c# pdf free : Add page number to pdf hyperlink application SDK tool html winforms windows online TRENCH_REAL_ANALYSIS17-part234

Section3.4
ImproperIntegrals
163
then
0g.x/2jf.x/j
and
R
b
a
g.x/dx<1,becauseofTheorem3.4.6andtheabsoluteintegrabilityoff.Since
f Djfjg;
Theorem3.4.4impliesthat
R
b
a
f.x/dxconverges.
ConditionalConvergence
Wesaythatf isnonoscillatoryatb.D D 1
if bD1/iff isdefinedonŒa;b/and
doesnotchangesignonsomesubintervalŒa
1
;b/ofŒa;b/. Iff changessignonevery
suchsubinterval,f isoscillatoryatb. . ForafunctionthatislocallyintegrableonŒa;b/
andnonoscillatoryatb,convergenceandabsoluteconvergenceof
R
b
a
f.x/dxamount
tothesamething(Exercise3.4.16),soabsoluteconvergenceisnotaninterestingconcept
inconnectionwithsuchfunctions. However,anoscillatoryfunctionmaybeintegrable,
butnotabsolutelyintegrable,onŒa;b/,asthenextexampleshows.Wethensaythatf is
conditionallyintegrableonŒa;b/,andthat
R
b
a
f.x/dxconvergesconditionally.
Example3.4.14
WesawinExample3.4.13thattheintegral
I.p/D
Z
1
1
sinx
xp
dx
isnotabsolutelyconvergentif0<p1.Wewillshowthatitconvergesconditionallyfor
thesevaluesofp.
Integrationbypartsyields
Z
c
1
sinx
xp
dxD
cosc
cp
Ccos1p
Z
c
1
cosx
xpC1
dx:
(3.4.8)
Since
ˇ
ˇ
ˇ
cosx
xpC1
ˇ
ˇ
ˇ
1
xpC1
and
R
1
1
xp1dx <1ifp >0,Theorem3.4.6impliesthatxp1cosxisabsolutely
integrableŒ1;1/ifp>0.Therefore,Theorem3.4.9impliesthatxp1cosxisintegrable
Œ1;1/ifp>0.Lettingc!1in(3.4.8),wefindthatI.p/converges,and
I.p/Dcos1p
Z
1
1
cosx
xpC1
dx if p>0:
Thisand(3.4.7)implythatI.p/convergesconditionallyif0<p1.
ThemethodusedinExample3.4.14isaspecialcaseofthefollowingtestforconvergence
ofimproperintegrals.
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164 Chapter3
IntegralCalculusofFunctionsofOneVariable
Theorem3.4.10(Dirichlet’sTest)
Supposethatf iscontinuousanditsan-
tiderivativeF.x/ D
R
x
a
f.t/dt isboundedonŒa;b/:Letg
0
beabsolutelyintegrableon
Œa;b/;andsupposethat
lim
x!b
g.x/D0:
(3.4.9)
Then
R
b
a
f.x/g.x/dxconverges:
Proof
ThecontinuousfunctionfgislocallyintegrableonŒa;b/. Integrationbyparts
yields
Z
c
a
f.x/g.x/dx DF.c/g.c/
Z
c
a
F.x/g
0
.x/dx; ac<b:
(3.4.10)
Theorem3.4.6impliesthattheintegralontherightconvergesabsolutelyasc!b,since
R
b
a
jg
0
.x/jdx<1byassumption,and
jF.x/g
0
.x/jMjg
0
.x/j;
whereMisanupperboundforjFjonŒa;b/.Moreover,(3.4.9)andtheboundednessofF
implythatlim
c!b
F.c/g.c/D0.Lettingc!bin(3.4.10)yields
Z
b
a
f.x/g.x/dx D
Z
b
a
F.x/g
0
.x/dx;
wheretheintegralontherightconvergesabsolutely.
Dirichlet’stestisusefulonlyiff isoscillatoryatb,sinceitcanbeshownthatiff is
nonoscillatoryatbandF isboundedonŒa;b/,then
R
b
a
jf.x/g.x/jdx<1ifonlygis
locallyintegrableandboundedonŒa;b/(Exercise3.4.14).
Example3.4.15
Dirichlet’stestcanalsobeusedtoshowthatcertainintegralsdi-
verge.Forexample,
Z
1
1
x
q
sinxdx
divergesifq >0,butnoneoftheotherteststhatwehavestudiedsofarimpliesthis. It
isnotenoughtoarguethattheintegranddoesnotapproachzeroasx !1(acommon
mistake),sincethisdoesnotimplydivergence(Exercise4.4.31). Toseethattheintegral
diverges,weobservethatifitconvergedforsomeq > 0,thenF.x/ D
R
x
1
x
q
sinxdx
wouldbeboundedonŒ1;1/,andwecouldlet
f.x/Dx
q
sinx and g.x/Dx
q
inTheorem3.4.10andconcludethat
Z
1
1
sinxdx
alsoconverges.Thisisfalse.
Section3.4
ImproperIntegrals
165
ThemethodusedinExample3.4.15isaspecialcaseofthefollowingtestfordivergence
ofimproperintegrals.
Theorem3.4.11
SupposethatuiscontinuousonŒa;b/and
R
b
a
u.x/dxdiverges:Let
vbepositiveanddifferentiableonŒa;b/;andsupposethatlim
x!b
v.x/D1andv0=v2
isabsolutelyintegrableonŒa;b/:Then
R
b
a
u.x/v.x/dxdiverges:
Proof
Theproofisbycontradiction. Letf D D uvandg D D 1=v,andsupposethat
R
b
a
u.x/v.x/dxconverges.ThenfhastheboundedantiderivativeF.x/D
R
x
a
u.t/v.t/dt
onŒa;b/,lim
x!1
g.x/D0andg
0
Dv
0
=v
2
isabsolutelyintegrableonŒa;b/.Therefore,
Theorem3.4.10impliesthat
R
b
a
u.x/dxconverges,acontradiction.
IfDirichlet’stestshowsthat
R
b
a
f.x/g.x/dxconverges,thereremainsthequestionof
whetheritconvergesabsolutelyorconditionally.Thenexttheoremsometimesanswersthis
question. ItsproofcanbemodeledafterthemethodofExample3.4.13(Exercise3.4.17).
Theideaofaninfinitesequence, whichwewilldiscussinSection4.1, , entersintothe
statementofthistheorem. Weassumethatyourecalltheconceptsufficientlywellfrom
calculustounderstandthemeaningofthetheorem.
Theorem3.4.12
SupposethatgismonotoniconŒa;b/and
R
b
a
g.x/dxD1:Letf
belocallyintegrableonŒa;b/and
Z
x
jC1
x
j
jf.x/jdx; j j 0;
forsomepositive;wherefx
j
gisanincreasinginfinitesequenceofpointsinŒa;b/such
thatlim
j!1
x
j
Dbandx
jC1
x
j
M;j 0;forsomeM:Then
Z
b
a
jf.x/g.x/jdxD1:
Change ofVariableinanImproperIntegral
Thenext theoremenablesustoinvestigateanimproperintegralbytransformingitinto
anotherwhoseconvergenceordivergenceisknown. ItfollowsfromTheorem3.3.18and
Definitions3.4.1,3.4.2,and3.4.3.Weomittheproof.
Theorem3.4.13
Supposethatismonotonicandislocallyintegrableoneither
ofthehalf-openintervalsI DŒc;d/or.c;d;andletx D.t/mapI ontoeitherofthe
half-openintervalsJ DŒa;b/orJ D.a;b:Letf belocallyintegrableonJ:Thenthe
improperintegrals
Z
b
a
f.x/dx and
Z
d
c
f..t//j
0
.t/jdt
166 Chapter3
IntegralCalculusofFunctionsofOneVariable
divergeorconvergetogether;inthelattercasetothesamevalue. Thesameconclusion
holdsifand
0
havethestatedpropertiesonlyontheopeninterval.a;b/;thetransfor-
mationxD.t/maps.c;d/onto.a;b/;andfislocallyintegrableon.a;b/:
Example3.4.16
ToapplyTheorem3.4.13to
Z
1
0
sinx
2
dx;
weusethechangeofvariablexD.t/D
p
t,whichtakesŒc;d/DŒ0;1/intoŒa;b/D
Œ0;1/,with
0
.t/D1=.2
p
t/.Theorem3.4.13impliesthat
Z
1
0
sinx
2
dxD
1
2
Z
1
0
sint
p
t
dt:
Sincetheintegralontherightconverges(Example3.4.14),sodoestheoneontheleft.
Example3.4.17
Theintegral
Z
1
1
x
p
dx
converges ifandonlyifp > > 1(Example3.4.3). Defining.t/ D 1=t t andapplying
Theorem3.4.13yields
Z
1
1
x
p
dxD
Z
1
0
t
p
jt
2
jdtD
Z
1
0
t
p2
dt;
whichimpliesthat
R
1
0
t
q
dtconvergesifandonlyifq>1.
3.4Exercises
1. (a)
Letf belocallyintegrableandboundedonŒa;b/,andletf.b/bedefined
arbitrarily.Showthatf isproperlyintegrableonŒa;b,that
R
b
a
f.x/dxdoes
notdependonf.b/,andthat
Z
b
a
f.x/dxD lim
c!b
Z
c
a
f.x/dx:
(b)
Statearesultanalogousto
(a)
whichendswiththeconclusionthat
Z
b
a
f.x/dxD lim
c!aC
Z
b
c
f.x/dx:
2.
ShowthatneithertheexistencenorthevalueoftheimproperintegralofDefini-
tion3.4.3dependsonthechoiceoftheintermediatepoint˛.
Section3.4
ImproperIntegrals
167
3.
Prove:If
R
b
a
f.x/dxexistsaccordingtoDefinition3.4.1or3.4.2,then
R
b
a
f.x/dx
alsoexistsaccordingtoDefinition3.4.3.
4.
Findallvaluesofpforwhichthefollowingintegralsexist
(i)
asproperintegrals
(perhapsafterdefiningf attheendpointsoftheinterval)or
(ii)
asimproperinte-
grals.
(iii)
Evaluatetheintegralsforthevaluesofpforwhichtheyconverge.
(a)
Z
1=
0
px
p1
sin
1
x
x
p2
cos
1
x
dx
(b)
Z
2=
0
px
p1
cos
1
xCx
p2
sin
1
x
dx
(c)
Z
1
0
e
px
dx
(d)
Z
1
0
x
p
dx
(e)
Z
1
0
x
p
dx.
5.
Evaluate
(a)
Z
1
0
e
x
x
n
dx .nD0;1;:::/
(b)
Z
1
0
e
x
sinxdx
(c)
Z
1
1
xdx
xC1
(d)
Z
1
0
xdx
p
1x2
(e)
Z
0
cosx
x
sinx
x2
dx
(f)
Z
1
=2
sinx
x
C
cosx
x2
dx
6.
Prove:If
R
b
a
f.x/dxexistsasaproperorimproperintegral,then
lim
x!b
Z
b
x
f.t/dtD0:
7.
Prove:Iff islocallyintegrableonŒa;b/,then
R
b
a
f.x/dxexistsifandonlyiffor
each>0thereisanumberrin.a;b/suchthat
ˇ
ˇ
ˇ
ˇ
Z
x
2
x
1
f.t/dt
ˇ
ˇ
ˇ
ˇ
<
wheneverrx
1
,x
2
<b.H
INT
: SeeExercise2.1.38.
8.
Determinewhethertheintegralconvergesordiverges.
(a)
Z
1
1
logxCsinx
p
x
dx
(b)
Z
1
1
.x
2
C3/
3=2
.x4C1/3=2
sin
2
xdx
(c)
Z
1
0
1Ccos
2
x
p
1Cx2
dx
(d)
Z
1
0
4Ccosx
.1Cx/
p
x
dx
(e)
Z
1
0
.x
27
Csinx/e
x
dx
(f)
Z
1
0
x
p
.2Csinx/dx
168 Chapter3
IntegralCalculusofFunctionsofOneVariable
9.
Findallvaluesofpforwhichtheintegralconverges.
(a)
Z
=2
0
sinx
xp
dx
(b)
Z
=2
0
cosx
xp
dx
(c)
Z
1
0
x
p
e
x
dx
(d)
Z
=2
0
sinx
.tanx/p
dx
(e)
Z
1
1
dx
x.logx/p
(f)
Z
1
0
dx
x.jlogxj/p
(g)
Z
0
xdx
.sinx/p
10.
LetL
n
.x/betheiteratedlogarithmdefinedinExercise2.4.42.Showthat
Z
1
a
dx
L
0
.x/L
1
.x/L
k
.x/ŒL
kC1
.x/p
convergesifandonlyifp>1. HereaisanynumbersuchthatL
kC1
.x/>0for
xa.
11.
Findconditionsonpandqsuchthattheintegralconverges.
(a)
Z
1
1
.cosx=2/
q
.1x2/p
dx
(b)
Z
1
1
.1x/
p
.1Cx/
q
dx
(c)
Z
1
0
x
p
dx
.1Cx2/q
(d)
Z
1
1
Œlog.1Cx/
p
.logx/
q
xpCq
dx
(e)
Z
1
1
.log.1Cx/logx/
q
xp
dx
(f)
Z
1
0
.xsinx/
q
xp
dx
12.
Letf andgbepolynomialsandsupposethatghasnorealzeros. . Findnecessary
andsufficientconditionsforconvergenceof
Z
1
1
f.x/
g.x/
dx:
13.
Prove:IffandgarelocallyintegrableonŒa;b/andtheimproperintegrals
R
b
a
f2.x/dx
and
R
b
a
g
2
.x/dxconverge,then
R
b
a
f.x/g.x/dxconvergesabsolutely.H
INT
:.f˙
g/
2
0:
14.
Supposethatf islocallyintegrableandF.x/D
R
x
a
f.t/dt isboundedonŒa;b/,
andletf benonoscillatoryatb.LetgbelocallyintegrableandboundedonŒa;b/.
Showthat
Z
b
a
jf.x/g.x/jdx<1:
15.
SupposethatgispositiveandnonincreasingonŒa;b/and
R
b
a
f.x/dx existsas
aproperorabsolutelyconvergentimproperintegral. Showthat
R
b
a
f.x/g.x/dx
existsand
Section3.4
ImproperIntegrals
169
lim
x!b
1
g.x/
Z
b
x
f.t/g.t/dt D0:
H
INT
:UseExercise3.4.6:
16.
ShowthatiffislocallyintegrableonŒa;b/andnonoscillatoryatb,then
R
b
a
f.x/dx
existsifandonlyif
R
b
a
jf.x/jdx<1.
17. (a)
ProveTheorem3.4.12.H
INT
:SeeExample3.4.13:
(b)
ShowthatgsatisfiestheassumptionsofTheorem3.4.10ifg
0
islocallyinte-
grable,gismonotoniconŒa;b/,andlim
x!b
g.x/D0.
18.
Findallvaluesofpforwhichtheintegralconverges
(i)
absolutely;
(ii)
condition-
ally.
(a)
Z
1
1
cosx
xp
dx
(b)
Z
1
2
sinx
x.logx/p
dx
(c)
Z
1
2
sinx
xplogx
dx
(d)
Z
1
1
sin1=x
xp
dx
(e)
Z
1
0
sin
2
xsin2x
xp
dx
(f)
Z
1
1
sinx
.1Cx2/p
dx
19.
Supposethatg
00
isabsolutelyintegrableonŒ0;1/,lim
x!1
g
0
.x/D0,andlim
x!1
g.x/D
L(finiteorinfinite). Showthat
R
1
0
g.x/sinxdxconvergesifandonlyifLD0.
H
INT
:Integratebyparts:
20.
LethbecontinuousonŒ0;1/.Prove:
(a)
If
R
1
0
es
0
xh.x/dx convergesabsolutely, then
R
1
0
esxh.x/dx converges
absolutelyifs>s
0
.
(b)
If
R
1
0
e
s
0
x
h.x/dxconverges,then
R
1
0
e
sx
h.x/dxconvergesifs>s
0
.
21.
Supposethatf islocallyintegrableonŒ0;1/,lim
x!1
f.x/D A,and˛>1.
Findlim
x!1
x
˛1
R
x
0
f.t/t
˛
dt,andproveyouranswer.
22.
SupposethatfiscontinuousandF.x/D
R
x
a
f.t/dtisboundedonŒa;b/.Suppose
alsothatg>0,g
0
isnonnegativeandlocallyintegrableonŒa;b/,andlim
x!b
g.x/D
1.Showthat
lim
x!b
1
Œg.x/
Z
x
a
f.t/g.t/dt D0; >1:
H
INT
:Integratebyparts:
23.
InadditiontotheassumptionsofExercise3.4.22,assumethat
R
b
a
f.t/dtconverges.
Showthat
lim
x!b
1
g.x/
Z
x
a
f.t/g.t/dt D0:
H
INT
:LetF.x/D
R
b
x
f.t/dt;integratebyparts;anduseExercise3.4.6:
170 Chapter3
IntegralCalculusofFunctionsofOneVariable
24.
Supposethatf iscontinuous,g
0
.x/0,andg.x/>0onŒa;b/.Showthatifg
0
is
integrableonŒa;b/and
R
b
a
f.x/dxexists,then
R
b
a
f.x/g.x/dxexistsand
lim
x!b
1
g.x/
Z
b
x
f.t/g.t/dt D0:
H
INT
:LetF.x/D
R
b
x
f.t/dt;integratebyparts;anduseExercise3.4.6:
25.
Findallvaluesofpforwhichtheintegralconverges
(i)
absolutely;
(ii)
condition-
ally.
(a)
Z
1
0
x
p
sin1=xdx
(b)
Z
1
0
jlogxj
p
dx
(c)
Z
1
1
x
p
cos.logx/dx
(d)
Z
1
1
.logx/
p
dx
(e)
Z
1
0
sinx
p
dx
26.
Letu
1
bepositiveandsatisfythedifferentialequation
u
00
Cp.x/uD0; 0x<1:
.A/
(a)
Prove:If
Z
1
0
dx
u
2
1
.x/
<1;
thenthefunction
u
2
.x/Du
1
.x/
Z
1
x
dt
u
2
1
.t/
alsosatisfies(A),whileif
Z
1
0
dx
u
2
1
.x/
D1;
thenthefunction
u
2
.x/Du
1
.x/
Z
x
0
dt
u
2
1
.t/
alsosatisfies(A).
(b)
Prove:If(A)hasasolutionthatispositiveonŒ0;1/,then(A)hassolutions
y
1
andy
2
thatarepositiveon.0;1/andhavethefollowingproperties:
y
1
.x/y
0
2
.x/y
0
1
.x/y
2
.x/D1; x>0;
y
1
.x/
y
2
.x/
0
<
0; x>0;
and
lim
x!1
y
1
.x/
y
2
.x/
D0:
Section3.4
ImproperIntegrals
171
27. (a)
Prove:IfhiscontinuousonŒ0;1/,thenthefunction
u.x/Dc
1
e
x
Cc
2
e
x
C
Z
x
0
h.t/sinh.xt/dt
satisfiesthedifferentialequation
u
00
uDh.x/; x>0:
(b)
Rewriteuintheform
u.x/Da.x/e
x
Cb.x/e
x
andshowthat
u
0
.x/Da.x/e
x
Cb.x/e
x
:
(c)
Showthatiflim
x!1
a.x/DA(finite),then
lim
x!1
e
2x
Œb.x/BD0
forsomeconstantB.H
INT
:UseExercise3.4.24:Showalsothat
lim
x!1
e
x
Œu.x/Ae
x
Be
x
D0:
(d)
Prove:Iflim
x!1
b.x/DB(finite),then
lim
x!1
u.x/e
x
D lim
x!1
u
0
.x/e
x
DB:
H
INT
:UseExercise3.4.23:
28.
Supposethatthedifferentialequation
u
00
Cp.x/uD0
.A/
hasapositivesolutiononŒ0;1/,andthereforehastwosolutionsy
1
andy
2
withthe
propertiesgiveninExercise3.4.26
(b)
.
(a)
Prove:IfhiscontinuousonŒ0;1/andc
1
andc
2
areconstants,then
u.x/Dc
1
y
1
.x/Cc
2
y
2
.x/C
Z
x
0
h.t/Œy
1
.t/y
2
.x/y
1
.x/y
2
.t/dt .B/
satisfiesthedifferentialequation
u
00
Cp.x/uDh.x/:
Forconveniencein
(b)
and
(c)
,rewrite(B)as
u.x/Da.x/y
1
.x/Cb.x/y
2
.x/:
172 Chapter3
IntegralCalculusofFunctionsofOneVariable
(b)
Prove:If
R
1
0
h.t/y
2
.t/dtconverges,then
R
1
0
h.t/y
1
.t/dtconverges,and
lim
x!1
u.x/Ay
1
.x/By
2
.x/
y
1
.x/
D0
forsomeconstantsAandB. H
INT
:UseExercise3.4.24withf Dhy
2
and
gDy
1
=y
2
:
(c)
Prove:If
R
1
0
h.t/y
1
.t/dtconverges,then
lim
x!1
u.x/
y
2
.x/
DB
forsomeconstantB. H
INT
:UseExercise 3.4.23withf D D hy
1
andg D
y
2
=y
1
:
29.
Supposethatf,f
1
,andgarecontinuous,f >0,and.f
1
=f/
0
isabsolutelyinte-
grableonŒa;b/.Showthat
R
b
a
f
1
.x/g.x/dxconvergesif
R
b
a
f.x/g.x/dxdoes.
30.
Letgbelocallyintegrableandf continuous,withf.x/>0onŒa;b/. . Sup-
posethatforsomepositiveMandforeveryrinŒa;b/therearepointsx
1
andx
2
suchthat
(a)
r < < x
1
< x
2
< b;
(b)
gdoesnotchangesigninŒx
1
;x
2
;and
(c)
R
x
2
x
1
jg.x/jdx M. Showthat
R
b
a
f.x/g.x/dxdiverges. H
INT
:UseExer-
cise3.4.7andTheorem3.3.7:
3.5 A MORE ADVANCED LOOK AT THE EXISTENCE OF
THEPROPERRIEMANNINTEGRAL
InSection3.2wefoundnecessaryandsufficientconditionsforexistenceoftheproper
Riemannintegral,andinSection3.3weusedthemtostudythepropertiesoftheintegral.
However, itisawkwardtoapplytheseconditionstoaspecificfunctionanddetermine
whetheritisintegrable, sincetheyrequirecomputationsofupperandlowersums and
upperandlowerintegrals,whichmaybedifficult. Themainresultofthissectionisan
integrabilitycriterionduetoLebesguethatdoesnotrequirecomputation,buthastodo
withhowbadlydiscontinuousafunctionmaybeandstillbeintegrable.
Weemphasizethatweareagainconsideringproperintegralsofboundedfunctionson
finiteintervals.
Definition3.5.1
Iff isboundedonŒa;b,theoscillationoff onŒa;bisdefinedby
W
f
Œa;bD
sup
ax;x
0
b
jf.x/f.x
0
/j;
whichcanalsobewrittenas
W
f
Œa;bD sup
axb
f.x/ inf
axb
f.x/
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