DEFINITIONS
INTERPRETATIONS
• !leuristies. The defi nite integral captures the idea of
addin g the va lues or a run clion over a continuulll .
• Riemann sum. A su itably we ig hted sum of values. A
de fin ite illlcgra l is the li mi t ing value of sueh sums. A
Ri e man n s um of a funct ion f de f incd on [II ,h] is
de te rmincd by a partitio n , which is a fini te division of
[{I .h·1 into subintervals, ty p ically exprcssed by
{I =xu
• Area under a cun·e. I ('lis nonnegati ve and cont inuous on
fro m each sub intcrval , say c; fro lll [x; " x ;]. The associated
" i)(Xi  X;
Rie mann sum is: ~f(c
j
d.
I
A regula r pa rtitio n has sub intervals all the samc Icngth,
{\. x =( h ~{I)/II , x ; = {I + i {\'x. A parti ti on 's norm is its
max im um subintc rva l length. A left s um takes the left
e nd poi nt Cj=Xj_, o f each , ubinterval ; a r ight su m , thc
ri ght endpoin t. An uppe r sum of a continuous f takes a
po int <"; in each subint c rva l w here thc max imu m \alue off
is ac hieved; a lowe r sum , thc m in imum val ue . E.g., tl e
uppe r Rie ma nn su m of cos x on [0,3] with a reg ul ar
partitio n of II int cr va ls is the left sum (since the cosine is
decreas ing o n the inte rva ll :
f ICOs(U I):t)+
I]..:i.
i I
11
Il
A '(x )=f(x ) (valid
endpoints).
[a,h], the n ( >tf ed dx !; ivcs the a rea hetwee n the xa xi s
• I>
a nd the graph. T he area func tion A (x) =
.c'f (n dl gives
       ..
A(x)
' 1,
a .(/
x
U
A rough estimatc o f a n integ ral may be made by est ima ting
the ave ra ge va lue ( by in spec tin g th e gra ph ) a nd
multip ly ing it by the le ng th o f the illl erva!' (See M el/II
Ii' /l/I! Theorem (MVT) jiJl" illtegra /s. in the Theon ' sccti on. )
• Accumulated Change. The int egra l of a ra te o f c hange of
a q ua ntity over 3 time interva l gives the tOlal change in th e
q uanti ty over the time inte rva l. E.g., if v (l) =s '(I ) is a
ve locity (the rate o f e ha nge of pos it ion), then v (l )1'lt is the
approxim ate displ acement occ urring in th e time inc re me nt
/ to t +{\./ ; addi ng the di sp lace me nts for all time inc rc m c n.ts
gives the approx imate change in pos ition ovcr the enti re
time inte rva l. In the lim it of small time in crcme nts, o ne
."
gc ts the exac t to ta l di splacement: / v(f) dl =s (h)~'(IIl .
."
• Differentiation of integrals. r unctions a rc oftcn de fined
a" integra ls. E.g., the "si ne in tegral fu nction" is
Si(x)= ./,:"( Si;l I )d l.
To dirfe re n t iatc s uch, u se t he ' ccon d pa rt of the
fundam ent al theo rem: Si '(x)= sin x /.\'. A fu nction
such
as
( 'f (t)(l/
•a
l 'j (x )dx=
lim ~f( C i)Llsj.
11 .lxll .() i
T he lim it is sa id to cx ist ifsome number S (to be called the
integra l) satis fies the roll ow ing: Every £> 0 admits a 6
such tha t all Riem a nn sums on partitions of [tI ,h] with
norm less than 6 dilTer from S by less than £ . If there is
suc h a valuc S. the I'unct ion is sa id to be int egra ble a nd the
value is de noted
l 'j(x )dx or I f .Thc function must be
(I
(l
bounded to be integra bl e. The fu nction f is ca ll ed the
integrand a nd the points {I a nd h arc called the lo,ver lim it
a nd uppe r li mit o f integralion, respectively. T he word
integral rerers to thc rormation of
jaI>f
from f and [(I .h], as
\Ve il as to the resuiting value ifl here is one .
• ·\ ntiderhative. A n "ntidc riva tivc of a fi.lI1ction
f is ,[
func ti o n A whose de rivative is f : A'(x )=f(x) for all x in
some do ma in (usua lly a n inter va l) . A ny t'.I'O antiderivati ves
of a fu nctio n o n an in terva l differ by a co nstan t (a
consequence of the Mea n Va lu e T henrem ). E.g. , bot h
J1.. j" j
W dl=
dx "
J1.. A (x ')= A ' (x') 2x = 2.xj(x'1.
2I (x 
a )' a ll(I iIX "., 
(j .lil''
c rln g
de noted
I.
(IX
.
a re ant 'd
i c n.
vatlvcs
0f
j)·(x){I (x)dx=/(;)j~b(l (x)Clx.
1
jj(x)dx,' ,s
a lll ide rivatives o n a typica l (often
inten a!' E.g. , ( lo r x < ~ I , or for x > I).
Ml~ ~~ ~~~~~~B""~I~
~b
a
unspecifi ed )
X dx =! x '  1 + C.
j '!x'  i
Th e consta nt C, w hich may have any real valuc, is the
constant of integr a tion . (Com puter prog ra ms, and this
cha rt . may om it the constant, it being understood by the
kn owledgea hle user that the g iven a nl idcrivative is just one
representative of a ra m ily.)
is f ; and
if (x) ldx.
• Fundamental theorem of calculus. O ne part o f the
theore m is use d to evaluate integrals: Iffis continuo us o n
[(I, h]. a nd A is a n ant idcr iva ti ve off n n th at inte rval , then
j
·"f (x) dx=A(x )I"C= A (b)  A(a).
(,
{/
The other part is used to construct antide rivativcs:
If f is conti nu ous o n [a , h) . th en the fun cti on
A (x)l= ( j U)d l is an a ntiderivative of f on [(I ,h] :
. a
~
b
apprehend or cvaluate. In effect, the "a rea" is smoothly
rcd istributed w it hout changing the integral;' value. !I' g is
a fun ction w ith cont inuous deriva tive andf i, continuous.
then
" lI )dn = ...I'df (.9(f ))g ' (f )dl,
1"f(
where c. d are
points with g (C)= il a nd g(rI )= h.
In pract ice, su bstit !!!r lI = g(l); compu te t11/ = g'(I)dl; and
find what t is when 1/ = 0 and I/ = h. E.g.. I/ =sin t e ffe cts the
transrorm ation
I> ( )
Iff is non negative , then .(".f
x dx is no nnegati ve.
i[ ' j(x )dx i< t
II!I
/ .,'
lnnnunuu : ~ ~
whkh becomes
X  {I ,
an exp ression for the fam ily or
n
integra tion can be cha nged to make an in tegral eas ier to
Use this
in tegra l evalu at ions w ith rough
ove restimates o r u nderes tima tes.
l/.hJ, the n so
,
• Cbange of variable formula. An integrand and limit, of
"
to ch ec k
U
C
is attained
i\1VT for Integra h
L · (b alS l "f (xldX S, M . (/) a) .
Iff is int egrablc on
f
somewhere on the inLe rval : b  a . a .f(x) dx =.f( ; ).
\ u du= I' " "~
!  s inI co~ I til.
/.",....,;
of
of
on
·
I
Oil
by 2a.
I ., T he indefinite in tegra l ora runction!
'
dx
In thc case g  I , the a\eragc value of
• Integrability & inequalities. A co ntinuo us fun cti on o n a
closed inter val is integra bl e. Integ rab ility On [tI .h] imp li es
int egra bi lity on c losed sub inte rva ls of [tI .h]. Ass umin gfi s
in teg rable , if L "'f(x) 5. M for a ll x in [{I ,hJ, th e n
~
IIiII
·Mean value theorem for integrals. Ir f an d ~ a rc
conti nuo us o n [II ,h], then there is a S in [a.h] such Lhat
f«,
L_
in\o l vin g
. a
THEORY
II
c OI11[l osition
A( u )= ( "f (tldt. T o d i frcre nt ia te, u sc the c hai n rule
x~
• Definite integral. The definite inte gra l off frol11 {I to b
may bc described as
is a
and the fundamcn ta l th eorem :
(x )= y ,, + ('fll )d l.
.
;.o...
::;.
 ;.;.
~

Illay bc defined by average VII[lIe = 'h= / f(xl dx.
Integral curve. Imag ine that a run cti on f determines a
slope fix) lor eac h x . Plac ing linc segments w ith s lo pe
fi x) at po ints (x.y) fo r va ri ousy, and doi ng thi s lor va rio us
x , one gets a slope field . A n antidc ri vati ve o ffi s a fu nc tion
whose graph is tange nt to th e slope field at each po int. T he
gra ph o f the a nt ide ri va ti vc is called a n integ ral curve of
the slope fi e ld.
• Solution to initial value problem . The so lution to the
diiTerential eq umiony ' ",/"( x) with initia l va luey(xnl = y" is
3
F unda m e nta l T h e orem
fl.x)
the arca accu lllul ated up to x. Iffis negati ve, the integral
is the negat ive o f the area .
• Average value. The awragc va llie 011 over a n int erval [a ,h]
i
for o nesi de d de rivatives
."
(l
2
cos ' l dl , si nce ! 1 s in 'l = cos I
for () < 1< 1[/ 2. T he for mul a is ollen used in revcrse,
staJ1ing w ith ], "1'(.9 (x ))O' Cddx.
ec Technic/iII!s on pg. 2.
• i'atural logarithm . A rigorous defini tion is In x =
(' Ida
. T he change of variable for mula with 11 = I I I
U
. ,
. , ., I
yiel ds ./ ,
/.,  I
/., \
a rill = . , I cc1 d l =  _, . I dt sho\\ ing that
In ( f Ix) = ~I nx. The other elementary propertie, of the
natural log can likewise be easily derived from thi,
definition. in this approach, a n inverse function is deduced
a nd is derined to be the natural exponent ial function .
•,
C
n
INTEGRATION FORMULAS
Other routine integrationbyparts integrands arc arcsinx,
XCOSX, and xe UX ,
• Rational functions . Every rational function may be
written as a polynomial plus a proper rational function
(degrec of numerator less than degree of denominator). A
proper rational function with real coefficients has a partial
fraction decomposition: It can be written as a sum with
each summand being either a constant over a power of a
lincar polynomial or a linear polynomial over a power ofa
quadratic. A factor (XC)k in the denominator of the
rational function implies there could be summands
Inx. xnInx. xsinx,
• Basic indefinite integrals. Each lormula gives just one
antidcrivativc (all others dillering by a constant from that
given). and is valid on any open interval where the
integrand is defined:
~
1/ !
1
*
J x"dx=~(n
n+l
f 1dx=lnlxl
x
1)
fe"xdx=e;x(k * O)
feos x dx=sin x
fsin x dx= cns x
fl'~:' =arctan x
x
f~=81'Csin
Iix'
• Further indefinite integrals. The above conventions hold:
J~()t x dx=ln lsin xl
ftan x dx=lnlsec xl
.A.!...+",+~"
xc
(xc)"
A factor (x'+bX+C)k (the quadratic not having real roots)
in the dcnominator implies there could be summands
Isec x dx=lnlsec x + tan x l
x'+bx+c
".
J~sc x dx=ln lcsc x+cot xl
Math software can handle the work, but the following case
l xal
f~=lln
x'a' 2a x+a
Iix ldx= lxlxl
X
f l x~+\: a.,=lnlx+lx'+a' l=sinh
dx=cosh x
1~+lna
.,=Inlx+lx'a'icosh 1~+lna
fld~
xa
(take positive values for cosh· l )
flx' ± a'dx=!xl x'a'
± ~Inlx+ Ix' ± a' i
(Take same sign. + or . throughout)
fla
2
2
• Common definite integrals:
I
l'
1" x"dx= n+l
II
fU  Idll,ju"du(tI > O,fll(U'+ll "du (handled with
substitution
and fill'+l) "du (handled with
w=1I 2 +1) ,
II
= tan t).
,l
th~ integrand is
),dx= ) , + 2 < 2.4.
GEOMETRY
• Areas of plane regions, Consider a plane region admitting
an axis slIch that sections perpendicular to the a.xis \ur) in
lcngth according to a known function L(p). tl"'I'''' b. The
area of a strip of width I'!.p perpendicular to Ih~ axis at p i,
M = L(pll'!.p, and the total area is
IMPROPER INTEGRALS
Over [1I.bJ is
IIJ
on [II,B] for all B>II, then I" j(x)dx ~ ,!i lll " jlx)dx
provided the limit exists.
• Substitution. Refers to the Change of variable formula
(see the Thmrv section), but ollen the formula is used in
reverse. For an integral recognized to have the form
f."F(y(x»)y' (x)dx (with F and g' continuous), you can put
tllI=g'(x)dx, and modify the limits of integration
[ "F(,q(xl)y' (x)dx= j~(I("1
. F(u)dll.
,1/((1)
In effect, the integral is over a path on the IIaxis traced out
by the flJJlction g. (I I' g(b) = g(a) [the path returns to its
start], then the integral is zero.) E.g., u= I+x' yiclds
l_ x _ 1 1 [I _ l _ ? d 11 (1+ x,
')
l+x'cx 2 . o l+x'_x x 2 n
Substitution may be used lor indefinite integrals.
E.g..
l we
A= .(,L(p )dp. E.g"
/J .
iCJ
I "j(x)dx = lim 1."j(x)c/x.
d,1
;1 •
•t
In each case, if the limit exists, the improper integral
converges. and otherwise it diverges. For f defined on
(_ 00,00) and integrable on every bounded interval,
·
f .~jlx)dx
= ,[i~ l'j(x)dx+ ,li~.,["j(x)dx
deI
(the
.4
choice of c being arbitrary), provided each integral on the
right converges.
• Singular integrands. Iff is defined on (lI ,b] but not at
x=a and is integrable on closed subintervals of(a,b], then
),"" j(x)dx (hI
= }i!~
Ih, j(x)dx
provided the limit exists. A
similar definition holds if the integrand is defined on
[1I,b).
E.g.,
l',,4x'
".!.,dx
lim
" '
is
II
l',,4x'
~dx =
1J
,lim
., arcsin(f)=!f,
2
2
' , f~
E.g.
l+x 2 dxlfdllllnalln(l+x')
u  2
2
'
' 2
f y(x)",1I (x)dx,z:tl,I I fY'(x)
y(x)dxln1y(x)l,
"~
_lI(X)"
I
.
(C)
t
• Volumes of solids
Consider
a
solid
admitting an axis slich
that
crosssections
perpendicular to the
axis vary in arca
according to a known
function A(p), a"'p",b.
The volume of a slab of'
thickness I'!.p pl'rpcndi"ular to
(l
For indefinite integration, fll dv=uu fv dll,
The procedure is used in derivations where the iemetions
arc gencral, as well as in explicit integrations. You don't
need to usc "II" and "v." View the integrand as a product
with one factor to be integrated and the other to be
differentiated; the integral is the integrated factor times the
one to be differentiated, minus the integral of the product
of the two new quantities. The t~lctor to be integrated may
be I (giving v=x).
E.g., farctan x dx=x arctan x f 1':x ,dx
ih~
axis at p is III ' =.1(plt!.{',
and the total I'olumc is V = ."('"i\(p)dp, Lg.. a pyramid
a
('
V=j""/I.(z)dz= ("rrrlz)'dz.
ct
The common formula is j'''1t dU=llul" j'''u dll.
bp
: IIp
a
·Solids of re\olution. Consider a solid of r~I'()lution
det~rmined by a known radius ilmction r(:) , a"' :'" h, along
it s axis of revolution. The area of th~ cross·,,,ctional
"disk" at : is A (:)=nr(:)l. and the I'olum e is
feu rxly' (x)dx=e yrxl ,
['u(x)u' (X)dX=Il(X)U(x)l~  ['U(X)Il' Cddx.
~3"~
V= f~'( l  ~r dz = S~lh ,
n
'/4x' dx=arcsm 22
• Integration by parts. Explicitly,
[a,b]. Sometimcs it i. simpler to I' iew a region as bounded
by two graphs "over" the y·axis, in which case the
integration variabk isy.
having square horiLollwi Cfllssscctions. with bottom ~ide
length !\ and height Jr. has crossst:L:tional area
A(:)=[.\'( I://r)]' at height ... Ih ,,)!tUllC is thu>
Singular Integrand
.,
.I"." if/(x)  jlx)idx. prm i(kd K(X) "f(x) on
~
xdx= Iim(1e 1J)=1
u
Some general formulas are:
(J
convergcs since
lIn lxhi).
Ji x  a i( x~ndx=....l....b(lnlxa
a
def
TECHNIQUES
a
0'
)
APPLICATIONS
In gencral, the indefinite integral of a proper rational function
can be broken down via partial Ihlction decomposition and
linear substitutions (of lorm II =ax+b) into the integrals
(X)
IJ
(
Thus r(
Likewise, lor appropriate}:
l
1 l':x' . dx
bounded by 1/2.112: on [fI , l.! and is always less Ihan 1/.\"'12.
It
convcrges
to
a
numhcr
lcss
than
a1
• Unbounded limits. III is defined on [11,00] and integrable
cos 26) equals cos'6 or
r/
II
x~ a + x Dh
IJ
II
appropriately:
1 x
1
where C. D are seen to be C=  D= ....l....
l .
l "sin xdx=2
d8= R
l ""sin' (1d8=1";'1cos28
,,2
4
II =g (x),
1 x
The above integrals arc llseful in comparisons to
establish convergence (or divergence) and to get bOllnJs.
the arca of the region bounded by thc graphs or!, and K
1+cos20 d8= TI
j"'rr"cos'8d8=1"n
,,2
4
±
x=oo, p=O or < 1 div~rgL's at x=oo.
1
'11 .
.
;j dx converges to I and
~ dx dl\crg~s.
E.g..
2:~
(l
.,
j 'Ir'x'dx= IT"4
To remember which of II, (I
sin'6, recall the value at zero.
A"+B,,x
(x'+hx+()'"
shoulcJbel'lmiliar.lfa"b,( xa ~ X1
l)
substitution
lxla' x' +~arcsin'!
2 x'dx=
N
tfdXI
I
oe: x(lnx)"   (1I  1l0nx)" I'P> cOJ1\crges at
'1
+ +
fSinh
I·
)/dx convcn.!.cs
for p> I. divcr!.!cs otherwi se.
.; x ( 'I~
nx
~
....
E.g. ,
Al +Blx
fcosh x dx=sinh x
1
1d
"lor p < I ,(I Iverge s 01I
'
u'1 x"
x converg~s
lcrWlse.
rtf is not defined at a finite number of points in an interval
)a
intervals between such points, the integral f.'1' is defined
Ifth" solid lies betw"~n two radii rl(:) and r2(:) at each
point: along Ih" axis of revolution. the crosssections
are "washers," and the volume is th~ ob\ iOllS
as a sum of lell and righthand limits of integrals over
appropriate closed subintervals, provided all the limits exist.
differcnce or volumes like that above. Sometim es
a radial coordinate r, a", r'" b, along an a~i s
[II,b], and is integrable on closed subintervals of open
E.g.,
fl ~dx= lim
IX'
a ' II
1" ~dx+ lim [I ~dx
 IX'
b ' 0 ·1,
X'
if the
limits on the right werc to exist. They don't, so the integral
diverges.
• Examples & bounds .
1 ~dx
x
I
converges for p > 1, diverges otherwise.
2
perpendicular to the axis of revolution .
paramctri/es the heights /r (r) of cylindrical
sections (shells) or the solid parallel to ihc ax is or
revolution. In this case, the area of the shell at r is
A(r)=2nr/r(r), and the volum e of the solid is
V=
["A (r)dr = ~("2ITrh(r
)dr.
,
~o
• \rc lenl1;th A graphY=I(x) between x=a andx=h has length
V= I"/1
+f' (x)"dx.
f(a) " I
.
f ib»)
.'
T,,= ( :r+ ;~f(aIh ) + :r h. T im IS al so the
A curve C parametrized by «x(1).
1 I"
I"
,
y(t i), aSlsh. has length c ds= " "X'
(t) ( y' d( t) dt .
• \rea of a surface of re".lution The surface generated by
revolving a graph y=I(x) between x=a and x=h about the
xaxis has area 1"2rr.f(x)/I+f'(xl'dx. If the
TAYLOR'S FORMULA
'Ta~lor pol~nomials .
The nth degree Taylo r polynomial of
average of the left sum and right sum fo r the g il'en partition.
T he approx imation re mains val id iff is not positive.
• \lidpoint rul~ T his e va luates the Riemann sum on a
regu lar parti tion with the sampling g ive n by the midpoints
1c2rr.yds= I"
I,
I
.,
.,
2rr.yU) x' (t) + y' (t) dl.
PHYSICS
j 'a(u)du, X(t)=X(tIl) + .,"['v(u)du. E.g .. the height X(I)
~
of an object thrown at time 1,,=0 li'om a height x(O) =x"
with a vertical velocity 11(O)=VO undergoes the acceleration
g due to gravity. Thus v(l) = I'(VII) +
and x(l) = x" +
1(
l' ( u)du
= v,,I:t
II
VII
,qu)du = x" + I,,,ti,ql'.
• \\orL If F(x) is a variable force acting along a line
parametrized by x. the approximate work done over a sIIlali
displacement ~x at x is ~W=F(x)~" (force times
displacement), and the work done over an interval [a.h] is
In a nuid lifting problem, often ~W=~F'''(Y), where
h(y) is the lifting height for the "slab" of fluid at y with
crosssectional arca A(y) and width ~y. and the slab's
weight is ~F=pA(y)~y , p being the fluid's weightdensity.
T hen W= tpA(y)h(y)dy.
0
1..,
f U' I(c )(x  c)"
n.·
(provided the derivati ves ex ist). When
c=O, it's also called a M a cLaurin polynomial.
'Ta~lor 's I'ormula
Assume I has 11+1 co ntinuous
derivatives on open interval and that c is a point in the
interval. Then lor any x in the interval.f(x)= P,,(x) +R,,(x).
(n ~l!!.1"''' I II(q).
(X C),,+I for somc
ite m Solution to initial value problem; an example of that
type is in Motion in one dimension. In those, the expression
for the derivative involved only the independent variable. A
basic DE involving the dcpendent variable is y'=ky. A
gcneral DE where only the firstorder derivative appears
and is linear in the dcpendent variable is y'+p(l)y =q(t).
Generally more difTicult arc equations in which the
independent variable appears in a Ihlt{ nonlinear} way;
c.g.. .1" = y2  x . Common in applications are secondorder
DEs that are linear in the depcndent variabJe; c.g ..
y"=ky. x2y"+xy'+x2y=O.
·Solutions. A solution of a DE on an interval is a function
that is dillcrentiable to the order of the DE and satisfies the
equation on the interval. It is a general solution if it
describes virtually all solutions. if not all. A general
solution to an 11th order DE generally involves II constants.
each admitting a range of real values. An initial value
problem (IVP) for anlllh order DE includes 11 specification
of the solution's valuc and III (krivativcs at some point.
Generally in applications, an IVP has a unique solution on
the remainde rs fo r the Maclaurin polynomial s of I(x) =
(1)"
I n (l +x), 1I • X"+I.
(n + 1)(1+;)"
T he re is a
S between
0 a nd x such that In( I + x)
firstord~r
.
=
x ,£ + _ _1_ _
111" =
interval :
linl' ar DE . Thc equation y'=ky,
dy
I
(
1
Eac h
• Slmp,on s rule The weighted sum:{1'1 ,+
, :1211
' I (In the
int erval
[a , h ]
yield s
rule
S impso n's
s= b;;a (f(a)+4f(a ! h)+.f(b»).
A
T his is also the integral of the Simpson' s ~ ;?
quadra tic that interpo la tcs the
fun ction at the three points. For
a reg ul ar pa rtitio n of [a, h] into
a n even num be r /1= 2", of ~':::;:""L'\:
i!f1'
U
b
interval s, a formula is:
h
til
I
III
1
S2m=::r U (a)+4 ;~/(a )+[ 2i+ 1 ]" + 2 ;~/(a+ 2i'/I )+ fib)}
whe re " = (h  a)llI.
2 3(1+;)'"
"
impson 's r ul e is e\ act 011 cubics.
• . ... ror bounds As x approaches c. the remainder ge ne rall y
becomes smallcr, and a given Taylor po lyno mial provides a
bettcr approximation of the fun ct io n value. With the
assump tio ns and notation above, if !f'''
by
M
on
th e
"(fl~l!! Ixc l" i I
interval,
II
then
(x)1 is bOlmded
If(x)  P,. (x )1
tor all x in the interva l. E.g .. fo r
~"I<\.
because the thi rd derivative o f eX is bounded by 3 on (1.1).
• Big 0 notation T he statement f (x)=p(x)+O(x m )
f(x)  p ix) .
(as x .....O) me ans tha t
x '"
IS bounded near x=o.
(Some authors require that the limit of thi s ra tio as x
approac hes 0 ex is!.) That is, f(x)p (x) approaches 0 at
essentially the same rate as x'" E.g.. Taylor's for mula
has
continuous thi rd derivative on an open interva l containing
O. E.g.. sinx=x +O(x J). [Si m ilar relations can be infe rred
fro m the ide ntit ies in the item Basic MacLaurin Series.]
'I'llupital's rule. This resol ves indeterminate ratio s or
(H or ~). IfFI~f(x)=
0 =
F~g(x)
and if IJt.nJ(x) = 0
= IJ'~: g(x) are defined and glx)"O. ror x ncar a (but not
'1
I
I'
fix)
nccessan y at a), t 1en }~ g(x) =
()
I'
=}~
F( x )
'd d
g' (x) proVI e
the latter li mi t exi sts, or is infin ite . The rule also holds
whe n the limits ofIand g are infinite. No te that/'(a) and
g'(ll ) are not required to exist. To resol ve an in deter m inate
dill"e re nce (00 _ 00) . try to rewrite it as an indcterminate
ratio and apply l ' Hli p ita l's ru le . To resol ve a n
indeterminate exponential (O".loo. oroo") , take its logarithm
to get a product and rewrite this as a suitable indeterminate
ra tio: apply L: II"pita l 's rul e; the expone ntial of the result
resol ves the original indeterminate exponential.
F'or}I~~
r II"
Inlx
l/x
0
x yougctan dfdr
111 Imx01
/ x L rml.\«)_l
/x~'
where lim
some interval containing the initial value point.
• Basic
S
(s
between c and x
varying with x). The expression fo r
R ,,(x ) is called the Lagrange form ofthe remainder. E.g ..
implies I (x )=f (O) of'(0)x+!f"(0)X 2+O(X3 ) if I
• F'.. mples. A ditkrcntial equation (DE) was solved in the
I' 2"IIh)h.
summand is the area o f a trapc/oid \vh o~l.· top is the
tangent line segment th ro ugh the midpoin t.
<,x=l+x+x2/ 2, w ith error no more than ftlxl" = 0 .5 Ix !",
W= t'f'(x)dx.
a+
o f ea ch
w he re R,,(x) =
• '\Iotion in one dimen.ion . Suppose a variable
displacement x(1) along a line has velocity v(t)=x'(t) and
acceleration a(l)=v'(t). Since v is 3n antidcrivative of a,
the fundamental theorem implies: v(t) = V(III) +
M, f
Iat c is P,,(x ) = I(c) + f'(c)( x c) + !!f"(c)(xC)2 + ... +
I,
../ g enerating curve C is parametri7cd by «x(1), .1'(1».
u:stsb, anti is revolved about the x axis, the area is
x 0
Ixlx =
ell = I.
SEQUENCES
·S~qu~ncc
SequC'llces an: rllnction~ whose domain~
consist of all integers greater than or equal to somc initial
integer, usually 0 or I. The integer in a sequelll"e at /I i<
usually denoted with a subscripted symbol like a" (rather
than with a functional notation a(/I» and is ealkd a term
of the sequence. A sequence is olien referred to with an
expression for its terms. e.g. , 1//1 (with the domain
understood), in lieu of
fuller notation like :
{l / n},;"" ,orn l , l / n(n = l, 2, ... ).
.) I
I~nt.
SCllu,
An arithmetic sequence
un has a
difference d bdwccn slIccc"s ih! \alll e~ :
u,,=a,,_I+d=Uo+d·/I. It is a scqucllIial \(,fsion of n linear
C0l111110n
l[ll1ction. the common diflerence in the mil' of slope. A
geometric sequence, with terms Nfl" has a common ratio r
hetwcen successive values: gn=I:",lr=g"r". e.g.. 5.0. 2.5.
1.25, 0.625. 0.3125 ..... It is a sequential "'r,ion of an
exponential function. the common ratio in the rok of base.
·ConH~rl!: 'nee A sequcnce (un] Co"\'erges ifsuml.! number
L (called the limit) satisfies the j(,lIolVing: l::.Iery £ >0
admits all N such that la,,LI < £ I'"·ullll " .v. Ifa limit L
e xists, the re is only one: on(' says i u,,: t..:onn~rgc~ to L. and
writes a,,L, or ,!i!" all = L. If a sequence does not
converge, it diverges. If a seqllclll'L' a" di"'~rgcs in 'lh::h a
way that every M>O admits all N suth that a,, >Jl h,r all
"" N. then one writes a,, + oo. E.g .. if Irl < I then r"..... O: if
r= I then rll+I; otherwise r" diverges. and ifr> 1. r" x ,
• Boun 'd n unotone
Sl ph Il
;\11 iilcr~a:.ing
sequence'
that is bounded above converges (to OJ limit less thall or
equal to any hound). This is a fundamental "let about the
real numhcrs, and is basi~ to series convcrgt.:llcc tL·~t S .
dy
rewntten lit =ky suggests y =kdt where lyFkt+c. In
this way, one finds a solution y=CeAt. On any open
interval. every solution must have that form, because
y'=ky implies
1ft (ye
M),
where yr kl is constant on the
interval. Thusy=Ce kl (C real) is the general solution. The
unique solution with y(a)=y" is y=y"ek(l "J. The trivial
solution isy .. O. solving any IVP y(a)=O.
• (.eneral
firstorder
line r
DE
Consider
y'+p(t).I' =q(l). The solution to the associated
homogeneous equation h'+p(t)1r =0 (dhlh =p(t)dt)
with h(a)=\ is hU)=expl I'p(u)dul.
If .I' is a solution to the original DE. then (ylh)'=qlh.
where
y(t)=Y
y=h fq / h. The
solution
 [q(u)h(u) 'dul.
with
y(<<)=y"
is
NUMERICAL INTEGRATION
• General notes . Solutions to app lied p ro blems often
involve definite integral s tha t cannot be evaluated easily. if
at all. by finding antiderivativcs. Readily available
so ftware using refined algorithms can evaluate many
integrals to lleeded preci sio n. T he fo ll owi ng methods for
approximating l"f(x)dx are elementary. Thro ughout.
II
is
the number of intervals in the underlying regular partition
and Ir=(ha )/II.
·Trapezoid rule. The line connecting two points on the
graph o f a positive func tion together with the underlying
illterval on the x axis l'lrl11 a trapezoid whose area is the
average of the two func tion values times the length of the
interval. Adding these areas up over a regular partition
gives the trapezoid ru le approximati on
3
SERIES OF REAL NUMBERS
A ~cri cs is a scqucn..:c ohtained by adding the
." 'r
\
values of another sequence L;a" = ao+...+a,. The Hduc
" u
of the serics at N is the sum ofvalucs lip to a,· and is ~3lkd
\.
La" "n+...+a\. The scries itself is
La". The an arC called the terms of the series.
,.
t' A series L:a" converges if the sc:tJucnce or
a partial sum:
II
II
denoted
II
(00\
II
"
II
partial slims converges. in which case the limit o f the
sequence of partial SUI11S is called the sum "I' the If the series converges. the notation I'.)r the ~eric s it.. e lf
stands also for its sum:
La" =
1/
II
\
lim
.\ .
"
Lna"
.
Series continued
La
An equation suc h as
~
Z
fI
a nd it s
III
.01I1III
"11l1lI
La" may stand for
looks like a pseries. but is not directly comparable to it.
II
form L;a,.". where r is a real number and a" O. A key identity
I)
1/
\
1 .S ,
L;,," = I+r+r2+ +r;\=_I__ (,.* 1) It implie s
" "
...
1 ,.
.
is
L;,." = _ I _ (ifl ,.l< ll( a lsoL; a l"" =a(_ L  1)). and
I r
" ,
I'I"
that the scries diverges if Irl > I. The serics diverges if
r =± 1. T he l'~Hlvcrge ncc and ross iblc Sum l)f any geometric
series can be determined lIsing thl.' pl"cccding k1J"l11ll1a.
""
r~a l
L:
num ber.
,
"
" 1
IV
are unbollnded: L: Ii ~ 1 + ~ .
1
1/
...
strictly decrease in ahsolutc valuc and approach a limit of
zero, then the series (:onvcrgcs. Moreover. the truncation
error is less than the absolute value of the first omitted
I
L:(  I Y1 a ll  ±(  1)fl a" I'< a\ f I ,
term :
"
I
I
If
(assuming
,
It:.
• Basic consideration" For any
if
L:a"
converges, then
" A
co nverges. and conversely. If a" 7t H, then
ll
L:a
di verges. (Equivakntly. irL:u lI co nvl'rgcs, then
says nothing about. e.g., L;
"
1 , ;\
(In .....
O). Thi s
series of positive terms is
h II
un inCfl!as ing sequence of parti a l sums; if the sequence of
partial sum s i ~ hounded the ;;eli e~ COI1\crges. This is the
fo undation of all the follow ing criteria for co nvergence.
• Integral te,t & e~timat< Assume I is continuou s,
positive. and decreas ing on (K.oo). Then L;/(II) converges
if and o nl y if
1 /Ix)dx .
" A
converge s. If th e se rie s
J. /(x)dx .
\
L; /(II )S L; f(n) +
co nverges, then
1I
ri gh t s ide
A.
A
II
ovcr~stimating
.,
,,\11
II
l..
III
the
'I., X
len
s ide
underestimating the sum with error less thanfl N+ I).
Integr al test
,
"
".
"
".
f( ·\'+I)
~(II!)"" = ~,
1
"
;\ +
1~ln'
12
• Po\\er series A power series in x is a sequence of
\
polynomial s inxofthc Ilmll L;a"x" (N=O.I. 2, ",J,

1
1:\
1.2018 .. " an
I,,1
L; a ni S ,,1
L;l a" l.
ll
n
II
A power series inx c(or "centered at cO' or "about c") is written
()
Replacing x with a real number q in a power series yields
a series of real numbers. A power series converges at q if
Ixl" l \
x",
co nverges, and
2"n'!. _ lxl
,
'~ ) .,. Iun  ,  " 11" = 2 <1 =>l x l<2.
/~I:'''lr /I . 2 n ! (n+])x
which. with the ratio test. shows that the radius of
conve rgence is 2 .
Geometric po\\"er \eries , A power series determines a
function on its interval of convergence:
x l • f(x)= L;a" (xc)", One says the series converges
IR,,(l)I= (11+ 1)(11+ ;)"
1 1 . ().
' <; n+
(  I)" ,
so In2= L;  n  '
,
11~I .. r
and
R>O
If
f(x)= L;a,,(x c)"(lx  clII
necessarily the Taylor coetficients: Q,, = f''''(l')/n!. This
means Taylor series may be found other than by direetlv
computing coctficients. Diffe rentiating. the geometri c
(I  x)·
"
'= L;(// + llx"
I
"
(Ix i< n
()
• Ba ic I\l:tcLaurin wri,
_ 1_= I+x+\,2+ = L;x" (lxlI x
. . ., 11 II
arctanx=x~ ~+~  "' = L: (  J)1/ ~ ~"
,{
~
" "
The li)lIowing hold tor all real x:
(l xj '5 ])
I
2// + 1
.
x~ x: l
" " x"
e" = I + x + ?T+ ",+"'= L.. i
....
oJ,
II
un.
x:!
(  1)"x:!rI
Xl
cos x = lx+?T+4t  " ' = L; (,)),
......
II II
_n.
•
x: i
x!i
SInx=x  ;l,+~,"' = "L;
• ,.),
II
(1)"x:.!"
I
_n + Jl'
,
(?
I)
• Binomial \crie. For P" O. and ti". Ixl < I .
to the tl1l1et,on. The series L;x", i.e., the sequence of
"
u
p(p
(l+x)I' =I+px + ?,  I) x'+ .. , = L; (") x·,
\
1 V i '
L;x" =1 +X+X2+ ... +X\= ·~(x * 1).
polynomials
1/
X
()
converges for x in the interval (1, I) to 1/( Ix) and
II
II
~X
L;2·;{ "x" = 2~i=(~)"=2x. _ ~,_.
"
II
3" t) 3
3 1 x / .{
for
f (x) = L;(I" t.t:c)" ,
/I
J;.",
The binomial coefficients are
P(~ l),
(")=
k
()
II
(]=l. C)=p,
(J =
and I"p choose k ")
P(p1)(p2!'" (pk+ 1)
k.
Ifp is a positive integer,
C~O Illr k >p.
A ll rig ht!l "'·_""rI CII. 'f' part 01 till'
rn.l~ b..: rerro.lu('cu or
l(':1n~l1l1t\o:dl!\nn ... fom l ,orInJn)~,
c lcdr'(lI1IC or II1cchalm:al. n1<:ILlJIn~
[lhuIOCOP).rcc:orJUll:,l.oran\ InfOlll\lI1tl~tOr.lgC
(I
a,,£b,,(II ~ N)
derivativc there is f' (x)= L;na" (xc)" "
"
,
The differentiated series has radius of convergence R, but
may diverge at a n endpoint where the original converged.
Such a tl111etion is integrable on (e  R. e + R). and its
integral vani shing at cis:
converges. but not abso lt1!cly.
·(ompari ~m te\1 Assll m~ u".b,,>O.
has a limit, thell L;a" converges.
•
If e=O. it is also ca lled a Maclaurin series. The Ta)lor
series at x may cOJl\'ergc without converging to f(x). It
converges to fix) if the remainder in Tay lor's f(mnu la.
series g ives __1_ ., = L;nx"
·Intenal ofcomerJ.:ence The set ufrealnu1l1bcrs at which
a power series converges is an interval , called the interval
of convergence. or a point. If the power series is centered
at c. this set is either (i) (00,00); (ii) (cR ,e+R) lor some
R>O. possibly together with one or both endpoints; ur (iii)
the point (' alonc. In case (ii), R is called the radius of
convergence 01' the power series, which may be 00 and 0
t(, r cases (i) and (ii i). respectively. Convergence is absolute
for Ix cl < R. You can often determine a radius of
convergence by solving the inequality that puts the ratio
(or root) test limit les s than
I. E.g. , for
~
1"( ,)
,)
,
11
L;a" (xc)" =all +a, (x  c) +a, (x  c)' + "'.
"
A serics converges conditionally if it
If L;b" converges and either
I)
·Computinl!
Such a function ·is differentiahle on (eR ,c+R). and its
comcrgence I r L;ia" i converges, that is, if
L:a
.. .f k '(
L; T(xcl h = f(c)+f'(c)(x e )+ T ( x  c)'+ .. ·.
It
Jlublica ri o!l
4.
tcol)vergcs absolutciy: , then
argument (see below) implies equality for x=1.
• Taylor and \lac! durin 'ric The Taylor series
about e of an infinitely diiTercntiable functionIis
"
Lanx".
The power series is denoted
n
'~' for Ixl<1; a remainder
,
remainders at x=1 tllr the Maelaurin polynomial s ot
Taylor's
formula
above) satisfy
In( I +x) (in
Ix/31< I. The interval of convergence is ( 3,3).
·(alculus or po\\er series. Consider a function given by a
power series centered at c with radius of convergence R:
X>
The ini tial (geometric) series converges on (1.1). and the
integrated serics converges on (1,1). The integration says
R (x)= __I {'" I II('')·(X  c)''' 11 (I: between c and \'
"
(//+1l!'"
.,
.,
1; varying with x and II), approaches 0 as " .... 00. E.g.. the
POWER SERIES
E.g.,
.'\'+1
.'V
' 1
"
;\
1~l n '
iU •Absolute
"11l1lI
root test: ,!imn"' " = I (any p) and ,!im(II!)'" = = , More
diverges otherwise. That is. L; x" = 1 1 (lx l
1 X";\dx
. "..
Z underestimate
with error < 13.1<5.10
~
then
geometric series may be identified through this basic one.
K
III
,Jiml.~~ >I()r ,!i!" lan l' /" >1.
....
ll
converges
geometric serics. The following are useful in applying the
the slim with error less than
J _ "1
. 1
_
L;"  L; ,, +
" dx  1.2011l .. "
Eg
L:u"
L;a" diverges. These tests arc derived by comparison with
"
the
2
E"
."'., _1_
l+x = Ix+x 2 "'illll,li"c .s_1_
l+x =Il:+t·
. . .. , ,
"
the resulting series of real numbers converges.
0 ill a s trictly Llecreasing manner).
CONVERGENCE TESTS
D.
If
, Thcse are series \\!hosc terms alternate
in (nonzero) sign. If the term s of an alternating series
La
(ahsolutely).
T he integrated series has radius o f convcrgence R. and may
converge at an endpoint where the original diverged.
In(I+x)=L;(J)"
M
,, "'· 0
'\eri~
• \lternafin
tl
11n
is ca lled the I)series.
II)
I~ diverges. li,r the partial
be low ). T he harmonic se ries L;
La
,
1'1 a tl 1' /" < 1 , then
If }!"
I· I~an
I II
The /Ise ries diverges if ps i and converges if p>1 (hy
comparison with harmonic sl'ri~s and the inh:graJ lest,
"
n  ~'
h
precisely. ,!im
If
(11/ .....
"
• Ratio & 1'00t tests. Assume an" O.
1.1
• pscn', For P , a
SUIllS
V,.
I "
.
. .
'
sin(I / II')
E.g.. L;SlIl (1 / n) converges smee Hm
.., = 1.
A (numerical) geometric series has the
II
__
The pseries and geometric scries are otien used tl".
comparisons. Try a "limit" comparison when a series
E.~ .. L;I :f,, =4(1 .! ,<, 1)=2.
~
lfL;b" diverges and either b,,£an (1I ~ N) or a,,Ib n has
a 110112('/'0 limit (or approaches 00), then L;a" diverges.
L;a ,, =S.
/I
...
..
== S means the series conve rges
isS. In general statemcnts,
SlIlll
iU •Geometric series
A
/I
4)
or a"lb"
f /W dt= L; _a+
"I (x  cY' ' (ixc l< R).
C
1/
lin
4
and
rClnl"\'31 s~)I("m ..... Uhvul
\HllIen rcrnli"'10n frolll the rubh.JlcT
f\ 2Ui.1IZ007 UnCh ll r U.lnc. 0 108
~u lt, : Due IU lIS condensed
fqfTnQI,
plc,uc II!>C thl
\)uid.)tudv
nOI h
a~ II
111,111.1.... bUl
a rep lacement for
"" lgncJ ,· l.t$S\\ ' Ir\;
U,S, $4,95 CAN. $7.50
Author: Gerald Harnet. PhD
Customer Hotline # 1,800,230,9522
ISBN13' 9781572224759
ISBN  1D: 1572224754
9 1 ~ lll)llli ~1I1!1!1!IJIJ~l l l rl l lil l l l