
14.264
s
(a2-
x2)3/2
-
x2)5/2
+
a2x(a2--2)3/2
6 x2)7/2
=
(a2 -3x2)3'2
dx
=
-(a2-x2)3/2
+
_
a2(a2-
=
+
x2)5/2
a2dm
3x&z%
-
2
a3 ln
_
(a
+ y)
;a2sin-1~ a
_
“7
+ gain
a+&PZ X
.
+ igsin-l;
5
_
_ ta2 ;x;2)3’2
a6
16
X
dx
a4xjliGlF
24
7
dx
x2
la2 -x;2)3’2
x(a2
(a2 -
x3(&2 -
14.262
5
>
x
INDEFINITE
INTEOiRALS
INTEGRALS
71
ax2 f bz + c
LNVULWNG
2 14.265
s
&LFiP
dx bx + c
ax2+
=
2ax + b - \/b2--4ac $-z
If results
14.268
14.269
14.270
14.271
14.272
14.273
14.274
14.275
s
xdx ax2 + bx + c
=
&
s
x2 dx ax2 + bx + c
=
--X a
s
ax2-t
x”’ dx bx+c
S s
dx + bx + c)
xz(ax2
S S S S S
xn(ax2
ax2 + bx + c
(
dx + bx + c)
14.277
14.278
14.279
X2
1 =
-(n
- l)cxn-l
-- b c
b 2c
--
( ax2 + bx + c )
&ln
2ac
~“-1 dx ax2 + bx + c
I b2 - 2ac 23
x”-l(ax2
(4ac -
x dx (ax2 + bx + ~$2
=
- (4ac -
=
2c (b2 - 2ac)x + bc f4ac - b2 a(4ac - b2)(ax2 + bx + c)
=
- (2n - m - l)a(ax2
2ax + 6 2a +b2)(ax2 + bx + c) 4ac - b2, f
x”’ dx
+ bx + c)n--l
(n - m)b (2n - m - 1)a s
dx ax2 + bx + c
S
xnp2(ax2
dx ax2 + bx + c
S S
dx ax2 + bx + c
(m - 1)~ (2n-m1)a s
’
~“‘-2 dx (ax2 + bx + c)n
xm-1 dx (ax2 + bx + c)fl
+bx+c)n= $S(a392f~~3~~)“-I - $S(ax:";;:!+ -iS S S S S S S .I S Sx~-~(ccx~ s
x2n--1 dx (m2
dx x(ax2 + bx f
x2n-2
dx
(ax2 + bx -t- c)n
c)~
dx x2(ax2 f bx + c)~
xn(ax2
dx + bx + c)
dx ax2 + bx + c
b
-4ac
xWL-l
(ax2 + bx f CP
S
dx -- a + bx + c) c
S
=
$2 dx
use
S
dx (ax2 + bx + c)2
(ax2 + bx + c)2
b = 0
dx ax2 + bx + c
J
_ 1 cx >
bx + 2c b2)(ax2 + bx + c)
If
dx ax2 + bx + c
s
x”-2 dx -- b ax2 + bx + c a
s
60-61 can he used.
dx ax2 + bx + c b2 -
X2
$1,
=
a
:i
on pages
+ T
C
--
(m-l)a
=
s
&ln(ax2+bx+c) x?T-l
=
dx + bx + c)
x(ax2
In (ax2 + bx + c) - $
-
14.276
i( 2ax + b + dn
b2 = 4ac, ax2 + bx + c = a(z + b/2a)2 and the results on page 64. If a or c = 0 use results on pages 60-61.
14.266
14.267
In
dx f bx $
1 -2c(ax2 + bx + c)
=
1
=
- cx(ax2
+ bx + C)
b 2c
-- 3a c
dx +$ (ax2 + bx + c)2
dx -- 2b (ax2 + bx + c)2 c
1
c)~
=
-(m
- l)cxm-l(ax2
_ (m+n-2)b (m - 1)~
+ bx + c)n--l
-
(m+2n-3)a (m - 1)c
dx + bx + c)n
dx x(ax2 + bx + c)
x(6x2
dx + bx + c)2
x-~(ux~
dx + bx + c)”
72
INDEFINITE
INTEGRALS
In the following results if b2 = 4ac, \/ ax2 + bx + c = fi(z + b/2a) and the results be used. lf b = 0 use the results on pages 67-70. If a = 0 or c = d use the results $
ax
14.280
=
ax2+bx+c
a
In (2&dax2
-&sin-l
on uaaes 60-61 can on pages 61-62.
+ bx + e + 2ax + b) (J;rT4ic)
or
&
sinh-l(~~~c~~2)
14.281 14.282
x2 dx s,
ax2+bx+c
14.283
dx
14.284
=
-
ax2 + bx + c
14.285
ax2+bx+cdx
(2ax+
=
14.286
b)
ax2+ 4a
bx+c
+4ac-b2
16a2
14.288
14.289 14.290 14.291 14.292 14.293
=
6az4a25b
bx+c
(ax2 + bx + c)~/~ +
“““,,,“”
J
d ax2f
bx+c
dx
ax2+bx+c
S“
X
ax2+bx+c X2
S S ax2 Scax2 x2 S+x2+%+c)3’2 = cdax2 : bx+e+: SJ s S, S dx (ax2 + bx + c)~‘~
2(2ax + b)
(4ac - b2)
x dx
(ax2 + bx + dx + bx +
x2(aX2
ax2 + bx + c
2(bx + 2c)
~)3’~
(b2 - 4ac) \/
43’2
a(4ac - b2)
+ bx + c
(2b2 - 4ac)x
+ 2bc
dx + bx +
c)~‘~
=
ax2 + 2bx + c - &?xdax2 + bx + c +
2c2
(ax2 + bx + c)n+1/2dx
=
dx
1~x2 + bx + c
-- 3b
14.295
ax2 + bx + c
axz+bx+c
x
14.294
.
(ax2 + bx + c)3/2 b(2ax + b) dp ~ ax2+ 3a 8a2 dx - b(4ac - b2)
=
14.287
dx 8a
ax2+bx+c
S +ifif+
dx
(QX~
axz+bx+c
b2 -
26
2ac
Scax2
dx + bx +
43’3
dx
x
ax2+bx+c
(2ax + b)(ax2 + bx + c)n+ 1~2 4a(nf 1) + (2% + 1)(4acb2) (a&+ 8a(n+ 1)
S
bx + c)n-1’2dx
4312
.
INDEFINITE
14.296 14.297
S s’(ax2-t
x(uxz + bx + C)n+l/z dx
=
(ax2 + bx + C)n+3'2 cq2n+ 3) 2(2ax
dX
bx + ~)n+l’~
=
dx + bx + ++I’2
x(ux2
s
73
.
_ $
(ax2 + bx + ~)~+l’zdx
s
+ b)
(2~2 - 1)(41x - b2)(ax2 + bx + +--1/z 8a(n1) dx (2~2 - 1)(4ac - b2). (‘ (61.x2 + bx + c)n--1E
+
14.298
INTEGRALS
1 =
(2~2 - l)c(ux’J
+ bx + c)n--1’2 dx
JPJTEORALS Note 14.299
14.300
14.301 14.303
14.304
14.305
that
14.308 14.309 14.310
dx
~
s
involving
=
+
X3
x2 - ax + c-9 + 1 (x + c-42
=
__ = x3 + CL3
$ In (x3 +
ClX
s
.(
s
x2(x3
u3)
=
'(z3yu3)2
'
1
-+
u3)2
%(X3
+ a3)2
s
x2(x3
dx +
+ &In
14.312
=
1 -
--
=
CL62
xdx + u4
=
x4
S
x3
dx
~ x4 + a4
3u5fi
(x + a)2
x2 xm-3
-
m-2
a3
~
x3
-1 1)x+-’
c&3@-
-
1
In
4u3fi
&
-L 4ufi 14.314
2x-u a \r 3
x2 - ax + a2
2x +
=
tan-l
In
-4-.---
3u6 s
3a6(x3 + u3)
xm-2 -
=
a3)
=
~
tan-l
3utfi3 tan-’
3
&,3(x3 + as)
=
u3)2
x4 + a4
S
2
-
1
u3
dx
+
F
- 3(x3 + US)
dX
I'
+3tanP1
x2 - ax + a2
+ &n
a3)
x2 + axfi x2 - uxfi
x3
x dx + u3
[See 14.3001
dx + a3 dx
-2
JNTEORALS
14.311
-
(x + a)2
=
x(x3+u3)
(xfcp
x2 3a3(x3 +
=
x-’ dx
s
G-4 In
3u3(s3 +a3)
dx
x9x3+
dx
s
1
s
s
43
x2 - ax + u2
1
-
x2 dx (x3+
+
2x-u 7
tan-l
X
=
s
x3
a by --a.
14.302
~3)
a32
xdx (x3 + c&3)2 =
~
(ax2 + bx + c),+ l/i
3ea+ a3
a\/3
x2 dx
s
x3 - u3 replace
2”~ s
x2 - ax + cl2
u3
~ x dx x3 + a3
s
14.306 14.307
for formulas
JNVOLVING
dx
--
x(ux2 + bx + c)“-~‘~
s
u3
s
+
xn-3(x3
INVOLYJNG + a2
c?+* a* 1
--
u3)
tan-1
2aqi
+ c&2
-!!tC-LT
22 - CL2
$
x2 - axfi
+ a2
x2 + ax&
+ u2
$ In (x4 + a4)
--
1 2ckJr2
tan-1
-!!G!- 6
x2 - a2
a
74
14.315 14.316
INDEFINITE
INTEGRALS
dx x(x4 + d)
s s
dx x2(x4
14.317
+ u4)
+-
=
1 2a5&
dx x3(x4
.
+ a4)
=
14.322
14.323
14.324
14.325
14.326
14.327
14.328
14.332
dx
.I’ x(xn+an) fs
=
S
xm dx (x”+ c&y
I’
dx xm(xn+ an)’
xn + an
‘, In (29 + an)
=
s
xm--n dx (xn + (yy-l 1
=
2
x”’ dx s-- (xn - an)’
14.333
14.334
&nlnz
=
=
an S
s
-
an s
x”’ --n dx (xn + an)T
dx xm(xn + IP)~--~
xm--n dx (~“-a~)~
1 an -s
xm--n dx
+ s
(xn-an)r-l
=
S
dx = m..?wcos-~
!qfzGG
m/z
dx xmpn(xn + an)r
tan-l
CiXfi ___ x2 -
a2
INDEFINITE
14.335
xp-1 dx
INTEGRALS
75 x + a cos [(2k - l)d2m]
1 ma2m-P
I‘----=xzm + azm
a sin [(2k - l)r/2m]
x2 + 2ax cosv where 14.336
xv- 1 dx X2m
s
-
m-1
1
a2m
=
2ma2m-P PI2
cos kp7T In km sin m
x
(’
2*
ka
x -
tan-l
+ a2 a cos (krlm)
a sin (krlm)
k=l
+
.
2ax ~0s;
m-1
1
where
x2 -
m
k=l
-&pFz
14.337
+ a$!
0 < p 5 2m.
{In (x - 4
+ (-lJp
> ln (x + 4)
0 < p 5 2m.
x2m+l
xP-ldX + a2m+l 2(-l)P--1 (2m + l)a2m-P+1k?l
=
m
sin&l
x + a cos [2kJ(2m
a sin [2krl(2m
+ l)] + l)]
m
(-1p-1 (2m + l)az”-“+‘k?l
-
tan-l
cossl
In
x2 + 2ax cos -$$$+a2
+ (-l)p-l In (x + a) (2m + l)a2m-P+ l where
14.338
O
m
+
s
14.340
sinaxdx
=
=
%sinax+
14.342
=
(T-
siyxdx
=
14.345 14.346
14.347
=
-$)sinax
sin ax
s
dx
+ a S Ydx
= =
sin2 ax dx
+ (f-f&--$)
cosax
5*5!
X
S sin ax xdx S sin ax
s
cos ax
3*3!
dx
sin ax
ax-(aX)3+(a2)5-...
s sinx;x
lNVOLVlNC3
x cos ax ___ a
y-
14.341
14.344
2ax cos
-- cos ax a
=
‘ssinaxdx
14.343
x2 -
O
X--
a (
s
14.513
85
e""
s
s
INTEGRALS
1 5 a S
- b2) sin bx - 2ab cos bx} (a2 + b2)2
_ eaz((a2 - b2) cos bx + 2ab sin bx} (a2 + b2)2
dx
eu sinn bx dx
=
e”,2s~~2’,~
eaz co@ bx dx
=
em COP--~ bx (a cos bx + nb sin bx) a2 + n2b2
in sin bx - nb cos bx)
+
+
n(n
- l)b2
a2 + n2b2 S n(n - l)b2 a2 + n2b2
S
eu sin”-2
em
bx dx
cosn--2 bx dx
86
INDEFINITE
INTEGRALS
HWEOiRA1S 1NVOLVfNO 14.525 14.526 14.527 14.528 14.529 14.530 14.531 14.532 14.533 14.534 14.535 14.536
s
14.538 14.539
=
S S S$Qx
xlnx
xlnxdx
=
xm lnx
dx
-
$1
=
2
nx-4)
--$ti
1 m+1
-
lnx (
=
14.541 14.542
see 14.528.1
;lnzx
P 1+x dx J
=
x ln2x
~Inn x dx
=
-lP+lx
s
dx
xln
=
x
-
lnnx
[If
In (lnx)
=
In x
+ 2x
n = -1
+ lnx
ln(lnx)
dx
xlnnx
=
+ $$
+ s
* .
-
n
m = -1
see 14.531.
S S
S
Inn-1
=
x ln(x2+&)
In (x2 - ~2) dx
=
x In (x2 - u2) xm+l
=
sinh ax dx
x sinh ux dx
x2 sinh ax dx
+ (m+3!)~~x
x dx
In (x2 + ~2) dx
xm In (x2 f a9 dx
.*a
.
+ (m+2t)Iyx
n xm+l Inn x -m+l m+1
=
+ l
+ (m+l)lnx
xmlnnxdx
S S S
see 14.532.1
In (lnx)
=
xm dx
2x lnx
nfl
X
S Sf& S S S
-
-
In (x2* m+l
INTEGRALS
14.540
[If m = -1
s
If
14.537
lnxdx
Inx
xm Inn-1
s
2x + 2a tan-1
&)
--
2
m+1
!NVOLVlNO
~
=
x cash ax -- sinh ax U
=
u2
coshax
z
2x + a In
cash ax a
=
x dx
-
$sinhax
S
Y$gz
sinh (cx
c-lx
+ a**
INDEFINITE
14.543
'14.544
14.545
14.546
14.547
14.548
14.549
14.550
sinLard
14.552
14.553
14.554
sinizax
*
i In tanh 7
xdx sinh ax
=
1 az
ax
sinhz ax dx
=
sinh ax cash ax 2a
-
s
s
x sinha ax dx
,I'
dx sinh2 ax
~
I‘
I
.(
cash 2ax 8a2
x2 4
a
px dx
sinh (a + p)x %a+p)
=
p)x aa - P)
sinh (a -
' sinh ax sin px dx
=
a cash ax sin px -
' sinh ax cos px dx
=
a cash ax cos px +
p sinh ax sin pz a2 + p2
ax+p--m qeaz + p + dm
1
s
(p +
=
ad~2
dx
S
p +
q sinh ax
=
dx -
’ sinh” ax dx
dx
S sinhn ax ~ x dx
.I’ sinhn ax
xrn
cash a
ux
--
m a
+ -n-l
=
- cash ax a(n - 1) sinhnP1
ax
=
- x cash ax a(n - 1) sinhn--l
ax -
dx
=
I’
p + dm
tanh ax
p - dm
tanh ax
xm--l
sinhn--l ax coshax _ -n-1 n an - sinh ax (n _ l)xn-’
sinh ax Xn
2apGP =
=
In
1
=
q2 sinh2 ax
xm sinh ax dx
~
a(p2 +
dx
q2 sinh2 ax
p2 +
-
=
>
- q cash ax +” q2)(p + q sinh ax) P2 + 92
dx
q sinh ax)2
p sinh ax cos px
c&2+ p2
dx
p + q sinhax
S
2
a = *p see 14.547.
s
14.558
-~--
X
--
-- coth ax
=
sinh ax sinh
.I'
x sinh 2ax 4a
=
[See 14.5651
=Fdx
s
=
87
,. . . .
I a
dx sinh ax
I‘ p” S
14.561
=
-
S
14.556
14.560
dx
x
S
14.559
I jJ$: / 05 * . 5*5!
s
14.555
14.557
ax
s
For
14.551
=
INTEGRALS
cash ax dx
S
sinhnP2
cash ax
a
S QFr -- n-2 92-l
[See 14.5851
ax dx
[See 14.5871
dx dx
S sinh*--2
as(n - l)(n
ax
1 - 2) sinhnP2
ax
-- n-2 n-l
~- x dx
S sinhnP2 ax
88
INDEFINITE
INTEGRALS
INTEGRALS
14.562
cash ax dx
14.563 14.564
cash -& ax
14.565
s
a
x sinh ax -- cash ax a a2
=
- 22 cash ax a2
=
z
*
dx
=
- dx cash ax
14.570 14.571 14.572 14.573 14.574
14.575 14.576 14.577
14.580
14.581
=
xcosh2axdz
s
dx
cosh2 ax
s
=
s
4+
P)
%a + P)
=
a sinh ax sin px - p cash ux cos px a2 + p2
cash ax cos px dx
=
a sinh ax cos px + p cash ax sin px a2 + p2
dx
s
dx cash ax - 1
s
cash ax + 1
=
$tanhy
=
-+cothy
=
!? tanh a
xdx
x dx
cash ax - 1
--$coth
=
dx
(cash ax + 1)2 dx
(cash ax - 1)2
7
7
-$lncosh + -$lnsinh -
&tanh3y
=
& coth 7
-
&
coths y
tan-’ ln
s war + p - fi2
( qP
s
7
&tanhy
p + q cash ax
dx (p + q cash ax)2
f
=
S dx =
14.582
+
sinh (a - p)z + sinh (a + p)x
=
cash ax sin px dx
cash ax + 1
s
. . . + (-UnE,@42n+2 (2%+2)(272)!
~tanh ax a
2(a -
s
S
5(ax)6 + 144
x sinh 2ax cash 2ax 4a -8a2
X2
=
+
(ad4 8
sinh ax cash ux 2a
;+
cash ax cash px dx
s
[See 14.5431
s
-
S
14.570 14.579
cosh2 ax dx
s
. . .
= -
14.569
+
; a
X
s
(axP 6*6!
4*4!
.
cash ax
s
14.567
+
lnz+$!!@+@+-
X
cos&ax
14.566
-
x2 cash ax dz
.
cash ax
sinh ax
=
x cash ax dx
.
INVOLVING
=
+ p + @GF
q sinh ax -a(q2 - p2)(p + q cash as)
)
P 42 -
P2
dx p + q coshas
S
***
INDEFINITE
In
1
14.583
p2 -
s
dx q2 cosh2 ax
INTEGRALS
p tanh ax + dKz p tanh ax -
2apllF3
=
89
I
14.584
dx
!
p tanh ax + dn
In
p tanh ax - dni
2wdFW
=
s p2 + q2 cosh2 ax
1
1
tan
--1 p tanhax
dF2
14.585 14.586
xm cash ax dx
.
coshn ax dx
s
coshnax
14.587
dx
coshn--l
= =
ax sinh ax
14.591 14.592 14.593
s s s
14.594 s 14.595
I
ax +
(n-
=
sinh2 ax ~ 2a
sinh px cash qx dx
=
cash (p + q)x 2(P + 9)
sinhn ax cash ax dx
=
sinhn + 1 ax (n + 1)a
coshn ax sinh ax dx
=
coshn+ l ax (n + 1)a sinh 4ax ~ 32a
dx sinh2 ax cash ax
=
14.597
S
______ dx sinh ax cosh2 ax
zz -sech a2 + klntanhy
S
14.600
S
14.601
S
z
dx
=
sinh
;s,hh2;;
dx
=
cash ax + ilntanhy a
dx cash ax (1 + sinh ax)
[See 14.5591
i tan-1
ax
- 2,‘a2 coshn--2
ax ’
cash (p - q)x
[If
n = -1,
see 14.615.1
[If
n = -1,
see 14.604.1
ax _
- 2 coth 2ax a
-
,jx
n-2 -n-l
sinh ax AND c&t USG
a
a
ax dx
-- x 8
S
=
coshn--2
ax
_ t tan - 1 sinh
[See 14.5571
2(P - 9)
14.596
14.599
1 In tanh a
+
dx sinh ax cash ax
dx sinh2 ax cosh2 ax
l)(n
INVOLVCNG
S
14.598
S
dx coshnPz
sinh ax cash ax dx
=
n-1 n ?$!?
x sinh ax a(n - 1) coshn--l
=
sinh ax dx
s
ax
sinh2 ax cosh2 ax dx
f-
xn--l
a n-1
sinh ax a(n - 1) coshn--l
INTEGRALS
,('
_ m a s
an
-cash ax (n - l)xn-1
s
14.590
l.h=7
xm sinh ax a
=
>
sinh ax
csch ax a
J
~- xdx coshn--l:
ax
.:,".'
INDEFINITE
90 14.602 14.603
S S
dX
sinh ux (cash ax + 1) dX
sinh ax (cash
14.604
14.605
14.606
14.607
14.608
14.609
14.610
14.611
14.612
14.613
14.614
14.615
14.616
14.617
14.618
14.619
14.620
S S S S S S S S S S S
tanhax
dx
x
=
tanhs ax dx
=
=
=
ax
tanhn + 1 (72 + 1)a
1 2
1
X2
=
=
(ax)3
3
- 2
-
bxJ5 +
-
-2k47 105
15
ax _ k!$
dx
=
+ ?k$
tanhn ax dx
cothax
dx
=
- PX
P2 -
_
dP2 - q2)
- tanhn--l ax + a(?2 - 1)
=
x -
coths ax dx
=
i In sinh ax -
cothn ax csch2 ax dx
- dx coth ax
dx
=
S
=
-
=
-
-coth2 ax 2a
cothn + 1 ax (n + 1)a
- i In coth ax
$ In cash ax
...
...
(-l)n--122n(22n - l)B,(ax)2n+ (2n + 1) !
(-l)n--122n(22n - l)B,(ax)2n-’ (2% - 1)(2?2) !
In (q sinh ax + p cash as)
tanhnw2 ax dx
coth ax a
coth2 ax dx
s
Q
-
42
i In sinh ax
=
-
x tanh ax + -$ In cash ax a
X
S S
1 2a(cosh ux - 1)
tanh2 ax 7
k In cash ax -
=
p+qtanhax
S S S
-
‘, In sinh ax
xtanhzaxdx
s
-&lntanhy
ilntanhax
xtanhaxdx
tanh ax dx ___
=
1 2a(cosh ux + 1)
+
tanhax a
tanhn ax sech2 ax dx
=
7
i In cash ax
tanhe ax dx
~ dx tanh ax
klntanh
- 1)
ux
=
edx
=
INTEGRALS
-t . . .
1
+
... >
INDEFINITE
14.621
14.622
14.623
s
s
x coth ax dx
1 i-2
=
x coth2 ax dx cothaxdx
1
ax
x2 -
=
-
2
x coth ax + +2 In sinh ax a b-d3 135
-$+7-v
X
14.624 14.625
14.626
14.627
14.628 14.629
14.630
14.631
14.632 14.633
14.634
14.635
14.636
14.637
14.638 14.639
S S
dx
p+
qcothax
cothn ax dx
S S S S S S S S Sq + p S
- PX
=
sech ax dx
cothn--l ax + a(n - 1)
-
=
+
i tan-l
. . . (-l)n22nBn(ux)2n--1 (2n- 1)(2n)!
9 In a(P2 - q2)
-
P2 - !I2
=
cothn-2
tanh ax ___ a
sech3 ax dx
=
sech ax tanh ux + &tan-lsinhax 2a
xsechaxdx
na
+ 5(ax)s + 144
=
x sech2 ux da
x tanh ax a
=
=
sechn ax dx
=
=
“-2 9
9
S
Gus 4320
dx
i In tanh y coth ux a
csch2 ax dx
=
- -
csch3 ax dx
=
- csch ax coth ax 2a =
cschn ax na
- -
+
. . (-lP~,kP 2n(2?2)!
[See 14.5811
p+qcoshax
sechnP2 ax tanh ax + n-2 a(n - 1) m-1
cschn ax coth ax dx
. . . (-1)n~&X)2”+2 (2n + 2)(2n)!
+
...
$ In cash ux 5(ax)4
lnx--m++-- (ad2
=
dx sechas
csch ax dx
- ~sechn ax
=
sinh ax a
.A!-= sech ax
S S S S
ax dx
eaz
=
sechn ax tanh ax dx
+ ---
(p sinh ax + q cash ax)
sech2 ax dx
“e”h”“,-jx
91
INTEGRALS
$lntanhy
ssechnm2
ax dx
+ ** *
INDEFINITE
92
14.640 14.641
14.642
14.643
S S
ds= csch ax
i cash ax
x csch ax dx
S Sq + p S
csch*xdx
1 2
=
x csch2 ax dx
s
= =
X
14.644 14.645
14.646 14.647 14.648
dx csch ax
cschnax
S S S
sinh-1
=
S
a
-
$+f
=
sinh-1
S
(x/a)
dx
I
14.650
14.651
14.652
14.653
14.654
sinh;~W*)
dx
S S
E dx
S
; dx
S S
14.656
14.657 14.658
cash;:
S S S r
(u/x)2 2.2.2
--
- ln2 (-2x/a) 2 -
1 3 5(a/xY 2*4*6*6*6 l
+
1x1 < a
+
l
...
l-3 * 5(alx)6 2*4*6*6*6
_
x>a
...
*Jr&F2
:In
X
(
)
(x/a)
-
d=,
cash-1
(x/a)
> 0
i x cash-1
(x/a)
+ d=,
cash-1
(x/a)
< 0
&(2x2 - a2) cash-1
(x/a)
-
i a(222 - a2) cash-1
(x/a)
+ $xdm,
=
f
(x/a) > 0, dx
E dx a
= =
x tanh-19
dx
x2 tanh-1
z dx Il.
ix@??,
4x3
cash-1 (x/a)
-
$x3
cash-1
+ Q(x2 + 2a2) dm,
-
C f
ln2(2x/a)
if cash-1
_ cash-1
(x/a) X
tanh-1
...
cash-1
(x/a)
> 0
cash-1
(x/a)
< 0
3(x2 + 2~2) dm,
cash-1 (x/a)
> 0
cash-1
< 0
=
dx
(da)
_
1*3(a/x)4 2*4*4*4
-
+
l
+ 1. 3(a/x)4 2.4.4.4
+ __ (a/~)~ 2.2.2
(x/a)
1.3 5(x/a)’ 2*4*6*7*7
x cash-1
i
cosh-;W*)
_
l
=
x2 cash-1 E dx
+ if cash-1 14.655
+ 1 3(x/a)5 2.4~505
(xlaJ3
2.3.3
_ sinh-1
=
&FT2
9
=
a
x cash-’
cschn--2 ax dx
x m x 4 +a
-
X
cash-1
S
(2a2 - x2)
z +
ln2 (2x/a) 2
=
X
...
[See 14.5531
)
g sinh-1
+
dm~ sinh-1;
(
f dx
dX
p + q sinhax
xsinh-1: =
a
Q
cschnm2 ax coth ax -- n-2 a(n - 1) n-l
-
z dx
x2 sinh-1
E-P Q
=
g dx a
x sinh-1
x coth ax + -$ In sinh ax a - 1)B,(ax)2n-1 v*x)3 + . . . (-l)n2(22n-1 e&-y+1080 (272 - 1)(2n) ! -
=
dx
ax
X --a
14.649
INTEGRALS
x tanh-1 = =
7 F
(x/a)
+(a/5)2 +
1. 3(a/x)4 2-4-4-4
292.2
+ 1.3 * 5(a/x)6 2*4*6*6*6
+
...
1
(x/a) < 0 r
1 ln a + v a X (
z + % In (a2 - x2)
+ # x2 - ~2) tanh-1: + $tanh-1:
(x/a)
a
+ $ln(a2-x2)
[- if cash-1 (x/a) > 0, + if coshk1 (x/a) < 0]
x < -a
INDEFINITE
tanh-1
14.659
14.660 14.661 14.662 14.663
14.664
14.665
S S S S S S
tanhi:
14.669 14.670
=
“+@$+&f$+... a
(z/u)
dx
=
_ tanh-1
!! dx a
x coth-’
'Oth-i
(x/u)
=
7
a
(xia)
' sech-'2
a
x sech-1
+ +(x2 - ~2) coth-’
dx
=
F
+ fcoth-1:
dx
=
_
;
dx
=
_ coth-1
dx
J? dx
(x/a)
(x/u)
+ a sin-l
(x/u),
sech-1
(x/u)
> 0
r x sech-1
(z/u)
-
(x/u),
sech-1
(x/u)
< 0
=
dx
(x/u)
-
+a~~,
sech-1
(x/u)
> 0
+x2 sech-1
(x/u)
+ +ada,
sech-1
(x/u)
< 0
-4
=
14.674 14.675
4 In (a/x)
S S
csch-1
” dz
=
x csch-1
U
x ds a
x csch-’
S
csch-;
(x/u) dx
S
xm sinh-15
s
xm cash-’
S S
a
x”’ coth-’
dx
E
U
U
T
=
=
14.677
xm sech-1
S
1 * 3Wu)4
_
...
2.4.4.4
’
sech--1
(s/u)
z+--
=
xm csch-’
: dx a
U
5
> 0
if x > 0, -
if x < 0]
[+
if z > 0, -
if
1. 3(d44
+
-
sech-1
(x/u)
x < 0] ...
O<x a
-
cash-’
E -
cash-’
i
--&s$=+
xmfl
coth-’
+
? U
-
dx
~
a
mt1
E -
-J?m+l
+ am
xm+1
m+lswh-‘s
U
cash-1
(x/a)
> 0
cash-1
(x/u)
< 10
S x2 SCL2- x2 S Zm+l
dx
u2 -
Zm+l
+ 1
xm dx ~~
dx
seckl
(da)
> 0
sech-1
(s/a)
< 0
xm+l
m+l
csch-1:
c a
< 0
- $T$$ + ' '3(x/u)4 -.... -u<x 0, -
if x < 0]
15
DEFINITE
DEFINITION
OF A DEFINITE
INTEGRAL
a)/n.
Let f(x) (b -
INTEGRALS
the interval into n equal parts be defined in an interval a 5 x 5 b. Divide Then the definite integral of f(x) between z = a and x = b is defined as
of length
Ax
=
b
15.1
f(x)dx
s a
The limit If
will f(x)
=
certainly
S
if f(x)
f(x)dx
S
dx
=
lim
dx
S S S S f(x)
b-tm
dx
=
b-m
continuous. theorem
=
g(x)
calculus
the above
definite
integral
a
=
c/(b)
-
in the interval, the definite limiting procedures. For
integral example,
s(a)
dx
f(x)
dx
dx
=
lim t-0
f(x)
dx
=
lim
f(x)
a
dx
if b is a singular
point
if a is a singular
point
b
f(x)
c-0
dx
a+E
F6RMULAS
INVOLVING
b
DEFINITE
INTEGRALS
b
{f(x)“g(s)*h(s)*...}dx
S
=
a
f(x)
dx *
a
b g(x) dx * s a
Sb h(x) dx a
2
* **
b
b
cf(x)dx
=
c
S
where
f (4 dx
c is any constant
cl
a
15.9
of the integral
a
GENERAL
15.8
f(a + (n - 1) Ax) Ax}
b--c
f(x)
a
S S Sa S Sb Sb
. . . +
a
iim n-r--m
b
15.7
+
b
Cc f(x) -m
a
15.6
Ax
or if f(x) has a singularity at some point and can be defined by using appropriate
b
15.5
+ 2Ax)
b
m f(x)
a
15.4
f(a
b
b d -g(x) (I dx
=
If the interval is infinite is called an improper integral
S S S S
is piecewise
f
the result
a
15.3
Ax + f(a + Ax) Ax
then by the fundamental
by using
b
15.2
{f(u)
exist
= &g(s),
can be evaluated
lim
n-m
f(x)
dz
=
0
=
-
a
b
15.10
f(x)dx
a
15.11
a f(x)dx
b
f(x)dx
=
a
15.12
S
f(z)dx
=
SC f(x) a
dx + jb
(b - 4 f(c)
f(x)
dx
c where
c is between
a and b
a
This aSxSb.
is called
the mearL vulzce theorem
for
94
definite
integrals
and is valid
if f(x)
is continuous
in
DEFINITE
b s
15.13
f(x) 0)
dx
=
$
This is a generalization g(x) 2 0.
of 15.12 and is valid
LEIBNITZ’S
RULE FOR DIFFERENTIATION
S
a
a and b
* a
dlz(a)
15.14
95
where c is between
f(c) fb g(x) dx
a
and
INTEGRALS
if j(x)
and g(x)
are continuous
in
a 5 x Z b
OF lNTEGRAlS
m,(a) aF
F(x,a)
dx
S
=
xdx
f
F($2,~)
2
-
F(+,,aY)
2
m,(a)
6,(a)
APPROXIMATE
FORMULAS
FOR DEFINITE
INTEGRALS
In the following the interval from x = a to x = b is subdivided into n equal parts by the points a = ~0, . . ., yn = j(x,), h = (b - a)/%. Xl, 22, . . ., X,-l, x, = b and we let y. = f(xo), y1 = f(z,), yz = j(@, Rectangular
formula b
S(I f (xl dx
15.15 Trapezoidal
=
h(Y, + Yl +
i=
$(Y,
Yz
+ ..*+
Yn-1)
formula b
S
15.16
j(x)
dx
+ 2yi
+
ZY,
+
...
+
%,-l-t
Y?J
a
Simpson’s
15.17
formula
(or
b
I‘ a
f(z)
dz
DEFINITE
15.18
15.19
15.21
INTEGRALS
xp-ldx
1+x
=
-
o
for
--?i
sin p7r ’
=
RATiONAl
+ 4Yn-l f
Yn)
OR IRRATIONAL
Oq>o
d2
apl2
=
S S
9)
X
0
15.36
+
p=o
=
m sin px cos qx dx
0
15.35
m=l,2,...
X
0
S
2*4*6..*2m ... 2m+l’ 1.3.5
)...
p > 0
-%-I2
15.34
=
m=1,2
2’
UP) r(4)
=
x cos29--1z dx xl2
0
dx
0
0
s
=
0
15.31
15.33
= ;
cot325 dx
0
15.30 s
n
0
a/2
S
=
and m =
2mf (m2 - 4)
II
mx cos nx dx
FUNCTIONS
indicated.
i 7~12 m, n integers
T/2 s
otherwise
m, n integers
0
15.29
TR10ONOMETRIC
0
=
dx
0
15.28
unless
=
ii sin mx sin nx dx
D cos
INTEGRALS
dx
a + b cosx
$2-3
(ala)
DEFINITE
15.46
2r; S S S S o
(a + b sin x)2
257
15.47
0
15.48
S
=
o
dX
o
dx l-2acosx+az
=
cos mx dx l-2acosx+a2
Ial
ram l-a2
> 1
a2 < 1,
m = 0, 1,2, . .
r
sin ax2 dx
S naYn
cos ax2 dx
=
=
i
S S S S S S
w sinaxn
=
-
m cos axn dx
=
---&
0
15.52
1
dx
2 II-
0
15.51
r(lln)
sin & ,
rfl/n)
cos
jc sin
0
15.54
S
m cos x dx 6
dx=
-@/dx
=
0
15.55
-!?$i?dx
=
0
15.56
=
0
6
2Iyp)
Sk (pn/2)
’
2l3p)
c,“, (pa/2)
’
m sin ax2 cos 2bx dx
=
k
=
i
Ol n>l
2,
0
15.53
b’)312
laj < 1
77 In (1 + l/a)
=
(az-
O
{F(vii)(n) F(~P-l,(O)}
- F(vii)(O)) +
. . .
20
TAYLOR
TAYLOR
f(x)
20.1
=
SERIES
FOR
f@&) + f’(a)(x-
SERIES
FUNCTIONS
a) + f”(4(2z,-
OF
42
+
ONE
1
.
VARIABLE
. . . + P-“(4(x
-4n-’
+ R,
(n-l)! where R,, the remainder 20.2 20.3
Lagrange’s Cauchy’s
after
n terms,
form form
R,
=
R,
=
is given
by either
f’W(x
of the following
forms:
- 4n n!
f’“‘([)(X
-p-y2
- a)
(n-l)!
The value 5, which may be different in the two forms, continuous derivatives of order n at least.
lies between
a and x.
The result
holds
if f(z)
has
If lim R, = 0, the infinite series obtained is called the Taylor series for f(z) about x = a. If tl-c-3 a = 0 the series is often called a Maclaurin series. These series, often called power series, generally converge for all values of z in some interval called the interval of convergence and diverge for all x outside this interval.
BINOMIAL
20.4
(a+xp
=
&I
+
nan-lx
=
an
+
(3
Ek$a
an--15 .
Special
+
+
20.5
(c&+x)2
=
a2 + 2ax + x2
20.6
(a+%)3
=
a3 +
3a2x
+
3ax2
20.7
(a+x)4
=
a4 +
4a3x
+
6a2x2
20.8
(1 + x)-i
=
1 -
x + x2 -
x3 + 24 -
20.9
(1+x)-2
=
1 -
2x
-
20.10
(1+x)-3
=
1 -
3x + 6x3 -
20.11
(l$
20.12
(1 fx)i’3
20.13
(1 +x)-l'3
20.14
(l+z)'/3
=
x)-l'2
=
=
+
dn--
1,‘,‘”
an--2z2
+
(‘;)
@--3X3
23
+ +
4ax3
+
x4
...
4x3 + 5x4 -
-l<x 1, p = 1 if x < -11 0:
20.31
see-l x
=
cos-‘(l/x)
=
E2 -
20.32
csc-1 x
=
sin-1
=
k+‘-
(l/x)
I4 > 1 2-3x3
+
2
*l-3 4 * 5x5
+
...
14 > 1
/
TAYLOR
112
SERIES
SERIES FOR HYPERBOLIC 20.33
sinh x
=
x+g+g+g+
20.34
cash x
=
l+$+e+e+...
20.35
tanh x
=
x-if+z&rg+...
20.36
cothx
=
~+fA+E+
20.37
sechx
=
l-~+~x&+
20.38
cschx
=
1 -
FUNCTIONS
-m<x<m
***
--m<x<m (-l)n-l22n(22n
...
(-I)*-
122nBnx2n-
+
. . . (-l)nEnx2n (2n) !
; + g
X
-
E.
sinh-lx
-
G
+
2.4.5
1
+
, l
+
1x1 <x
l)B,Gn--1
...
+
20.40
cash-1x
20.41
tanh-1~
=
20.42
coth-1s
=
=
-r-
lnj2xl
+ A--
1x1 < 1.
‘*’
k{In(2x)-
l-3*5 + 2.4.6.6~6
1*3 204.4~~
(
esinz =
20.44
ecosz
[‘ii
L-
1
E~~~I:~~~:
:::I
I4 < 1
x2
SERtES
x5 + . . .
x4
--m<X<m
l-$+x!pz!+...
--m<x<m
(
)
20.45
etanz
20.46
ez sin x
20.47
e2
20.48
In lsin xl
=
In(x(
20.49
~nlcosxl
=
-$
20.50
In ltan x1
=
x2 7& In 1x1 + -py + g-
20.51
In - (1 +x) 1+x
=
x -
x
+ifxZl if x 5 -1
>
1x1 > 1
e
=
’ ‘.
x+$+g+$+...
1+x+;i--s-z
=
-
(&+&+,.::“,Y”,x6+.**))
MlSCELLAN(KMJS 20.43
2
0 < 1x1 < x
= 1
2
0 < /xl < a
(2n) !
1.3 ’ 5x7 + 2.4.6.7
3x5
...
...
(-l)n2(22”-l-
-0.
1x1