Section3.4
ImproperIntegrals
153
 AttemptingtoformulateDefinition3.1.1forafunctiondefinedonaninfiniteorsemi-
infiniteintervalwouldintroducequestionsconcerningconvergenceoftheresulting
Riemannsums,whichwouldbeinfiniteseries.
Inthissectionweextendthedefinitionofintegraltoincludecaseswherefisunbounded
ortheintervalisunbounded,orboth.
Wesayf islocallyintegrableonanintervalI I iff f isintegrableoneveryfiniteclosed
subintervalofI.Forexample,
f.x/Dsinx
islocallyintegrableon.1;1/;
g.x/D
1
x.x1/
islocallyintegrableon.1;0/,.0;1/,and.1;1/;and
h.x/D
p
x
islocallyintegrableonŒ0;1/.
Definition3.4.1
Iff islocallyintegrableonŒa;b/,wedefine
Z
b
a
f.x/dxD lim
c!b
Z
c
a
f.x/dx
(3.4.1)
ifthelimitexists(finite).ToincludethecasewherebD1,weadopttheconventionthat
1D1.
Thelimitin(3.4.1)alwaysexistsifŒa;b/isfiniteandf islocallyintegrableandbounded
onŒa;b/.Inthiscase,Definitions3.1.1and3.4.1assignthesamevalueto
R
b
a
f.x/dxno
matterhowf.b/isdefined(Exercise3.4.1). However, , thelimitmayalsoexistincases
whereb D1orb< 1andf isunboundedasxapproachesbfromtheleft. . Inthese
cases,Definition3.4.1assignsavaluetoanintegralthatdoesnotexistinthesenseofDef-
inition3.1.1,and
R
b
a
f.x/dxissaidtobeanimproperintegralthatconvergestothelimit
in(3.4.1).WealsosayinthiscasethatfisintegrableonŒa;b/andthat
R
b
a
f.x/dxexists.
Ifthelimitin(3.4.1)doesnotexist(finite),wesaythattheimproperintegral
R
b
a
f.x/dx
diverges,andf isnonintegrableonŒa;b/. . Inparticular,iflim
c!b
R
c
a
f.x/dx D˙1,
wesaythat
R
b
a
f.x/dxdivergesto˙1,andwewrite
Z
b
a
f.x/dxD1
or
Z
b
a
f.x/dxD1;
whicheverthecasemaybe.
Similarcommentsapplytothenexttwodefinitions.
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154 Chapter3
IntegralCalculusofFunctionsofOneVariable
Definition3.4.2
Iff islocallyintegrableon.a;b,wedefine
Z
b
a
f.x/dxD lim
c!aC
Z
b
c
f.x/dx
providedthatthelimitexists(finite). Toincludethecasewherea D 1,weadoptthe
conventionthat1CD1.
Definition3.4.3
Iff islocallyintegrableon.a;b/;wedefine
Z
b
a
f.x/dxD
Z
˛
a
f.x/dxC
Z
b
˛
f.x/dx;
wherea<˛<b,providedthatbothimproperintegralsontherightexist(finite).
Theexistenceandvalueof
R
b
a
f.x/dxaccordingtoDefinition3.4.3donotdependon
theparticularchoiceof˛in.a;b/(Exercise3.4.2).
Whenwewishtodistinguishbetweenimproperintegralsandintegralsinthesenseof
Definition3.1.1,wewillcallthelatterproperintegrals.
Instatingandprovingtheorems onimproperintegrals, wewillconsiderintegralsof
thekindintroducedinDefinition3.4.1. SimilarresultsapplytotheintegralsofDefini-
tions3.4.2and3.4.3. Weleaveittoyoutoformulateandusethemintheexamplesand
exercisesastheneedarises.
Example3.4.1
Thefunction
f.x/D2xsin
1
x
cos
1
x
islocallyintegrableandthederivativeof
F.x/Dx
2
sin
1
x
onŒ2=;0/.Hence,
Z
c
2=
f.x/dxDx
2
sin
1
x
ˇ
ˇ
ˇ
ˇ
c
2=
Dc
2
sin
1
c
C
4
2
and
Z
0
2=
f.x/dxD lim
c!0
c
2
sin
1
c
C
4
2
D
4
2
;
accordingtoDefinition3.4.1.However,thisisnotanimproperintegral,eventhoughf.0/
isnotdefinedandcannotbedefinedsoastomakef continuousat0. Ifwedefinef.0/
arbitrarily(sayf.0/D10),thenf isboundedontheclosedintervalŒ2=;0andcon-
tinuousexceptat0.Therefore,
R
0
2=
f.x/dxexistsandequals4=2asaproperintegral
(Exercise3.4.1),inthesenseofDefinition3.1.1.
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Section3.4
ImproperIntegrals
155
Example3.4.2
Thefunction
f.x/D.1x/
p
islocallyintegrableonŒ0;1/and,ifp¤1and0<c<1,
Z
c
0
.1x/
p
dxD
.1x/pC1
p1
ˇ
ˇ
ˇ
ˇ
c
0
D
.1c/pC1 1
p1
:
Hence,
lim
c!1
Z
c
0
.1x/
p
dxD
.1p/
1
; p<1;
1;
p>1:
ForpD1,
lim
c!1
Z
c
0
.1x/
1
dxD lim
c!1
log.1c/D1:
Hence,
Z
1
0
.1x/
p
dxD
.1p/
1
; p<1;
1;
p1:
Example3.4.3
Thefunction
f.x/Dx
p
islocallyintegrableonŒ1;1/and,ifp¤1andc>1,
Z
c
1
x
p
dxD
x
pC1
pC1
ˇ
ˇ
ˇ
ˇ
c
1
D
c
pC1
1
pC1
:
Hence,
lim
c!1
Z
c
1
x
p
dxD
.p1/
1
; p>1;
1;
p<1:
ForpD1,
lim
c!1
Z
c
1
x
1
dxD lim
c!1
logcD1:
Hence,
Z
1
1
x
p
dxD
.p1/
1
; p>1;
1;
p1:
Example3.4.4
If1<c<1,then
Z
c
1
1
x
log
1
x
dxD
Z
c
1
1
x
logxdxD
1
2
.logx/
2
ˇ
ˇ
ˇ
ˇ
c
1
D
1
2
.logc/
2
:
Hence,
lim
c!1
Z
c
1
1
x
log
1
x
dxD1;
so
Z
1
1
1
x
log
1
x
dxD1:
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156 Chapter3
IntegralCalculusofFunctionsofOneVariable
Example3.4.5
Thefunctionf.x/DcosxislocallyintegrableonŒ0;1/and
lim
c!1
Z
c
0
cosxdxD lim
c!1
sinc
doesnotexist;thus,
R
1
0
cosxdxdiverges,butnotto˙1.
Example3.4.6
Thefunctionf.x/ D D logx x islocallyintegrableon.0;1,butun-
boundedasx!0C.Since
lim
c!0C
Z
1
c
logxdxD lim
c!0C
.xlogxx/
ˇ
ˇ
ˇ
ˇ
1
c
D1 lim
c!0C
.clogcc/D1;
Definition3.4.2yields
Z
1
0
logxdxD1:
Example3.4.7
InconnectionwithDefinition3.4.3,itisimportanttorecognizethat
theimproperintegrals
R
˛
a
f.x/dxand
R
b
˛
f.x/dxmustconvergeseparatelyfor
R
b
a
f.x/dx
toconverge.Forexample,theexistenceofthesymmetriclimit
lim
R!1
Z
R
R
f.x/dx;
whichis calledtheprincipalvalueof
R
1
1
f.x/dx, doesnotimplythat
R
1
1
f.x/dx
converges;thus,
lim
R!1
Z
R
R
xdxD lim
R!1
0D0;
but
R
1
0
xdxand
R
0
1
xdxdivergeandthereforesodoes
R
1
1
xdx.
Theorem3.4.4
Supposethatf
1
;f
2
;...;f
n
arelocallyintegrableonŒa;b/andthat
R
b
a
f
1
.x/dx;
R
b
a
f
2
.x/dx;...;
R
b
a
f
n
.x/dx converge:Letc
1
;c
2
;...;c
n
beconstants:
Then
R
b
a
.c
1
f Cc
2
f
1
CCc
n
f
n
/.x/dxconvergesand
Z
b
a
.c
1
f
1
Cc
2
f
2
CCc
n
f
n
/.x/dxDc
1
Z
b
a
f
1
.x/dxCc
2
Z
b
a
f
2
.x/dx
CCc
n
Z
b
a
f
n
.x/dx:
Proof
Ifa<c<b,then
Z
c
a
.c
1
f
1
Cc
2
f
2
CCc
n
f
n
/.x/dxDc
1
Z
c
a
f
1
.x/dxCc
2
Z
c
a
f
2
.x/dx
CCc
n
Z
c
a
f
n
.x/dx;
byTheorem3.3.3.Lettingc!byieldsthestatedresult.
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Section3.4
ImproperIntegrals
157
ImproperIntegralsofNonnegativeFunctions
Thetheoryofimproperintegralsofnonnegativefunctionsisparticularlysimple.
Theorem3.4.5
Iff isnonnegativeandlocallyintegrableonŒa;b/;then
R
b
a
f.x/dx
convergesifthefunction
F.x/D
Z
x
a
f.t/dt
isboundedonŒa;b/,and
R
b
a
f.x/dxD1ifitisnot.Thesearetheonlypossibilities,and
Z
b
a
f.t/dtD sup
ax<b
F.x/
ineithercase:
Proof
SinceFisnondecreasingonŒa;b/,Theorem2.1.9
(a)
impliestheconclusion.
Weoftenwrite
Z
b
a
f.x/dx<1
toindicatethatanimproperintegralofanonnegativefunctionconverges. Theorem3.4.5
justifiesthisconvention,sinceitassertsthatadivergentintegralofthiskindcanonlydi-
vergeto1.Similarly,iff isnonpositiveand
R
b
a
f.x/dxconverges,wewrite
Z
b
a
f.x/dx>1
becauseadivergentintegralofthiskindcanonlydivergeto1. (Toseethis, , apply
Theorem3.4.5tof.)Theseconventionsdonotapplytoimproperintegralsoffunctions
thatassumebothpositiveandnegativevaluesin.a;b/,sincetheymaydivergewithout
divergingto˙1.
Theorem3.4.6(ComparisonTest)
Iff andgarelocallyintegrableonŒa;b/
and
0f.x/g.x/; ax<b;
(3.4.2)
then
(a)
Z
b
a
f.x/dx<1
if
Z
b
a
g.x/dx<1
and
(b)
Z
b
a
g.x/dxD1
if
Z
b
a
f.x/dxD1.
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158 Chapter3
IntegralCalculusofFunctionsofOneVariable
Proof (a)
Assumption(3.4.2)impliesthat
Z
x
a
f.t/dt 
Z
x
a
g.t/dt; ax<b
(Theorem3.3.4),so
sup
ax<b
Z
x
a
f.t/dt sup
axb
Z
x
a
g.t/dt:
If
R
b
a
g.x/dx <1,therightsideofthisinequalityisfinitebyTheorem3.4.5,sotheleft
sideisalso.Thisimpliesthat
R
b
a
f.x/dx<1,againbyTheorem3.4.5.
(b)
Theproofisbycontradiction.If
R
b
a
g.x/dx<1,then
(a)
impliesthat
R
b
a
f.x/dx<
1,contradictingtheassumptionthat
R
b
a
f.x/dxD1.
Thecomparisontestisparticularlyusefuliftheintegrandoftheimproperintegralis
complicatedbutcanbecomparedwithafunctionthatiseasytointegrate.
Example3.4.8
Theimproperintegral
I D
Z
1
0
2Csinx
.1x/p
dx
convergesifp<1,since
0<
2Csinx
.1x/p
3
.1x/p
; 0x<1;
and,fromExample3.4.2,
Z
1
0
3dx
.1x/p
<1; p<1:
However,Idivergesifp1,since
0<
1
.1x/p
2Csinx
.1x/p
; 0x<1;
and
Z
1
0
dx
.1x/p
D1; p1:
Iffisanyfunction(notnecessarilynonnegative)locallyintegrableonŒa;b/,then
Z
c
a
f.x/dxD
Z
a
1
a
f.x/dxC
Z
c
a
1
f.x/dx
ifa
1
andcareinŒa;b/. Since
R
a
1
a
f.x/dxisaproperintegral,onlettingc !bwe
concludethatifeitheroftheimproperintegrals
R
b
a
f.x/dx and
R
b
a
1
f.x/dx converges
thensodoestheother,andinthiscase
Z
b
a
f.x/dxD
Z
a
1
a
f.x/dxC
Z
b
a
1
f.x/dx:
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Section3.4
ImproperIntegrals
159
Thismeansthatanytheoremimplyingconvergenceordivergenceofanimproperintegral
R
b
a
f.x/dx inthesenseofDefinition3.4.1remainsvalidifitshypothesesaresatisfied
onasubintervalŒa
1
;b/ofŒa;b/ratherthanonallofŒa;b/. Forexample,Theorem3.4.6
remainsvalidif(3.4.2)isreplacedby
0f.x/g.x/; a
1
x<b;
wherea
1
isanypointinŒa;b/.
Fromthis,youcanseethatiff.x/0onsomesubintervalŒa
1
;b/ofŒa;b/,butnot
necessarilyforallxinŒa;b/,wecanstillusetheconventionintroducedearlierforpositive
functions;thatis,wecanwrite
R
b
a
f.x/dx < < 1iftheimproperintegralconvergesor
R
b
a
f.x/dxD1ifitdiverges.
Example3.4.9
Ifp0,then
xp
2
.x1/p.2Csinx/
.x1=3/2p
4x
p
forxsufficientlylarge.Therefore,Theorem3.4.6andExample3.4.3implythat
Z
1
1
.x1/
p
.2Csinx/
.x1=3/2p
dx
convergesifp>1ordivergesifp1.
Theorem3.4.7
Supposethatf andgarelocallyintegrableonŒa;b/;g.x/>0and
f.x/0onsomesubintervalŒa
1
;b/ofŒa;b/;and
lim
x!b
f.x/
g.x/
DM:
(3.4.3)
(a)
If0<M<1;then
R
b
a
f.x/dxand
R
b
a
g.x/dxconvergeordivergetogether.
(b)
IfM D1and
R
b
a
g.x/dxD1;then
R
b
a
f.x/dxD1.
(c)
IfMD0and
R
b
a
g.x/dx<1;then
R
b
a
f.x/dx<1.
Proof (a)
From(3.4.3),thereisapointa
2
inŒa
1
;b/suchthat
0<
M
2
<
f.x/
g.x/
<
3M
2
; a
2
x<b;
andtherefore
M
2
g.x/<f.x/<
3M
2
g.x/; a
2
x<b:
(3.4.4)
Theorem3.4.6andthefirstinequalityin(3.4.4)implythat
Z
b
a
2
g.x/dx<1
if
Z
b
a
2
f.x/dx<1:
160 Chapter3
IntegralCalculusofFunctionsofOneVariable
Theorem3.4.6andthesecondinequalityin(3.4.4)implythat
Z
b
a
2
f.x/dx<1
if
Z
b
a
2
g.x/dx<1:
Therefore,
R
b
a
2
f.x/dxand
R
b
a
2
g.x/dxconvergeordivergetogether,andinthelattercase
theymustdivergeto1,sincetheirintegrandsarenonnegative(Theorem3.4.5).
(b)
IfMD1,thereisapointa
2
inŒa
1
;b/suchthat
f.x/g.x/; a
2
xb;
soTheorem3.4.6
(b)
impliesthat
R
b
a
f.x/dxD1.
(c)
IfM D0,thereisapointa
2
inŒa
1
;b/suchthat
f.x/g.x/; a
2
xb;
soTheorem3.4.6
(a)
impliesthat
R
b
a
f.x/dx<1.
ThehypothesesofTheorem3.4.7
(b)
and
(c)
donotimplythat
R
b
a
f.x/dxand
R
b
a
g.x/dx
necessarilyconvergeordivergetogether. Forexample, , ifb b D D 1, , thenf.x/ / D 1=x
andg.x/D1=x
2
satisfythehypothesesofTheorem3.4.7
(b)
,whilef.x/D1=x
2
and
g.x/ D D 1=xsatisfythehypothesesofTheorem3.4.7
(c)
. However,
R
1
1
1=xdx D D 1,
while
R
1
1
1=x
2
dx<1.
Example3.4.10
Letf.x/D.1Cx/
p
andg.x/Dx
p
.Since
lim
x!1
f.x/
g.x/
D1
and
R
1
1
xpdxconvergesifp>1ordivergesifp 1(Example3.4.3),Theorem3.4.7
impliesthatthesameistrueof
Z
1
1
.1Cx/
p
dx:
Example3.4.11
Thefunction
f.x/Dx
p
.1Cx/
q
islocallyintegrableon.0;1/.Toseewhether
I D
Z
1
0
x
p
.1Cx/
q
dx
convergesaccordingtoDefinition3.4.3,weconsidertheimproperintegrals
I
1
D
Z
1
0
x
p
.1Cx/
q
dx and I
2
D
Z
1
1
x
p
.1Cx/
q
dx
Section3.4
ImproperIntegrals
161
separately.(Thechoiceof1astheupperlimitofI
1
andthelowerlimitofI
2
iscompletely
arbitrary;anyotherpositivenumberwoulddojustaswell.)Since
lim
x!0C
f.x/
xp
D lim
x!0C
.1Cx/
q
D1
and
Z
1
0
x
p
dxD
.1p/
1
; p<1;
1;
p1;
Theorem3.4.7impliesthatI
1
convergesifandonlyifp<1.Since
lim
x!1
f.x/
xpq
D lim
x!1
.1Cx/
q
x
q
D1
and
Z
1
1
x
pq
dxD
.pCq1/
1
; pCq>1;
1;
pCq1;
Theorem 3.4.7impliesthatI
2
convergesifandonlyifpCq > 1. . Combiningthese
results,weconcludethatI convergesaccordingtoDefinition3.4.3ifandonlyifp <1
andpCq>1.
AbsoluteIntegrability
Definition3.4.8
Wesaythatf isabsolutelyintegrableonŒa;b/iff f islocallyinte-
grableonŒa;b/and
R
b
a
jf.x/jdx<1.Inthiscasewealsosaythat
R
b
a
f.x/dxconverges
absolutelyorisabsolutelyconvergent.
Example3.4.12
Iff isnonnegativeandintegrableonŒa;b/, , thenf f isabsolutely
integrableonŒa;b/,sincejfjDf.
Example3.4.13
Since
ˇ
ˇ
ˇ
ˇ
sinx
xp
ˇ
ˇ
ˇ
ˇ
1
xp
and
R
1
1
x
p
dx<1ifp>1(Example3.4.3),Theorem3.4.6impliesthat
Z
1
1
jsinxj
xp
dx<1; p>1I
thatis,thefunction
f.x/D
sinx
xp
isabsolutelyintegrableonŒ1;1/ifp > > 1. . ItisnotabsolutelyintegrableonŒ1;1/if
p 1. . Toseethis,wefirstconsiderthecasewherep p D1. Letkbeanintegergreater
than3.Then
162 Chapter3
IntegralCalculusofFunctionsofOneVariable
Z
k
1
jsinxj
x
dx >
Z
k
jsinxj
x
dx
D
k1
jD1
Z
.jC1/
j
jsinxj
x
dx
>
k1
jD1
1
.jC1/
Z
.jC1/
j
jsinxjdx:
(3.4.5)
But
Z
.jC1/
j
jsinxjdxD
Z
0
sinxdxD2;
so(3.4.5)impliesthat
Z
k
1
jsinxj
x
dx>
2
k1
jD1
1
j C1
:
(3.4.6)
However,
1
j C1
Z
jC2
jC1
dx
x
; j j D1;2;:::;
so(3.4.6)impliesthat
Z
k
1
jsinxj
x
>
2
k1
jD1
Z
jC2
jC1
dx
x
D
2
Z
kC1
2
dx
x
D
2
log
kC1
2
:
Sincelim
k!1
logŒ.kC1/=2D1,Theorem3.4.5impliesthat
Z
1
1
jsinxj
x
dxD1:
NowTheorem3.4.6
(b)
impliesthat
Z
1
1
jsinxj
xp
dxD1; p1:
(3.4.7)
Theorem3.4.9
Iff is s locallyintegrableonŒa;b/and
R
b
a
jf.x/jdx < < 1;then
R
b
a
f.x/dxconvergesIthatis;anabsolutelyconvergentintegralisconvergent:
Proof
If
g.x/Djf.x/jf.x/;
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