Integrales impropias

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4. Y
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5. NKO F
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6. F
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9. GNOKUOQOQFQNCKPVGITCNFGWPCHWPEKÎP H CEQVCFCGPGNKPVGTXCNQ f f0 5SWG
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GU KIWCN C NC UWOC FG FKEJCU FQU KPVGITCNGU #JQTC DKGP EWCPFQ WPC FG NCU KPVGITCNGU FGN
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16. F
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18. F
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43. F F
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44. 1DUGTXCEKÎP %QOQUCDGOQUSWG
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45. F
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48. ,QWHJUDO GH XQD IXQFLyQ QR DFRWDGD HQ XQ LQWHUYDOR
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49. Y
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51. NKO F
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52. [UGFGPQVCRQT

53. F
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56. F F F
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57. F
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H Z Z³ GUEQPXGTIGPVGUKNQUQPNCUFQUKPVGITCNGU
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58. F
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66. F F
D D
C C
H Z Z I Z Zd³ ³

67. 5KNCKPVGITCNFG H GUPQEQPXGTIGPVGGP C D0 NCKPVGITCNFG I PQGUEQPXGTIGPVG
GP C D0
# RCTVKT FG NC KPVGITCN FG NC HWPEKÎP H Z
Z C
0 5 0 5
D EQPXGTIGPVG RCTC D [ PQ
EQPXGTIGPVGRCTCD t RQFGOQUGPWPEKCTNCUKIWKGPVGEQPFKEKÎPUWHKEKGPVGFGEQPXGTIGPEKC
i 5GC H Z0 5WPCHWPEKÎPFGHKPKFCGP C D0 VCNSWGRCTCVQFQ Z C D 0 UGXGTKHKEC
d d
H Z
Z C
0 5 0 5D EQPD GPVQPEGUNCKPVGITCN H Z Z
C
D
0 5I F GUEQPXGTIGPVG
i 5KRQTGNEQPVTCTKQRCTCVQFQ Z C D 0 UGXGTKHKECSWG
d
d
Z C
H Z
0 5 0 5D EQP
D t GPVQPEGUNCKPVGITCN

68. F
D
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H Z Z³ GUPQEQPXGTIGPVG

69. (MHPSOR 6HD I [
[ [
1 6 1 6
GHILQLGDHQ 1 /DLQWHJUDOGHHVWDIXQFLyQQRHVFRQYHUJHQWH
SXHV
d d
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GHILQLGDHQ 1 /DLQWHJUDOGHHVWDIXQFLyQHVFRQYHUJHQWHSXHV
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70. ,QWHJUDOHV(XOHULDQDV
8COQUCGUVWFKCTFQUGLGORNQUFGKPVGITCNGUKORTQRKCUSWGFGRGPFGPFGWPQQFQU
RCT¶OGVTQU TGURGEVKXCOGPVG UQP NCU KPVGITCNGU $GVC [ )COOC VCODKÃP NNCOCFCU KPVGITCNGU
'WNGTKCPCU
GHKPKEKÎP 5GFGHKPGNCHWPEKÎP$GVCFGRCT¶OGVTQUR[SEQOQNCKPVGITCN
E R S Z Z ZR S
0 5 0 5˜
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F
2QTNQVCPVQVGPGOQUWPCHWPEKÎP H Z
Z
Z
Z
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R
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FGHKPKFCGPIGPGTCN
GP 0 5RWGURCTC Z GNFGPQOKPCFQTFG H Z
Z
Z
S
R0 5 0 5
UGJCEGEGTQUK R [
RCTC Z GNFGPQOKPCFQTFG H Z
Z
Z
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S0 5 0 5
UGJCEGEGTQUK S
2TQRQUKEKÎP
'ZKUVGPEKC
5K R ! [ S ! NCKPVGITCNE
R S GUEQPXGTIGPVG
GOQUVTCEKÎP8COQU C GUVWFKCT NQU XCNQTGU FG R [ S RCTC NQU EWCNGU GUVC KPVGITCN GU
EQPXGTIGPVG'NGUVWFKQNQXCOQUCJCEGTRQTEQORCTCEKÎP[RQTNQVCPVQQDVGPFTGOQUWPC
EQPFKEKÎPUWHKEKGPVGFGGZKUVGPEKCQEQPXGTIGPEKC'PRTKOGTNWICTEQPUKFGTCOQU
Z Z Z Z Z Z Z Z ZR S R S R S
I I I˜ ˜ ˜
0 5 0 5 0 5F F F
2CTCGUVWFKCTNCEQPXGTIGPEKCGP 0 QDUGTXGOQUSWG
d
d
Z
Z Z
S
R R
0 5
0 5 RCTCVQFQ Z  0
2QT VCPVQ RCTC D R
R ! [ EWCNSWKGT S NC KPVGITCN FG H GP 0 GU
EQPXGTIGPVG
2CTCGUVWFKCTNCEQPXGTIGPEKCGP 5EQOQUGXGTKHKEC
d
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Z
Z Z
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UGFGFWEGSWGRCTCD ! S NCKPVGITCNGUEQPXGTIGPVG

71. #UÈ RWGU EQOQ Z Z ZR S
I ˜
0 5 F GU EQPXGTIGPVG RCTC R ! [ RCTC VQFQ S [
Z Z ZR S
˜ I
0 5 F GU EQPXGTIGPVG RCTC S ! [ RCTC VQFQ R GPVQPEGU
Z Z ZR S
I ˜
0 5 F GUEQPXGTIGPVGRCTC R ! [ S !
2TQRQUKEKÎP
2TQRKGFCFGU
E ER S S R 0 5 0 5
2CTCVQFQ S ! E S S0 5
2CTCVQFQ R ! S ! UGXGTKHKECSWGE ER S
S
R
R S 0 5 0 5
2CTCVQFQ R ! S ! UGXGTKHKECSWGE
S
R S V V VR S
EQU UGP0 5 ˜
I
F
GOQUVTCEKÎP
2CTCFGOQUVTCTNCKIWCNFCFDCUVCJCEGTGNECODKQFGXCTKCDNG Z V
.CEQORTQDCEKÎPGUKPOGFKCVCCRCTVKTFGNCFGHKPKEKÎPFGE S0 5
E R S Z Z ZR S
0 5 0 5˜
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0 5 0 50 5
0 5 0 5
F F
F F
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(MHPSOR +DOODU
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73. § ·
E¨ ¸
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74. GHKPKEKÎP 5GFGHKPGNCHWPEKÎP)COOCFGRCT¶OGVTQREQOQNCKPVGITCN
* R Z G ZR Z
0 5 ˜
f
I
F
2TQRQUKEKÎP
'ZKUVGPEKC
5K R ! NCKPVGITCN *
R GUEQPXGTIGPVG
GOQUVTCEKÎP6GPGOQU SWG GUVWFKCT NC EQPXGTIGPEKC FG NC KPVGITCN FG H Z G
Z
Z
R0 5
FGHKPKFCGPGNKPVGTXCNQ f0 5#PCNKCTGOQUNQUXCNQTGUFG R RCTCNQUEWCNGUNCKPVGITCNGU
EQPXGTIGPVG[RCTCGNNQXCOQUCEQPUKFGTCT f ‰ f0 5 0 5
5GC H Z G ZZ R
0 5 ˜
GPVQPEGUEQOQ
d
˜
d
G Z ZZ R R
RCTCVQFQ Z  0 NCKPVGITCN
FGNCHWPEKÎP H Z G ZZ R
0 5 ˜
GP 0 UGT¶EQPXGTIGPVGRCTC D R [RQTVCPVQRCTC
VQFQ R !
5GC H Z G ZZ R
0 5 ˜
WPC HWPEKÎP FGHKPKFC GP f5 UKGPFQ R WP PÕOGTQ TGCN
EWCNSWKGTC GPVQPEGU UG XGTKHKEC SWG Z G ZR Z
f
I ˜
F EQPXGTIG RCTC VQFQ R  [C SWG
RQFGOQUEQORCTCTGUVCHWPEKÎPEQP I Z
Z
0 5
EW[CKPVGITCNEQPXGTIGRWGU
Z
Z
f
I F
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d ˜ ˜ d
Z G
Z
G
Z
G Z Z
R Z
R
Z
R
Z
GPVQPEGUNCKPVGITCN Z G ZR Z
f
I ˜
F GUEQPXGTIGPVGRCTCVQFQ R 
2QTVCPVQ
Z G Z Z G Z Z G ZR Z R Z R Z
f
f
I I I˜ ˜ ˜
F F F
UGT¶EQPXGTIGPVGRCTC R !
2TQRQUKEKÎP
2TQRKGFCFGU
* 0 5
* *R R R0 5 0 5 0 5 RCTC R !
5K R GUWPPÕOGTQPCVWTCN R t UGXGTKHKECSWG * R R0 5 0 5
E R S
R S
R S
0 5 0 5 0 5
0 5
˜
* *
*

75. 2QT NCU RTQRKGFCFGU [ [ RQT GUVCT VCDWNCFC NC HWPEKÎP * R0 5 UG FGURTGPFG NC
KORQTVCPEKC FG NC RTQRKGFCF SWG PQU RGTOKVG ECNEWNCT EQP EKGTVC HCEKNKFCF GN XCNQT FG
CNIWPCUKPVGITCNGUKORTQRKCU
(MHPSOR RPRE S
S
1 6 I GW E
1 6 1 6
1 6
*
*
REWHQHPRVTXH * 1 6 S
(MHPSOR +DOODU [ H [[
f
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6ROXFLyQ [ H [[
f
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(MHPSOR +DOODU
OQ[
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6ROXFLyQ (IHFWXDQGRHOFDPELRGHYDULDEOH
OQ[ W [ H
[
[ W [ [ W H W
[ W
[ W
W
W
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%
KK
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G G G G G
VHWLHQHTXH
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[
[ W H WW
f
f
I I ˜G G *1 6

76. (MHUFLFLRVUHVXHOWRV
GVGTOKPCTNCEQPXGTIGPEKCFGNCUUKIWKGPVGUKPVGITCNGUKORTQRKCU%CNEWNCTGNXCNQTFG
NCKPVGITCNRCTCNQUECUQUGPNQUSWGNCKPVGITCNUGCEQPXGTIGPVG
C
FZ
Z
f
I
D G FZZ
fI
E
G
Z
FZ
Z
f
I
F
FZ
Z Z
f
f
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CTE VI
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FZ
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f
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FZ
Z
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I
FZ
Z
I
J
FZ
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FZ
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5QNWEKÎP
2CTCNCTGUQNWEKÎPFGNGLGTEKEKQCRNKECTGOQUNCFGHKPKEKÎPFGKPVGITCNKORTQRKCEQOQ
NÈOKVGFGWPCKPVGITCNFGHKPKFC2CTCGNNQPGEGUKVCTGOQUEQPQEGTWPCRTKOKVKXCFGNCHWPEKÎPC
KPVGITCT#WPSWGGPNCOC[QTRCTVGFGNQUCRCTVCFQUNCEQPXGTIGPEKCUGRWGFGCPCNKCTCRCTVKT
FGNCRTQRKCFGHKPKEKÎPGPCNIWPQUFGGNNQUUGTGCNKCGUVGGUVWFKQCRNKECPFQNQUETKVGTKQUFG
EQORCTCEKÎP
2CTCRTQEGFGTCNCKPVGITCEKÎPUGT¶PGEGUCTKQFGVGEVCTGNQRTQDNGOCRSWGJCEGSWGNC
KPVGITCN UGC KORTQRKC GU FGEKT GN RWPVQ Z VCN SWG NC HWPEKÎP H C KPVGITCT XGTKHKEC SWG
NKO
Z Z
H Z
o r
rf

77. HWPEKÎP PQ CEQVCFC GP PKPIÕP GPVQTPQ FGN RWPVQ Z Q GN GZVTGOQ FGN
KPVGTXCNQFGKPVGITCEKÎPEWCPFQGUGUVGGNPQCEQVCFQ
C5GVTCVCFGWPCKPVGITCNKORTQRKCRWGUGNKPVGTXCNQFGKPVGITCEKÎP=
fPQGUCEQVCFQ'N
KPVGITCPFQEWORNGNCEQPFKEKÎPPGEGUCTKCFGKPVGITCEKÎP[CSWG NKO
Z Zof
[RQTVCPVQ
CRNKECOQUNCFGHKPKEKÎP
FZ
Z
FZ
Z
Z
DD
D
D
D
D
of
f
of
ofI INKO NKO NKO
8GOQUCUÈSWGNCKPVGITCNGUEQPXGTIGPVG[UWXCNQTGU
DGPWGXQVGPGOQUWPCKPVGITCNKORTQRKCRQTPQUGTCEQVCFQGNKPVGTXCNQFGKPVGITCEKÎP'N
KPVGITCPFQEWORNGNCEQPFKEKÎPPGEGUCTKCFGKPVGITCEKÎP[CSWG NKO
Z
Z
G
of
[RQTVCPVQ
CRNKECOQUNCFGHKPKEKÎP

78. G FZ G FZ G G G GZ
C
Z
D C
Z
C
C
C
C
C
off of of ofII NKO NKO NKO NKO
.CKPVGITCNGUEQPXGTIGPVG[UWXCNQTGU
E5GVTCVCFGWPCKPVGITCNKORTQRKCRWGUGNKPVGTXCNQFGKPVGITCEKÎP=
fPQGUCEQVCFQ'N
KPVGITCPFQPQEWORNGNCEQPFKEKÎPPGEGUCTKCFGKPVGITCEKÎP[CSWG NKO NKO
Z
Z
Z
Z
G
Z
G
of of
f
WVKNKCPFQ.O*ÏRKVCN[RQTVCPVQNCKPVGITCNGUFKXGTIGPVG
6CODKÃP RQFTÈCOQU JCDGT RTQEGFKFQ WVKNKCPFQ ETKVGTKQU FG EQORCTCEKÎP
Z
G
Z Z
Z
! Ÿ
Z=f[EQOQ
FZ
Z
f
I GUFKXGTIGPVGVCODKÃPNQUGT¶
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FZ
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F5GVTCVCFGWPCKPVGITCNKORTQRKCRWGUGNKPVGTXCNQFGKPVGITCEKÎP
f
fPQGUCEQVCFQ
.CHWPEKÎPCKPVGITCTGUCEQVCFC[UGEWORNGNCEQPFKEKÎPPGEGUCTKCFGKPVGITCEKÎP[CSWG
NKO
Z Z Zof
[ NKO
Z Z Zof
[RQTVCPVQRQFGOQUFGUEQORQPGTNCKPVGITCNGPUWOCFGQVTCUFQUKPVGITCNGUFQPFGUÎNQWPQ
FGNQUGZVTGOQUFGNKPVGTXCNQFGKPVGITCEKÎPUGCPQCEQVCFQRCTCFGURWÃUCRNKECTNCFGHKPKEKÎP

88. NKO NKO
NKO NKO NKO CTEVI NKO CTEVI
NKO CTEVI CTEVI NKO CTEVI CTEVI
E D
C EC D
E D E D
C EC EC D C D
E
CC D
FZ FZ FZ
Z Z Z Z Z Z
FZ FZ
Z Z
Z Z
E C D E
f
f of of
of of of of
of of
S S§ ·
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³ ³
G 5G VTCVC FG WPC KPVGITCN KORTQRKC RWGU GN KPVGTXCNQ FG KPVGITCEKÎP PQ GU CEQVCFQ 'N
KPVGITCPFQEWORNGNCEQPFKEKÎPPGEGUCTKCFGKPVGITCEKÎP[CSWG

90. NKO CTE VI
CTE VI NKO
NKO
NKO
D
Z
D D
ZZ
Z Z Z
of
of
of of
S
GUEQORQPGOQUNCKPVGITCN
CTE VI
Z
FZ
Z
f
³ GPUWOCFGKPVGITCNGU
CTE VI
Z
FZ
Z³
[
CTE VI
Z
FZ
Z
f
³ .CHWPEKÎPCKPVGITCTGUEQPVKPWCGPGNKPVGTXCNQ=?[RQTVCPVQDCUVC
GUVWFKCTNCEQPXGTIGPEKCFG
CTE VI
Z
FZ
Z
f
³
7VKNKCOQUETKVGTKQUFGEQORCTCEKÎP

91. 2QT UGT NC HWPEKÎP CTEVIZ WPC HWPEKÎP ETGEKGPVG GP GN KPVGTXCNQ =
f [
NKOCTE VI
D
Z
of
S VGPGOQU

93. CTE VI
CTE VI
Z
Z
ZZ
S
S Ÿ
Z=
f
#FGO¶UUGXGTKHKECSWG
! Ÿ
Z Z
Z Z
NWGIQ
CTE VI
Z
Z Z
S
Z=
f
#UÈ RWGU EQOQ UCDGOQU SWG
Z
FZ
f
I GU EQPXGTIGPVG [ EQOQ

94. CTE VI
Z
FZ FZ Z
Z Z
f fS
 f
³ ³ RQT ETKVGTKQU FG EQORCTCEKÎP VGPGOQU SWG
CTE VI
Z
FZ
Z
f
³ VCODKÃPGUEQPXGTIGPVG[EQPENWKOQUSWG
CTE VI
Z
FZ
Z
f
³ NQGU
%CNEWNCOQUGNXCNQTFGNCKPVGITCNWVKNKCPFQNCFGHKPKEKÎP

95. CTE VI
CTE VI
NKO NKO CTE VI
D
D
D D
Z Z
FZ FZ Z
Z Z
f
of of
§ · S S§ ·
¨ ¸ ¨ ¸¨ ¸ © ¹© ¹
³ ³
H 5G VTCVC FG WPC KPVGITCN KORTQRKC RWGU NC HWPEKÎP PQ GU CEQVCFC GP GN KPVGTXCNQ FG
KPVGITCEKÎP[CSWG NKO
Z Zo
f
#RNKECOQUNCFGHKPKEKÎP
FZ
Z
FZ
Z
Z
o oI INKO NKO
H H H
H
0 5
[EQORTQDCOQUSWGUGVTCVCFGWPCKPVGITCNEQPXGTIGPVGFGXCNQT
I 5G VTCVC FG WPC KPVGITCN KORTQRKC RWGU NC HWPEKÎP PQ GU CEQVCFC GP GN KPVGTXCNQ FG
KPVGITCEKÎP[CSWG NKO
Z Zo
f
8COQUCCRNKECTETKVGTKQUFGEQORCTCEKÎPRCTCGUVWFKCT
UKUGVTCVCFGWPCKPVGITCNEQPXGTIGPVG%QOQ
d d Z Z Z
UG FGFWEG SWG

100. @
Z Z Z Z Z Z Z d ˜ ˜  [
GPVQPEGU
!

Z Z
Z 5
2QTUGT
FZ
Z
I FKXGTIGPVGVCODKÃPNQGU
FZ
Z
I

101. J 5G VTCVC FG WPC KPVGITCN KORTQRKC [C SWG NC HWPEKÎP PQ GU CEQVCFC GP GN KPVGTXCNQ FG
KPVGITCEKÎP'PGUVGECUQGNRWPVQRTQDNGO¶VKEQGUZRQTNQSWGFGUEQORQPGOQUNCKPVGITCN
GPFQUKPVGITCNGUKORTQRKCU
FZ
Z I
[
FZ
Z I
RCUCPFQGNRWPVQRTQDNGO¶VKEQCNGZVTGOQ
FGECFCWPCFGGNNCU
#RNKECOQUNCFGHKPKEKÎP
FZ
Z
FZ
Z
Z
o
o
I I
NKO NKO
H
H
H
H
0 5
FZ
Z I
[XGOQUCUÈSWGNCKPVGITCNGUEQPXGTIGPVG[UWXCNQTGU
0ÎVGUG SWG JCEKGPFQ GN ECODKQ FG XCTKCDNG VZL NC HWPEKÎP C KPVGITCT GU WPC
HWPEKÎPUKOÃVTKECFGVKRQH
V
H
VEQPKPVGITCNEQPXGTIGPVG[RQTGUQNCKPVGITCNTGUWNVCUGT
PWNC
K5GVTCVCFGWPCKPVGITCNKORTQRKCFGDKFQCSWGGNKPVGTXCNQFGKPVGITCEKÎPPQGUCEQVCFQ[NC
HWPEKÎP KPVGITCPFQ VCORQEQ GU CEQVCFC GP FKEJQ KPVGTXCNQ
RWPVQ RTQDNGO¶VKEQ Z
2TQEGFGOQUCFGUEQORQPGTNCKPVGITCNGPUWOCFGKPVGITCNGU
FZ
Z
FZ
Z
FZ
Z
FZ
Z
f f
I I I I
.CKPVGITCNSWGGUVWFKCOQUUGT¶EQPXGTIGPVGUKECFCWPCFGGUVCUNQGUOKGPVTCUSWG
UGT¶ FKXGTIGPVG UK CN OGPQU WPC FG GNNCU NQ GU 8COQU C XGT SWG NC UGIWPFC KPVGITCN GU
FKXGTIGPVGWVKNKCPFQETKVGTKQUFGEQORCTCEKÎP%QOQ
d Z Z
GPVQPEGU VGPGOQU
Z Z Z Z
Z
!
t 
0 50 5 0 5 0 [ EQOQ
FZ
Z I
GU
FKXGTIGPVG WVKNKCPFQ GN ETKVGTKQ FG EQORCTCEKÎP VCODKÃP NQ GU
FZ
Z
I [ RQT VCPVQ
FZ
Z
f
I FKXGTIG
GVGTOKPCTNCEQPXGTIGPEKCFGNCUUKIWKGPVGUKPVGITCNGUKORTQRKCU
C
UGP Z
Z
FZ
f
I
D
NP Z
Z
FZ
f
I
E
I Z
Z
FZ
( )
1/2

102. 5QNWEKÎP
C'PGUVGECUQPQRQFGOQUCRNKECTGNETKVGTKQFGEQORCTCEKÎPCNCHWPEKÎP H Z
Z
Z
UGP
RWGU GN UKIPQFGNCOKUOCECODKCKPVGPVCOQURQTVCPVQWVKNKCTETKVGTKQUFGEQORCTCEKÎP
RCTCGNXCNQTCDUQNWVQFGGUVCHWPEKÎP
#UÈVGPGOQU
d d  f
UGP
Z
Z Z
Z 0 5[EQOQ
Z
FZ
f
I GUEQPXGTIGPVGRQT
GNETKVGTKQFGEQORCTCEKÎP
UGP Z
Z
FZ
f
I VCODKÃPNQGUFGNQSWGUGFGFWEGSWGNCKPVGITCN
UGP Z
Z
FZ
f
I GUEQPXGTIGPVG
D 4GCNKCOQU GN ECODKQ FG XCTKCDNG ZGV
'PVQPEGU VGPGOQU SWG FZGV
FV %CODKCOQU
VCODKÃPNQUNÈOKVGUFGKPVGITCEKÎPCUÈZŸ V[Z
f Ÿ V
f
.CKPVGITCNSWGFCRQTVCPVQFGNCUKIWKGPVGHQTOC
G VG FV VG FVV V V
f
f
I I
4GEQTFCOQUSWGNCKPVGITCN'WNGTKCPC * R Z G FZR Z
0 5
f
I
GUEQPXGTIGPVGRCTCVQFQ
R 'PGUVGECUQWPCXGJGEJQGNECODKQFGXCTKCDNGTGEQPQEGOQUNCKPVGITCN'WNGTKCPC
* 0 5[GURQTVCPVQEQPXGTIGPVG
4GUQNXGOQUNCKPVGITCN VG FVV
f
I
* * *0 5 0 5 0 5 0 5
E'PRTKOGTNWICTPÎVGUGSWG
I I Z
Z
FZ Z Z FZ
( )1/2
0 5
4GEQTFCOQUSWGNCKPVGITCN
'WNGTKCPC E R S Z Z FZR S
0 5 0 5
I
GU EQPXGTIGPVG UK RS 'P GN ECUQ GUVWFKCFQ
VGPGOQU SWG R [ S FG NQ SWG FGFWEKOQU SWG R [ S CODQU
OC[QTGUSWG5GVTCVCFGWPCKPVGITCNEQPXGTIGPVG[RQFGOQUECNEWNCTUWXCNQT

107. Z Z FZ
§ · § · § ·§ · § ·
* * * *¨ ¸ ¨ ¸ ¨ ¸¨ ¸ ¨ ¸
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*§ ·© ¹ * ¨ ¸
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§ ·§ ·
*¨ ¸ S¨ ¸
S© ¹© ¹
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³

108. %CNEWNCTUKGURQUKDNGGN¶TGCFGNQUEQPLWPVQUSWGUGKPFKECP
C

109. +4 Z [ [ [
Z
­ ½
 t d® ¾
¯ ¿
D4GIKÎPNKOKVCFCRQTNCEWTXC [ Z [NCUTGEVCU [ Z [ Z
E4GIKÎPNKOKVCFCRQTNCEWTXC [ Z [NCUTGEVCU [ Z [ Z
F 4GIKÎPFCFCRQT

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