Ch-8, Integrations or Anti-Derivatives [ ch-8 ] + differentiations [ ch-6 ].

7/28/2019 Ch-8, Integrations or Anti-Derivatives [ ch-8 ] + differentiations [ ch-6 ].

http://slidepdf.com/reader/full/ch-8-integrations-or-anti-derivatives-ch-8-differentiations-ch-6- 1/125

@NK-ECHNOIL CGANGCCPNGA ASNHC.NGO ]96/DSL^/069?U

069?

]BNAB-_OBKKL-EIXBCEIXNO_,

NGXCAPIXNKG+HNJJCPC

GXNIXNKG,OB-2+OB-:U]NGXCAPIXNKG_ KP IGXN-HCPNZIXNZC_ KP

NGHCJNGNXC-NGXCAPIL]OB-2U+HCPNZIXNZC_]OB-

:UU

@^ EKBIEEIH-_NMIGHIP-MBIG-LKHBN

Q Q Q . E C H N O I L - N E I A C - V P K O C _ _ N G A . @ L K A _ V K X . O I @ ^ _ N M I G H C P - L K H B N




NGXCAPIXNKG+HNJJCPCGXNIXNKG,OB-2+OB-:U

Q Q Q . E C H N O I L - N E I A C - V P K O C _ _ N G A . @ L K A _ V K X . O I @ ^

_ N M I G H C P - L K H B N

Viac 0-KJ-99?

IGXN-HCPNZIXNZC 5 NGHCJNGNXC-NGXCAPIL 5 NGXCAPIXNKG4-Xcxt-@kkm 4 Eitbceitnos-90 ] Jkr Oliss-RNN , Vrc-cgangccrnga Arkup U Jkr _nghb Xcxt`kkm @kirh

Diesbkrk,_nghb. @y Ai`i Chuoitnkgil `kkms, Srhu-`iziir Miriobn,Vimnstig.

Gktc4 N lcirgch Xbns Obiptcr Ngtcaritnkg ng Arihc 90 ng prc-cgangccrnga okursc ng bnab-sobkkl Miriobn

Vimnstig ng `ctwccg 066?-0663 .

Xi`lc kj OkgtcgtsIGXN-HCPNZIXNZC 5 NGHCJNGNXC-NGXCAPIL 5 NGXCAPIXNKG4- ..................................................................... 0

2.9 IGXN-HCPNZIXNKG KP NGXCAPIXNKG4- ................................................................................................ ?

2.9.9 Jkreulis Jkr Ngtcarils Kj Xbc Jkre4- ............................................................................................. 3

2.9.0 Hcjngntc-Ngtcaril4- ............................................................................................................................ 3

] U= ............................................................ :

_tighirh Jkreulic Jkr Ngtcaritnkg 4- ...................................................................................................... :

2.0 NGXCAPIXNKG @^ _S@_XNXSXNKG4- ....................................................................................................... 2

2.0.9 NGXCAPIL_ KJ oks(ix`), IGH sng(ix`), (i,` @CNGA OKG_XIGX_ IGH i≦6 ). .............................. 2

2.0.0 _u`stntutnkg ng “Hcjngntc-Ngtcarils― 4- ............................................................................................ 99

2.0.? XPNAKGKECXPNO-_S@_XNXSXNKG_4 ................................................................................................... 9?

2.? NGXCAPIXNKG @^ “OKEVLCXNGA XBC _YSIPC_―4- ............................................................................. 93

Cx 2.: ........................................................................................................................................................... 92

Bkec wkrm .............................................................................................................................................. 92

2.8 NGXCAPIXNKG-@^-VIPX_ 4- .................................................................................................................. 92

JKPESLI KJ NGXCAPIXNKG-@^-VIPX_4 ................................................................................................. 92

_kec Nepkrtigt Jkreuli tk rcece`cr is jkllkwnga 4 ...................................................................... 9>

2.3 VIPXNIL-JPIOXNKG_4- .......................................................................................................................... 9>

2.3.0 ........................................................................................................................................................ 09

2.: NGXCAPIXNKG KJ PIXNKGIL-JPIOXNKG_ 4- ........................................................................................ 01

@isno- Ngtcaritnkg Vrk`lces ]Ob-2U,]2.9U4- .................................................................................................. ??

] U= ....................................... ?8

{4. } ................................................................................................... ?8

{4. } .................................................................................................... ?3

{4. } .................................................................................................... ?:







Viac ?-KJ-99?

{4. } ............................................................................................................................... ?:

{4. } ............................................................................................................ ?1

{4.

}................................................................... ?2

{4. ] U= } ................................................................... 86

{4. } ................................................................................... 86

_ct # 2 ]_tirt-bcrc + “NGXCAPIXNKG @^ S_NGA BILJ IGALC NGHNXNC_― 4- .................................................. 89

_CX # >(i) ]_XIPX-BCPC+ “NGXCAPIXNKG-@^-XPNAKGKECXPNO-_S@_XNXSXNKG_― 4 ................................... 38

_ct # >(`) ]_tirt-bcrcU ................................................................................................................................. :8

“ NGXCAPIXNKG-KJ-PIXNKGIL-JPIOXNKG„_ *Cx # 2.:+ _CX # 96 (i) *_tirt+ -― .......................................... 10

OB-: ] @I_NO MGKQLCHAC KG HNJJCPCGXNIXNKGU ]_XIPX-BCPCU4- ........................................................... 1:

Jkreulis ] Ob-: U 4 ............................................................................................................................... 1:

PSLC JKP HNJJCPCGXNIXNKG_4- .................................................................................................................. 1:

JKPESLI KJ HNJJCPCGXNIXNKG ]OB-:U4- ........................................................................................... 11

_CX # 99 (i) ]_tirt-bcrcU .............................................................................................................................. 26

_ct # 90 (i) ]stirt-bcrcU ]Ngtcaritnkg @y VirtsU4- ......................................................................................... 22

NGXCAPIXNKG-@^-VIPX_4- ....................................................................................................................... 22

] U= ............................................................ 22

]

U= ................................................................................... >8

_ct # 90 (`) ]_tirt-bcrcU Ngtcaritnkg @y Virts4 ............................................................................................ >>

_ct # 90 (o) ]_tirt-BcrcU ............................................................................................................................. >>

_ct # 90 (h) ]_tirtU Ngtcaritnkg @y Virts4- .................................................................................................. 969

_ct # 9? (i) ]_tirt-bcrc+ “IPCI-SGHCP-XBC-OSPZC― 4- ........................................................................... 961

Ob-: ]HnjjcrcgtnilU ]Jnrst-Vrngonplc U 4- ........................................................................................................ 998

2.9 IGXN-HCPNZIXNKG KP NGXCAPIXNKG4-

Ng tbc prcvnkus obiptcr, wc hnsoussch tbc prk`lce kj jnghnga tbc hcrnvitnvc

kj i anvcg jugotnkg .







Viac 8-KJ-99?

Bkwcvcr, eigy kj tbc ekst nepkrtigt ipplnoitnkgs kj Oiloulus lcih tk tbc

ngvcrsc prk`lce kj jnghnga tbc jugotnkg wbcg nts hcrnvitnvc ns anvcg .

Xbc rcqunrch jugotnkg ns oillch ig igtn-hcrnvitnvc kr ig nghcjngntc ngtcaril kj

tbc anvcg hcrnvitnvc.

Xbc prkocss kj jnghnga nt ns oillch igtn-hcrnvitnkg kr ngtcaritnkg.

Xk `c prconsc, lct jugotnkgs jrke I tk . tbcg J ns oillch ig

igtnhcrnvitnvc kj j nj J„(x) cxnst igh * U

jkr ill x I.

Kjtcg, wc sbill siy “J(x) ns ig igtn-hcrnvitnvc kr ig ngtcaril kj j(x) kvcr I― tk

ecig tbit J4I ns ig igtn-hcrnvitnvc kj j4I .

Krnangilly, tbc okgocpt kj igtnhcrnvitnvc kr ngtcaril kj i jugotnkg j, wis

hcvclkpch ng krhcr tk jngh i ocrting irci.

O5okgstigt kj Ngtcaritnkgs5Ir`ntriry-okgstigt, igh j(x)5ngtcarigh.

] U=

Xbc cxprcssnkg

] U=

Ns oillch tbc “nghcjngntc ngtcaril, kr dust tbc ngtcaril kj j(x) wntb rcspcot tk x, igh

ns hcgktch `y

] U=

Bcgocjkrtb, wbcrcvcr , wc wrntc nt wnll `c issuech tbit ig

igtnhcrnvitnvc kj j(x) cxnsts.

Jrke gkw kg wc sbill ilwiys usc “O― jkr tbc okgstigt kj ngtcaritnkg.

Qc ecgtnkg, wntbkut prkkj , tbc jkllkwnga nepkrtigt rcsult tk `c usch ngtbc cviluitnkg kj ngtcarils.

i) ] U=

`) ] U= Vrkvnhch tbit tbc igtnhcrnvitnvcs kj tbc jugotnkg j igh a cxnst.







Viac 3-KJ-99?

2.9.9 Jkreulis Jkr Ngtcarils Kj Xbc Jkre4-

] U=

Qbcrc tbc ngtcarighs bivc prkpcrly hcjngch hkeings.

n. Is ] U=wc bivc ]

U= prkvnhch

] U=

nn. Iaing , is ] U=wc

bivc ] U= prkvnhch ] U=

nnn. Jngilly, is ]

U= wc bivc

] U=prkvnhch ] U=

Cxieplc4

9. ] U=

0. ] U=

2.9.0 Hcjngntc-Ngtcaril4-

Nj “i― & “`― irc twk “rcil gue`crs] +― suob tbit i`,wc hcjngch tbc

quigtnty.

] U=

Is tbc “hcjngntc-ngtcaril kj j(x) `ctwccg i igh `, igh wrntc nt is

] U=

Vrkvnhch tbit ]i,`U ns okgtingch ng tbc hkeing kj j.

tbus nj

] U=

Xbcg

] U=] U=







Viac :-KJ-99?

] U=

]

U=

Xbc rcil gue`crs “i― & “`― irc oillch tbc Lkwcr & Sppcr lnents kj tbc

“hcjngntc-ngtcaril―.

Xbc Pcil-gue`cr rcprcscgts tbc irci “I― `kughch `y tbc ourvc y5j (x) igh

tbc lngcs x5i, x5`, scc jnaurc 2.9 `clkw .

Xbc sbihch pkrtnkg sbkwg ng jnaurc ns tbc irci “I― .

]

U=

Jna 2.9

_tighirh Jkreulic Jkr Ngtcaritnkg 4-

Ng tbc obiptcr kg hnjjcrcgtnitnkg wc act i lirac gue`cr kj stighirh

jkreulis jkr hnjjcrcgtnitnkg.

Ssnga tbcsc , wc gkw act tbc jkllkwnga sct kj stighirh jkreulic jkr

ngtcaritnkg.







Viac 1-KJ-99?

_.Gk j(x)

9

0

?

8

3

:

1

2

>

96

99

90

9?

98

93 9:

91

92

9>

06

09

00







Viac 2-KJ-99?

0?

08

03

0:

] U=

2.0 NGXCAPIXNKG @^ _S@_XNXSXNKG4-

Nts skectnecs jkugh tbit i obigac ng virni`lc “x― ng * U `y

ecigs kj i su`stntutnkg kj tbc jkre ] U

( ) `rngas tbc ngtcaril

tk i stighirh jkre , kr tk i jkre cisncr tk hcil wntb .

Xbns ns pirtnoulirly sk nj tbc anvcg ngtcaril ns kj tbc jkre ]

U, jkr tbcg jrke prkpcrtncs kj hnjjcrcgtnils,

wc mgkw tbit ] U, sk tbit wc act

] U,

K`vnkusly, tbc ngtcaril kg tbc rnabt ns euob cisncr tk oiloulitc tbig

tbit kg tbc lcjt. wc sbill nllustritc tbns ectbkh `y skec cxieplcs.

2.0.9 NGXCAPIL_ KJ oks(ix+`), IGH sng(ix+`), (i,` @CNGA OKG_XIGX_ IGH

i≦6 ).

9. ] U=

] U=

Lct, ] U=

Qc hnjjcrcgtnitnga w.r.t x

] U=







Viac >-KJ-99?

] U=

_k kur gcw ngtcaril ̀ cokecs

] U=

] U=

] U=

] U= igswcr

0. ] U4

-----------Cx # 94

Jngh

] U=

_klutnkg4

Lct4 ] U=

Hnjjcrcgtnitnga w.r.t x

]

U=

] U=

_k, ngtcaril `cokec

] U=

] U=

]

U=

] U= igswcr

--------------jngnsbch bcrc-

Cx# 0

Jngh ] U=







Viac 96-KJ-99?

_klutnkg4

] U=

Lct 4

] U=

Hnjjcrcgtnitnga w.r.t tbcti ] U

] U=

] U=

Kur anvcg ngtcaril `cokec

] U=

] U=

] U=

] U=

]

U=

] U= Igswcr.

---------------jngnsbch-bcrcc--------

Cxieplc # 8 4-

Jngh ] U=

_klutnkg4

]

U=

{4. }{4. }

Lct , ] U=

] U=

] U=

] U=







Viac 99-KJ-99?

] U=

] U=

] U= Igswcr.

------------jngnsbch-bcrc-------

Cx # 3 4

Jngh ] U=

_klutnkg4

] U=

Qc hk lnmc tbit

] U=

] U=

Lct4

] U=

] U=

] U=

]

U=

] U=

] U=

] U=

] U=

] U=

] U=

] U=

] U=

]

U=

] U=

]

U=

] U=

] U=

] U=

] U=

] U=

] U=

] U= Igswcr.

-------jngnsbch-bcrc--------------2.0.0 _u`stntutnkg ng “Hcjngntc-Ngtcarils― 4-

Qbcg i su`stntutnkg ns eihc ng i hcjngntc-ngtcaril, tbc lnents kj ngtcaritnkg sbkulh ilsk `c obigach.

Xbc gcw lnents irc k`tingch is tbc vilucs kj gcw virni`lc wbcg tbc klh virni`lc cquils tbc klh lnents.

Qc nllustritc tbns `y skec cxieplcs. Is jkllkw 4

Cx # 9 4- Cviluitc4 ] U=

_klutnkg4-







Viac 90-KJ-99?

Qc bivc i anvcg hcjngntc ngtcaril ng `clkw cq-I.

] U=

Lct ,

] U=

] U=

] U=

] U=

Jkr gcw Lkwcr lnent4- wbcg klh Lkwcr lnent ns ] U

] U=

]

U= nts kur gcw lkwcr lnent .

Jkr Gcw Sppcr-lnent4- Qbcg klh Sppcr-lnent ns ] U4-

] U=

] U= nts kur gcw Sppcr lnent .

_k, kur anvcg ngtcaril `cokec.

] U=

] U=

] U=

] U=

] U=

] U=

] U= Igswcr.

-----------Jngnsbch-bcrc-----------

Cxieplc # 0 4-

Jngh ] U=

_klutnkg4-

Qc bivc i anvcg hcjngntc ngtcarils







Viac 9?-KJ-99?

] U=

Lct, ] U=

Hnjjcrcgtnitnga kg ciob snhc wntb rcspcot tk *w.r.t+ “x―

] U=

] U=

Jkr gcw Lkwcr-Lnent , wbcg klh Lkwcr-lnent ns ] U4-

] U= ] U=

] U=

Jkr gcw Sppcr-Lnent , wbcg klh Sppcr-lnent ns ] U4-

] U=

] U=

] U=

] U=

]

U=

] U=

] U= Igswcr.

------------------Jngnsbch-bcrc-----------

2.0.? XPNAKGKECXPNO-_S@_XNXSXNKG_4

Nj tbc ngtcarigh okgtings cxprcssnkgs kj tbc jkre ng `clkw ti`lc tbcg wc usc tbc jkllkwnga Xrnakgkectrno su`stntutnkgs4

_.Gk Ngtcarigh Xrnakgkectrno-su`stntutnkgs4

9 0

?

-------------------

Cx # ? 4 Cviluitc tbc jkllkwnga 4

] U=







Viac 98-KJ-99?

_klutnkg4

] U=

Lct4 ]

U=

Hnjjcrcgtnitnga

] U=

Jkr Pihnoil ] U4-

] U=

] U=

] U=

] U=

] U=

] U=

] U= nts kur Pihnoil .

Xbc lnent jkr irc4

Jkr Gcw Lkwcr-Lnent4 ] wbcg Klh Lkwcr lnent ns ( ) U

] U= ] U=

] U=

Jkr Gcw Sppcr-Lnent4 ] wbcg Klh Sppcr-lnent ns ( ) U

] U= ] U=

] U=

Bcgoch kur ngtcaril `cokec.

] U=

] U=

] U=

] U=

] U=







Viac 93-KJ-99?

] U=

]

U=

] U=

] U=igswcr

----------jngnsbch-bcrc------

2.? NGXCAPIXNKG @^ “OKEVLCXNGA XBC _YSIPC_―4-

Nj tbc ngtcarigh ngvklvcs i quihritno cxprcssnkg ng tbc hcgkengitkr kr ughcr i rihnoil, i uscjul hcvnoc ns tk okeplctc tbc squirc.

Cx # 94 Jngh ] U=

_klutnkg4

Lct4

] U=

] U=

] U=

] U=

]

U=

] U=

] U=

] U=

Kur i`kvc cquitnkg `cokecs.

Lct4

]

U=

] U=

]

U=

]

U=







Viac 9:-KJ-99?

] U=

]

U=

]

U=

]

U=

] U= Igswcr.

-----------jngnsbch-bcrc----------

Cx # 0 4- Jngh ]

U=

_klutnkg4

] U=

_k,

] U=

Lct4 ] U=

Ximnga hnjjcrcgtnitnga kg ciob snhc w.r.t “x―.

] U=

] U=

] U=

] U=

Jkr 4-

] U=

{4. }

] U=

] U=

_k kur i`kvc ngtcaril `cokec 4

] U=

] U=







Viac 91-KJ-99?

] U=

] U=

] U=

Okgsnhcr 4-

] U=

Jkreuli 4-

] U=

] U=

] U=

] U=

Okgsnhcr 4-

] U=

Lct4 ] U=

Hnjjcrcgtnitnga w.r.t “y―

]

U=

] U=

] U=

_k kur ngtcaril `cokecs 4

] U=

] U=

-------------

Jkr “y―4

] U=

Hnjjcrcgtnitnga w.r.t “y―.

] U=

] U=







Viac 92-KJ-99?

---------

………hk nt litcr ………….

--------------jngnsbch bcrc-------

Cx 2.:

Bkec wkrm

--------------jngnsbch bcrc--------

2.8 NGXCAPIXNKG-@^-VIPX_ 4-

JKPESLI KJ NGXCAPIXNKG-@^-VIPX_4

] U=

Cxieplc # 8 4-

Jngh ] U= wbcrc i,` irc okgstigts igh `6.

_klutnkg4

Lct 4 ] U= & ] U=

] U=

Hnjjcrcgtnitnga w.r.t “x―

] U=

] U=

] U=

Ngtcaritnga w.r.t “ x―

] U=

] U=

_k `y plionga tbc viluc ng cq-I wc act=







Viac 9>-KJ-99?

] U=

] U=

]

U=

] U=

Hk nt litcr ………

------------jngnsbch bcrc-------------

_kec Nepkrtigt Jkreuli tk rcece`cr is jkllkwnga 4

] U= tk cviluitc ocrting ngtcarils.

Cx 3 oiloulitc ]

U=

_klutnkg4

Lct 4 ] U=

] U=

Hnjjcrcgtnitnga kg ciob snhc w.r.t.―x―.

] U=

]

U=

] U=

] U=

] U=

] U=

@y plionga ill i`kvc viluc ng cq-@ , wc act

]

U=

] U= Igswcr.

----------jngnsbch-bcrc------

2.3 VIPXNIL-JPIOXNKG_4-

Okgsnhcr tbc jkllkwnga cxprcssnkg







Viac 06-KJ-99?

] U=

Qbcrc igh irc gkg-zcrk.

Nj wc wnsb tk oke`ngc tbcsc tbrcc tcres tkactbcr ngtk i sngalc jriotnkg, wc oig cisnly hk sk `y tbc usuil ectbkh. Xbus

] += nts oillch is “Virtnil-jriotnkgs ]kr Okepkgcgts kj Pcsultigt-jriotnkg U―.

] U=

] U=

] += nt„s Oillch is “Pcsultigt-jriotnkg―.

Ng tbc prcscgt scotnkg, wc hnsouss ectbkhs `y wbnob i anvcg jriotnkgil cxprcssnkg n-c i “ritnkgil-jugotnkg― oig `c rcsklvch kr

hcokepksch ngtk nts “pirtnil-jriotnkgs―.

Xbns tcobgnquc wnll `c usch ng tbc gcxt scotnkg tk oirry kut ngtcaritnkg kj “Pitnkgil-Jugotnkgs―.

Cvcry “Jriotnkgil-cxprcssnkg― oig `c cxprcssch is i quktncgt kj twk pklygkenils . Qc rcstrnot kursclvcs tk cxprcssnkgs kj tbc jkre G(x)/H(x), wbcrc G(x) igh H(x) irc pklygkenils wntb rcil okcjjnoncgts igh H(x) ns

Gkg-zcrk.

Nj tbc hcarcc kj tbc Guecritkr G(x) ns lcss tbig tbc hcgkengitkr H(x), tbc cxprcssnkg ns oillch i “Vrkpcr-jriotnkg―, ktbcrwnsc

“Neprkpcr-Jriotnkg―.

Cvcry Neprkpcr jriotnkg oig `c rchuoch tk i sue kj i pklygkenil igh i prkpcr-jriotnkg, jkr nj tbc hcarcc kj tbc guecritkr ns cquil

tk kr arcitcr tbig tbit kj tbc hcgkengitkr, wc eiy hnvnhc G(x) `y H(x) k`tingnga i quktncgt- pklygkenil “Y(x)― igh i Pceinghcr-

pklygkenil “P(x)― wbksc hcarcc ns lcss tbig tbit kj H(x).

] U=

] U=

Xbus ,

]

U=

Qbcrc4

{4. ] U= }

Qbcrc irc tbc hcarcc kj pklygkenil ] kr bnabcst hcarcc kj pklygkenil U P(x) , H(x) igh G(x) rcspcotnvcly.

Bcgoc wc rcstrnot kursclvcs ng wbit jkllkws, tk prkpcr-jriotnkgs.

Qc mgkw tbit nt ns ilwiys pkssn`lc tk cxprcss pklygkenil H(x) is i prkhuot kj jiotkrs wbnob irc lngcir kr quihritno wntb rcil

okcjjnoncgts.

Xbc lngcir jiotkrs okrrcspkgh tk tbc rcil rkkts kj tbc cquitnkg H(x)56 wbnlc tbc quihritno jiotkrs okrrcspkgh tk okeplcx rkkts wbnobkoour ng okgduaitc pinrs.

Qc suppksc tbns hkgc , tbcg cvcry “prkpcr-jriotnkg― oig `c rcsklvch ngtk “pirtnil-jriotnkgs― ng iookrhigoc wntb tbc jkllkwnga

tbckrce wbksc prkkj ns `cykgh tbc sokpc kj tbns `kkm.

2.3.9 Xbckrce4-

Cvcry prkpcr jriotnkg G(x)p<g

/H(x)g oig `c rcsklvch ngtk ilac`rino sue kj Virtnil-jriotnkgs is jkllkws4







Viac 09-KJ-99?

n. Nj i “lngcir-jiotkr (ix+`)― koours kgoc is i jiotkr kj H(x) , tbcrc ns i pirtnil -jriotnkg kj tbc jkre *I/ix`+ wbcrc “I― ns i

okgstigt. ] U=

nn. Nj i “lngcir-jiotkr (ix`)― koours “m― tnecs is i jiotkr kj H(x), tbcrc irc “m― Virtnil -jriotnkgs kj tbc jkre 4

] U=

Qbcrc I9,I0,…., Im irc okgstigts.

nnn. Nj i “Yuihritno-jiotkr(ix0`xo)― koours kgoc is i jiotkr kj H(x), tbcrc ns i pirtnil-jriotnkg kj tbc jkre is

] U=

Qbcrc @,O irc okgstigt.

nv. Nj i Yuihritno-jiotkr ( ) koours m tnecs is i jiotkr kj H(x), tbcrc irc m pirtnil-jriotnkgs kj tbc jkre.

]

U=

Qbcrc @9,O9,@0,O0,……,@m,Om irc okgstigts.

------------jngnsbch‘

2.3.0

Qc nllustritc tbc ectbkh kj hctcrengnga tbc ugmgkwg okgstigts ng tbc jkllkwnga cxieplcs .

Ng ciob oisc, tbc rcsult eiy `c obcomch `y ihhnga tbc “pir tnil-jriotnkgs― tk act tbc krnangil cxprcssnkg.

Oisc-N4 wbcg tbc jiotkrs kj tbc hcgkengitkr H(x) irc ill lngcir igh hnstngot.

Cx # 9 4 Pcsklvc ngtk Virtnil-jriotnkgs.

]

U=

_klutnkg4

Is tbc hcgkengitkr ns

Lct4 ] U=

Qc eiy wrntc , `y pirt (n) kj tbc tbckrce (sco. 3.0.9).

] U=

] U=

] U=

] U=

] U=

-----------







Viac 00-KJ-99?

Lct 4 ] U=

] U=

Cntbcr4 ]

U=

Kr ] U=

Kr ] U=

-------

Jkr I4- ]wbcg x56 U4- ] U=

] U=

] U=

]

U=

] U=

] U=

Jkr @4- ] U=

] U=

] U=

] U=

] U=

] U=

] U=

] U=

Jkr O4- ] U=

] U=

] U=

{4. ] U= }

] U=

] U=

] U=

] U=







Viac 0?-KJ-99?

] U=

] U= ] U= ] U=

]

U=

Qc act =

] U=

] U= Igswcr.

-----------jngnsbch-bcrc----------

Oisc-NN4- wbcg ill tbc jiotkrs kj tbc hcgkengitkr H(x) irc lngcir `ut skec irc rcpcitch.

Cxieplc # 04- Pcsklvc ] U= ngtk Virtnil-Jriotnkgs ;

{4. Lngcir-jiotkrs5H(x)5(x+i)56=rcil-rkkts kj cq-7 H(x)56 }

{4. Yuihritno-jiotkrs5H(x)5(x+i)(x+`)56 --7 okeplcx-rkkts kj cq-7 H(x)56 }

_klutnkg4

] U=

Lct 4 ] U=

] U=

]

U=

] U=

] U=

] U=

Lct ] U=

] U=

Cntbcr 4- ] U= ] U=

Kr ] U= ] U= ] U= ] U= ] U=

] U=

Jkr @] U 4- ]wbcg ] U= U4-

] U=

] U=







Viac 08-KJ-99?

] U=

] U=

]

U=

Jkr O 4- ] QBCG ] U= U4-

] U=

] U=

] U=

] U=

Xk act “I― wc cquitc tbc okcjjnoncgts kj “I― kg `ktb snhcs kj cq -sts igh act

]65I+O U= ] -O 5I U=

] U= ] U= ] U=

] U=

] U= Igswcr

------------jngnsbch-bcrc----------

Oisc-NNN4- wbcrc H(x) bis “gkg-rcpcitch-nrrchuon`lc― quihritno jiotkrs.

Cxieplc # ?4 rcsklvc ] U= ngtk pirtnil jriotnkgs ;

_klutnkg4-

] U=

] U=

] U=

] U=

{4.] U=}

Xk jngh “@― & “O― , wc cquitc tbc okcjjnoncgts kj x0

igh tbc okgstigt tcres kg tbc twk snhcs kj cq-X tk act=

{4.] U=}

] U= ] U=

] U= {4. ] U= }







Viac 03-KJ-99?

] U= ] U=

] U=

Jkr I4-

Lct 4 ] U=

] U= ] U=

] U=

{4.] U=}

@y plionga tbc viluc kj x59 ng cq-X wc act .

] U=

] U=

] U=

] U= ] U=

] U=

] U= ] U= ] U=

] U=

]

U= Igswcr

----jngnsbch-bcrc--------

Oisc-NZ4- Qbcg H(x) bis “Pcpcitch nrrchuoni`lc quihritno jiotkrs4

Cxieplc4- Pcsklvc 57 ] U= ngtk Virtnil jriotnkg;

_klutnkg4-

@y pirts (n) igh (nv) kj tbc tbckrce (sco 3.0.9), wc eiy wrntc4

] U=

] U=

] U=

Lct4

] U=

Cntbcr4







Viac 0:-KJ-99?

] U=

] U=

Kr

] U=

] U=

] U=

] U=

] U=

] U=

QBCG ] x9 59 U4-

] U=

] U=

…..

…

] U=

Iaing , cquitnga tbc okcjjnoncgts kj x8,x

?,x igh tbc okgstigt tcres, kg tbc twk snhcs kj cq-9, wc act

] U=

] U=

] U=

] U=

Bcgoc tbc rcqunrch Virtnil jriotnkg irc 4

] U=

]

U=

] U=

] U= Igswcr.

-------------JNGN_BCH-------------







Viac 01-KJ-99?

2.: NGXCAPIXNKG KJ PIXNKGIL-JPIOXNKG_ 4-

Qbcg tbc ngtcarigh okgsnsts kj i ritnkgil jugotnkg, wc rcsklvc tbns jugotnkg ngtk ig ilac`rino sue kj pirtnil jriotnkgs `y tbc ectbkhs hcorn`ch

ng sco # 2.3 .

Nts tbcg acgcrilly pkssn`lc tk oirry kut tbc ngtcaritnkg is nllustritch ng jkllkwnga cxieplcs.

Cxieplc 94

Jngh ] U= ;

_klutnkg4

Qc bivc

] U=

] U=

] U=] U=

Lct4

] U=

] U=

Cntbcr4

] U=

] U=

Kr

] U=

] U=

Vut tbc viluc ng cq 9 wc act

] U=

] U=

{4. ] U= ] U= }

] U=

Qbcg x95i 4-

] U=

] U=







Viac 02-KJ-99?

] U=

] U=

]

U=

Qbcg 4-

] U=

] U=

] U=

] U=

] U=

] U=

] U=

_k,

] U=

{4. ] U= ] U= }

] U=

] U=

] U=

Ipplynga ngtcaritnkg wntb rcspcot tk “x― kg cq-z

] U=

] U=

] U=

] U= Igswcr.

------------Jngnsbch-bcrc-----------

Cxieplc 04

Jngh ] U= ;

_klutnkg4

Pcsklvnga ngtk pirtnil jriotnkgs, wc act







Viac 0>-KJ-99?

] U=

] U=

Lct4

] U=

Cntbcr4

] U=

] U=

Kr

] U=

]

U=

] U=

Kr

] U=

] U=

] U=

Vut tbc viluc igh act tbc igswcr .

] U=

{4. ] U= ] U= ] U= }

Qbcg ] U= 4-

] U=

] U=

] U=

] U=

] U=

Qbcg ] U= 4-

] U=

] U=







Viac ?6-KJ-99?

] U=

] U=

]

U=

] U=

Qbcg ] U= 4-

] U=

] U=

] U=

] U=

] U=

] U=] U=

_k,

{4. ] U= ] U= ] U= }

Vlioc i`kvc ng cq-I4

] U=

] U=

] U=

] U=

] U=

] U=

] U=

] U=

] U= Igswcr.

--------------Jngnsbch-----------

Cxieplc ?4-







Viac ?9-KJ-99?

Jngh ] U=

_klutnkg4-

]

U=

] +=57 Nts “Ne-prkpcr jriotnkg― .

Hnvnhnga tbc guecritkr `y tbc hcgkengitkr usnga tbc ectbkh kj lkga hnvnsnkg , wc act,

] U=57 nts Ne-prkpcr jriotnkg.

Bcgoc,

] U=

]

U=

] U=

] U=

] U= Igswcr

--------------jngnsbch--------

Cxieplc 84-

Jngh ]

U=

_klutnkg4-

] U=

_k, tbit

] U=

] U=

] U=

] U=

] U=

] U= Igswcr.







Viac ?0-KJ-99?

----jngnsbch----------

Cxieplc 3 4-

Jngh ]

U= ;

_klutnkg4-

] U=

_k tbit,

] U=

] U=

]

U=

] U=

] U=

] U=

]

U=Igswcr.

---jngnsbch-------

Cxieplc :4-

Jngh ] U= ;

_klutnkg4-

] U=

] U=

] U=

Lct,

] U=

] U=

] U=







Viac ??-KJ-99?

] U=

] U= Igswcr .

-----------jngnsbch---------

@isno- Ngtcaritnkg Vrk`lces ]Ob-2U,]2.9U4-

Y9) ] U=

_klutnkg4-

] U=

] U=

] U=

]

U=

] U=Igswcr

------------jngnsbch---------

Y8) ] U=

_klutnkg4-

]

U=

] U=

] U=

] U=

]

U=

]

U=Igswcr.

[------------------Jngnsbch----

Y3) ] U= ;

_klutnkg4-

] U=

] U=

-----------







Viac ?8-KJ-99?

4. Jkreuli 4

] U=

-----------------------

] U=

] U=

] U= Igswcr.

----------------jngnsbch-------

Y> ) ] U=

_klutnkg4-

] U=

] U=

] U=

] U=

] U=Igswcr.

------------Jngnsbch-------

Y96) ] U=

_klutnkg4-

] U=

] U=

] U=

] U= Igswcr.

------------jngnsbch-------

Y99) ] U= ;

_klutnkg4-

{4. }

] U=

] U=







Viac ?3-KJ-99?

] U=

] U=

]

U=

] U=

]

U=

]

U= Igswcr.

--------jngnsbch---------

Y90) ] U= ;

_klutnkg4-

]

U=

] U=

] U=

] U=

] U=

] U=

] U=

]

U=

]

U=

]

U= Igswcr.

-----------Jngnsbch------

Cviluitc4-

Anvcg4 ] U= ;

_klutnkg4-

-------------------

{4. }

------------------

] U=

] U=

] U=







Viac ?:-KJ-99?

] U=

]

U=

] U=

] U= Igswcr.

----------Jngnsbch-----

Y9?) ] U= ;

_klutnkg4-

] U=

--------------

{4. }

---------------

] U=

] U=

] U=

] U=

]

U=

] U=

---------

{4. }

Ng Pkeig-Srhu57 Hcgkengitkr]`/oU mi Hcgkengitkr]oU , Gkengitkr]iU si eultnply bkk di ti bin .

Ng Cgalnsb57 tbc Hcgkengitkr]oU kj tbc Hcgkengitkr]`/oU acts eultnply wntb tbc Gkengitkr]iU.

-----------

]

U=

]

U=

]

U=

]

U= Igswcr.







Viac ?1-KJ-99?

-------Jngnsbch------

……

Y98) Hcjngntc ngtcaril Vrk`lce4 ]

U= ;

_klutnkg4-

] U=

] U=

Eultnply igh hnvnhc `y ] U sk,

] U=

]

U=

----------

{4. }

---------------

] U=

]

U=

]

U=

] U=

] U=

] U=

]

U=

]

U=

] U=

] U=

] U=

] U=







Viac ?2-KJ-99?

………..

] U= Igswcr.

------------Jngnsbch------

Y9:) ] U=

_klutnkg4-

] U=

]

U=

] U=

] U=

] U=

] U=

] U=

] U=

] U=

]

U=Igswcr.

------------Jngnsbch------

Y91) ] U=

_klutnkg4-

] U=

{4. }

] U=

]

U=

]

U=

] U=

] U=







Viac ?>-KJ-99?

---------------

{4. }

{4.

}

------------

] U=

] U=

] U=

] U=

] U=

] U=

] U= Igswcr.

-------------Jngnsbch------

Y ) ] U= ;

_klutnkg4-

{4. }

] U=

] U=

]

U=

] U=

] U=

] U=

] U=

]

U= Igswcr.

-------Jngnsbch---------

Y92) ] U= ;

_klutnkg4-







Viac 86-KJ-99?

] U=

] U=

-----------

{4. ] U= }

--------

] U=

] U=

] U= Igswcr.

--------------Jngnsbch--------

Y9>) hcjngntc Ngtcaril4 ] U= ;

_klutnkg4-

{4. }

_k,

] U=

] U=

] U=

]

U=

] U=

] U=

……

] U=

@y Ipplynga ngtcaritnkg wc act .

……

] U= Igswcr.

---------Jngnsbch------sk cisy --------

Y) ] U= ;







Viac 89-KJ-99?

_klutnkg4-

] U=

]

U=

Lct,

] U= , ] U=

] U=

] U=

_k, kur ngtcaril `cokecs

] U=

] U=

] U=

] U=

] U=

] U=

] U=

] U= Igswcr.

------------Jngnsbch----------

Gktc4 Gkw wc hnvnhc tbc stuhy kj Ngtcaritnkg ngtk ocrting scts ng krhcr tk systceitnoilly lcirgch tbc Ngtcaritnkgs kr igtn-hcrnvitnvcs Obiptcr .

_ct # 2 ]_tirt-bcrc U “NGXCAPIXNKG @^ S_NGA BILJ IGALC NGHNXNC_― 4-

---------------------

JKPESLI_ KJ BILJ-IGALC4-

] U=

Hnvnhch `y 0 kg ciob snhc

] U=

] U=

KP







Viac 80-KJ-99?

] U=

Hnvnhc `y 0 kg ciob snhc

]

U=

] U=

------------------

Y9 ) ] U= ;

_klutnkg4-

] U=

---------------

{4. ]

U= ]

U= }

] U=

] U=

------------

] U=

{4. }

] U=

] U=

] U=

] U=

] U=

] U=

] U=

] U=

] U=

----------------







Viac 8?-KJ-99?

] U=

] U=

Xbns sbkw tbit tbc ngtcaritnkgs igh hnjjcrcgtnitnkgs irc ngvcrsc kpcritnkgs.

--------------

] U= Igswcr.

{4. ] U= }

------------Jngnsbch----------

Y0) _ct-2 , ] U= ;

_klutnkg4-

] U=

{4. }

{4. }

{4. }

] U=

] U=

] U=

] U=

] U=

] U=

] U=

] U=Igswcr

------------Jngnsbch-----------

Y? ) ] U= ;

_klutnkg4-

] U=







Viac 88-KJ-99?

{4. }

] U=

] U=

] U=

] U=

] U=

]

U=

] U= Igswcr.

------------Jngnsbch----------

Y8) ] U= ;

_klutnkg4-

] U=

----------------

4.

] U=

] U=

----------------

] U=

Jkr Pihnoil4-

]

U=

] U=

] U=

] U=







Viac 83-KJ-99?

] U=

] U=

] U=

] U= kur Pihnoil ̀ cokec cqunvilcgt tk tbns cxprcssnkg.

_k, @y plionga tbc cqunvilcgt viluc kj kur Pihnoil ngtk tbc i`kvc anvcg ngtcaril wc act.

Kur ngtcaril `cokecs.

] U=

] U=

] U=

] U=

] U=

] U=

-------------

4.

{4. }

{4. }

---------------

] U=

] U=

] U=

OKG_NHCP “ ― 4-

] U=

Lct ,

] U=

Hnjjcrcgtnitnkg kg ciob snhc w.r.t .







Viac 8:-KJ-99?

] U=

] U=

] U=] U=

] U=

] U=

_k, kur i`kvc ngtcaril `cokecs

] U=

] U=

] U=

] U=

] U=

] U=

--------

] U=

---------

] U=

] U=

] U=

@y plionga tbc viluc kj cq-@ ng cq-I , tbcg cq-I `cokecs ,wc act.

] U=

] U=

] U=







Viac 81-KJ-99?

OKG_NHCP 4-

] U=

-------------------

Pc-vncw list issueptnkgs is 4

Lct ,

] U=

Hnjjcrcgtnitnkg kg ciob snhc w.r.t .

] U=

]

U=

] U=] U=

] U=

] U=

----------------

Vlioc ill i`kvc viluc ng kur cquitnkg-nn , wc act .

] U=

] U=

-----------

] U=

-----------

] U=

] U=

] U=

@y plionga tbc viluc kj cq-nnn ng cq-O wc act .

] U=







Viac 82-KJ-99?

] U=

] U= Igswcr.

{4. -7 o„o 5 o„„ -

------------Jngnsbch-----------

Y3 ) ] U= ;

_klutnkg4-

] U=

Ssnga Bilj igalc nhcgtnty

----------------

4.

] U=

] U=

-----------------------

] U=

_k, lct,

Jkr Pihnoil 4-

] U=

] U=

] U=

] U=

] U= Kur rihnoil `cokec cqunvilcgt tk tbns gcw cxprcssnkg.

_k,

Kur i`kvc anvcg ngtcaril `cokecs.

] U=

] U=







Viac 8>-KJ-99?

] U=

] U=

Eultnply igh hnvnhc `y ] U=

]

U=

] U=

Lct,

] U=

Hnjjcrcgtnitch kg ciob snhc wntb rcspcot tk “x―.

]

U=

] U=

] U=

@y plionga tbc i`kvc viluc ng cq-pnss , tbig wc act.

]

U=

] U=

] U=

] U=

] U=

------------

]

U=

-------------

] U=

] U= Igswcr , `ut nt tbc rcqucst jrke ec tk ill vncwcrs tbit plcisc eust vcrnjy tbc igswcr jrke

tbc `kkm kr Yuilnjnch tciobcr, `coiusc enabt `c N hnh sk, eigy enstimcs bcrc ng ey ill gktcs .







Viac 36-KJ-99?

----------------Jngnsbch-bcrc-------------

Cx # 2.? ] sct-2 U

Y: ) kr Y3 (xnv) ) 57 ]

U= ;

_klutnkg4-

] U=

--------------

4. Ssnga Bilj Igalc Nhcgnty4-

{4. }

{4. }

-------------------

] U=

] U=

] U=

] U=

]

U=

] U=

] U=

] U= Igswcr.

--------------Jngnsbch--------

]2.?U sct-2

Y1) ] U= ;

_klutnkg4-

] U=

-------------

Ssch Bilj Igalc Nhcgnty4-

] U=







Viac 39-KJ-99?

-------------

] U=

]

U=

JKP PIHNOIL4-

] U=

] U=

] U=

] U=

] U=

] U= nts wbit kur list rihnoil `cokec.

@y plionga tbc viluc kj i`kvc Pihnoil ng cq-I wc act.

] U=

] U=

] U=

] U=

] U=

] U=

] U=

-----------

4.

] U=

]

U=

-----------------

OKG_NHCP 4-

] U=

Eultnply igh hnvnhc `y ] U4-







Viac 30-KJ-99?

] U=

] U=

] U=

Lct,

] U=

Ssnga hnjjcrcgtnil kg `ktb snhc w.r.t “x―

] U=

] U=

] U=

_k, `y plionga tbc i`kvc viluc ng i`kvc cq-_AA wc act.

] U=

] U=

] U=

] U=

----------

]

U=

--------

] U=

OKG_NHCP 4-

] U=

eultnply igh hnvnhc `y ] U

] U=

]

U=

Lct,

] U=

Ssnga hnjjcrcgtnil kg ciob snhc w.r.t “x―

] U=







Viac 3?-KJ-99?

] U=

] U=

@y plionga tbc i`kvc viluc ng cq-__A wc act.

] U=

] U=

] U=] U=] U=@y plionga tbc viluc kj cq-B igh cq-T wc act.

] U=

] U=

] U= Igswcr.

--------------------------Jngnsbch---------------

_ct-2

Y>) ] U= ;

_klutnkg4-

------------------

S_C BILJ IGALC NHCGXNX^4-

{4. }

] U=

_k,

---------------

]

U=

] U=

]

U=

] U=

] U=







Viac 38-KJ-99?

Eultnply & hnvnhc `y ] U=

]

U=

] U=

Lct ,

] U=

Hnjjcrcgtnitnga kg ciob snhc wntb rcspcot tk ]w.r.tU “x―

] U=

] U=

] U=

] U=

@y plionga ill i`kvc viluc ng cq-I

] U=

] U=

]

U= {4. ] U= }

] U= Igswcr .

------------Jngnsbch-bcrc-------

----------_ct # 2 ] Jngnsbch-bcrcU ----------------

_CX # >(i) ]_XIPX-BCPCU “NGXCAPIXNKG-@^-XPNAKGKECXPNO- _S@_XNXSXNKG_― 4

NGXCAPIXNKG-@^-XPNAKGKECXPNO-_S@_XNXSXNKG_4-

JKPESLI4-

] U=

Lct

] U= Kr ] U=

--------------jngnsbch---------

Y9) ] U= ;

_klutnkg4-







Viac 33-KJ-99?

] U=

Lct,

]

U=

Hnjjcrcgtnitnga w.r.t “tbcti―.

] U=

] U=

] U=

JKP PIHNOIL ] U4-

] U=

] U=

] U=

] U=

] U=

] U= nt„s tbc gcw cxprcssnkg kj kur Pihnoil .

_k, tbc i`kvc ngtcaril `cokec ng cq-I, .

]

U= {4.]

U=}

] U= {4. }] U=

] U=

] U= Igswcr.

---------------Jngnsbch-----------

_ct # >(i)

Y0) ] U= ;

_klutnkg4-

] U=

Lct,







Viac 3:-KJ-99?

] U=

Hnjjcrcgtnil kg `ktb snhc w.r.t “― .

]

U=

] U=

] U=

JKP PIHNOIL ] U4-

OKG_NHCP PIHNOIL ] U4-

] U=

] U=

{4.] U=}

] U=

] U=

] U=

] U=

] U= Nts kur gcw Pihnoil cxprcssnkg.

@y plionga tbc i`kvc viluc ng cq-I.

] U=

]

U=

] U=

] U=

] U=

] U=







Viac 31-KJ-99?

]

U= Igswcr.

-------------Jngnsbch--------

_ct # >(i)

Y?) ] U= ;

_klutnkg4-

] U=

Lct,

] U=

Hnjjcrcgtnitnkgil kg `ktb snhc w.r.t “― .

] U=] U=] U=]

U=

] U=

] U=

OKG_NHCP Pihnoil 4-

] U=

] U=

] U=

] U=

] U=

] U= nts kur Pihnoil.

----------

] U=

---------







Viac 32-KJ-99?

] U=

] U=

] U=

-----------

] U=

] U=

--------

] U=Igswcr.

-------------Jngnsbch---------

Y8) ] U=;

_klutnkg4-

] U=;

] U=

Lct,

] U=

Hnjjcrcgtnitnga kg ciob snhc w.r.t .

] U=

] U=

] U=

JKP PIHNOIL ] U 4-

] U=

{4. ]

U= }

] U=

] U=

] U=

] U=







Viac 3>-KJ-99?

] U=

] U=

]

U= nts kur rihnoil.

Vlioc ng cq-I,

] U=

] U=

] U=

] U=

]

U=

] U=-------------

] U= ] U=

----------

] U=]

U=Igswcr.

-------------Jngnsbch--------

Y:) ] U= ;

_klutnkg4-

] U=

Lct,

] U=

Hnjjcrcgtnitc wrt “tbcti―

] U=

] U=

] U=

JKP PIHNOIL ] U4-







Viac :6-KJ-99?

] U=

------------

]

U=

-----------

] U=

] U=

] U=

] U=

] U=

]

U= nts kur rihnoil .

JKP x?4-

] U=

] U=

Ximnga ou`c kg `ktb snhc .

] U=

Gkw , plionga ill viluc ng cq-I.

]

U=

] U=

] U=

] U=

] U=

] U=

]

U=

] U=

] U=

Lct,

] U=







Viac :9-KJ-99?

Ssnga hnjjcrcgtnil kg `ktb snhc w.r.t “tbcti―

] U=

]

U=

Vlioc ill i`kvc viluc ng cq-I„ , wc act.

] U=

] U=

] U=

] U=

]

U=

] U=

-------------

] U=

-----------

] U=

----------

] U=

---------

] U=

] U=

] U=

-----------

@coiusc4 57 ]

U=

------------

Xbcrcjkrc

] U=

] U=







Viac :0-KJ-99?

-------

] U=

]

U=

-------------

]

U=

------------

------_neplc Ilac`ri usch bcrc----------

---------

] U= Igswcr.

--------------Jngnsbch---------

_ct # >(i)

Y 0) ]

U= ;

_klutnkg4-

]

U=

]

U=

Lct,

] U=

Ssnga hnjjcrcgtnil kg `ktb snhc w.r.t tbcti.

] U=

] U=

JKP PIHNOIL 4-

]

U=

] U=

] U=

] U=

] U= nts kur Pihnoil viluc .







Viac :?-KJ-99?

-------------

JKP x0 4-

]

U=

] U=

] U=

------------

JKP GCQ-SVVCP LNENX4- ]QBCG x59 U4-

] U=

] U=

] U=

] U= nts gcw uppcr lnent kj kur gcw ngtcaril.

JKP GCQ-LKQCP-LNENX ]QBCG x56 U4-

] U=

] U= ] U=

] U= nts gcw lkwcr lnent kj kur gcw ngtcaril.

_k, tbc i`kvc ngtcaril `cokec.

]

U=

]

U=

]

U=

]

U=

]

U=

]

U=

]

U=

]

U=

]

U=







Viac :8-KJ-99?

]

U=

]

U=

] U=

]

U=

]

U=

]

U= Igswcr.

-------------Jngnsbch---------

--------_ct Gk >(i ) , Jngnsbch-bcrc-----------

_ct # >(`) ]_tirt-bcrcU

JKPESLI4-

] U= Kr ] U=

Lct,

] U=

-------------------

Y9) ] U= ;_klutnkg4-

] U=

] U=

Lct,

] U=

Hnjjcrcgtnitnga kg `ktb snhc w.r.t tbcti.

] U=

] U=

JKP PIHNOIL ] U4-] U=







Viac :3-KJ-99?

] U=

] U=

]

U=

] U=

Xbc i`kvc ngtcaril `cokec„s.

] U=

] U=

] U=

]

U=

] U=

] U=

] U=

Lct,

] U=

Hnjjcrcgtnitnga kg `ktb snhc w.r.t “tbcti―

] U=

] U=

Xbc i`kvc ngtcaril `cokec.

{4. U=

] U=

] U=

] U=

] U=

{4. ] U= }

] U=







Viac ::-KJ-99?

] U=

{4. }

] U=

] U=

……..

……_neplc ilac`ri usch bcrc…….

] U= Igswcr.

---------Jngnsbch--------

_ct # >(`)

Y0) ] U= ;

_klutnkg4-

] U=

Lct,

] U=

Hnjjcrcgtnitnga kg `ktb snhc wrt “Xbcti―

] U=

] U=

JKP PIHNOIL ] U4-

] U=

] U=

] U=

] U=


] U=

] U=

] U=







Viac :1-KJ-99?

] U=

] U=

JKP XBCXI]

U4-

] U=

] U=

_K,

] U=

] U=] U= Igswcr.

------------Jngnsbch--------

Y?) ] U= ;

_klutnkg4-

] U=

] U=

Lct,

] U=

Hnjjcrcgtnitngakg `ktb snhc wrt “tbcti― .

] U=

] U=

] U=

] U=

JKP PIHNOIL ] U4-

] U=] U=] U=


] U=







Viac :2-KJ-99?

] U=

] U=

] U=

] U=

----------------

] U=

] U=

-----------

]

U=

] U=

] U=…….

…._neplc Ilac`ri usch bcrc…..

………

]

U= Igswcr.

----------Jngnsbch---------

Y8) ] U= ;

_klutnkg4-

] U=

] U=

Lct,

] U=

Ssnga hnjjcrcgtnitnga kg `ktb snhc w.r.t “tbcti―.

] U=

] U=







Viac :>-KJ-99?

JKP PIHNOIL ] U4-

] U=] U=

] U=

] U=

] U=

] U= Nts kur viluc Pihnoil.

] U=

]

U=

]

U=

] U=

] U=

]

U=---------

] U=

] U=

---------

]

U= Igswcr.

---------------Jngnsbch-----------

Y3) ] U= ;

_klutnkg4-







Viac 16-KJ-99?

] U=

Lct,

]

U=

Hnjjcrcgtnga …….w.r.t tbcti=

] U=

] U=

JKP PIHNOIL ] U4-

] U=

{4.] U= }

]

U=

] U=

] U=

] U=

Xbc i`kvc cquitnkg `cokec„s.

] U=

] U=

] U=

] U=

] U=

{4. }

] U=

] U=

] U=

] U=

] U=

] U=







Viac 19-KJ-99?

------------------

] U=

]

] {4.

----------------

] U=

] U=

] U=

] U=

] U=

] U=

] U=

] U=

---------

] U=

------

] U=

……

….sneplc ilac`ri usch bcrc….

…..

] U= Igswcr.

-----------Jngnsbch------------

Bkec wkrm jkr sct > (i) 4

Y9 ) ] U= ;

_klutnkg4-







Viac 10-KJ-99?

] U=

……hk nt ykur sclj

------------Jngnsbch--------

---------------_ct # > (`) Jngnsbch-bcrc------------

“ NGXCAPIXNKG-KJ-PIXNKGIL-JPIOXNKG„_ ]Cx # 2.:U { _CX # 96 (i) ]_tirtU }―

Y9 ) Y98 (2.:)) ] U= ;

_klutnkg4-

]

U=

]

U=

]

U=

]

U=

Lct,

] U=

Hnjjcrcgtnil ….wrt “tbcti―.

] U=

] U=

] U=

] U=

JKP PIHNOIL ] U4-

] U=

{4. ] U=}

] U=

] U=

] U=

] U= Nts kur Pihnoil viluc.







Viac 1?-KJ-99?

JKP “x― 4-

] U=

]

U=

] U=

Xbc i`kvc anvcg ngtcaril `cokec4

]

U=

]

U=

]

U=

]

U=

]

U=

]

U=

]

U=

]

U=

]

U=

]

U=

]

U=

{4. ] U= }

]

U=

Kb Hie N hnh enstimc

]

U=

]

U=

]

U=

]

U=







Viac 13-KJ-99?

] U=

] U=

Xbc i`kvc anvcg ngtcaril `cokec.

] U=

] U=

] U=

] U=

] U=

plcisc hk nt ykur sclj gkw nts ykur tcst4

------------------Jngnsbch-------------

Y?) ] U= ;

_klutnkg4-

] U=

Lct,

]

U=


] U=

] U=

JKP PIHNOIL ] U4-

] U=

] U= nts kur gcw cqunvilcgt Pihnoil.

Xbc anvcg ngtcaril `cokec.

] U=

] U=

] U=







Viac 1:-KJ-99?

] U=

] U=

Lct,

] U=

Hnjjcrcgtnitnga wntb rcspcot tk *w.r.t+ “y―.

--------------Hk nt ykur sclj----------

-----------Jngnsbch----------

OB-: ] @I_NO MGKQLCHAC KG HNJJCPCGXNIXNKGU ]_XIPX-BCPCU4-

Jkreulis ] Ob-: U 4

Hnjjcrcgtnitnkg

jkreuli

Jnrst prngonplc jkreuli

jkr hnjjcrcgtnitnkg.

] U=] U=

] U=

--------------------------jngnsbch----------

PSLC JKP HNJJCPCGXNIXNKG_4-

Hnjjcrcgtnitnkg oig `c ipply ng tbcsc jkllkwnga stcps.

i. Nj anvcg ilac`rino jugotnkg ns ] U=

9. Hnjjcrcgtnitnkg kg pkwcr ] U=0. Hnjjcrcgtnitnkg kg rchuoch tbc pkwcr `y engus kgc ] U=?. Hnjjcrcgtnitnkg kg ìsc ] U=

`. Nj anvcg cxpkgcgtnil jugotnkg ns ] U=

9. Hnjjcrcgtnitnkg kg pkwcr] U=

0. Hnjjcrcgtnitnkg kg ìsc ] U=

o. Nj anvcg lka jugotnkg ns ] U=

9. Hnjjcrcgtnitnkg kg lka ] U=

0. Hnjjcrcgtnitnkg kg “ìsc *`x+― ] U=h. Nj anvcg jriotnkgil jugotnkg ns ]

U=

9. Hnjjcrcgtnitnkg kg pkwcr] U=







Viac 11-KJ-99?

0. Hnjjcrcgtnitnkg kg rchuoch tbc pkwcr `y engus kgc] U=

?. Hnjjcrcgtnitnkg kg `isc] U=

c. Nj anvcg jugotnkg ns ]

U=

9. Jkreuli ] U=

j. ]hnjjcrcgtnitnkg ns i lngcir kpcritkrU Nj anvcg jugotnkg ns ] U=

9. Jkreuli ] U=

a. Nj anvcg jugotnkg ns ] U=

9. Nts hcrnvitnvc ns ] U=

------------Jngnsbch-bcrc-----------

JKPESLI KJ HNJJCPCGXNIXNKG ]OB-:U4-

_.Gk. y hy/hx

9 o5igy-

okgstigt

6

0

?

8

3

:

1

2

>

96

99

9







Viac 12-KJ-99?

0

?

8

3

:

1

_.Gk y hy/hx

2

>

96

----------Jngnsbch bcrc---------------

Cx#2.3 , Y) ] U= ;

_klutnkg4-

] U=

Lct,

] U=


] U=

] U=







Viac 1>-KJ-99?

] U=

JKP “ y0 “4-

]

U=

] U=

JKP PIHNOIL PKKX ] U4-

] U=

] U=

] U=

JKP SVVCP-LNENX4-*QBCG “y50― +4-

]

U=

] U= nts uppcr-lnent.

JKP-LKQCP-LNENX4*QBCG “y56― +4-

] U=

] U=nts Lkwcr lnent.

Xbc i`kvc ngtcaril `cokec„s.

] U=

] U=

] U=

] U=

] U=

] U=

………

……….._neplc Mgkwg Ngtcaritnkg Ipplnch bcrc ,igh sneplc Ilac`ri ipplnch bcrc------

………..

] U= Igswcr.

-------------Jngnsbch-bcrc-----------







Viac 26-KJ-99?

--------------_ct # 96 (i) Jngnsbch-bcrc----------------

_CX # 99 (i) ]_tirt-bcrcU

JKPESLI4-

] U=

Lct,

] ----------------------

Y9) ] U= ;

_klutnkg4-

{4. ] U= }

_k, anvcg ngtcaril ns

] U=

] U=

Lct,

] U=

Hnjjcrcgtnitnkg kg `ktb snhc w.r.t “y―

] U=

] U=

JKP PIHNOIL ] U4-] U=] U=] U=

] U=

] U=

] U= Nts kur rihnoil.

{4. }

Xbc i`kvc ngtcaril `cokec .







Viac 29-KJ-99?

] U=

] U=

] U=

Eultnply igh hnvnhc `y ] U =

] U=

] U=

] U=

] U=

Lct,

] Hnjjcrcgtnitnga ……―y―

] ] ] U=

@y plionga tbc i`kvc viluc ng i`kvc cq-@

] U=

] U=

] U=

] U=

Jkr “y― 4-

]

U=

] U=

] U=

JKP “ ―4-

] U=







Viac 20-KJ-99?

] U=

Jkr “ ―4-

]

U=

] U=

] U=

] U=

] U=

…….

…..

] U= Igswcr.

---------------Jngnsbch--------------------------

Y0) ] U= ;

_klutnkg4-

] U=

Lct,

] U=


] U=

] U=

JKP PIHNOIL ] U4-

] U=

] U=

] U=

] U=] U=] U= nts kur rihnoil.







Viac 2?-KJ-99?

Xbc i`kvc ngtcaril ng cq-I `cokecs .

] U=

]

U=

] U=

] U=

] U=

] U=

] U=

----------

] U=

] U=

] U=

-------------------

] U=

]

U= Igswcr.

-----------Jngnsbch---------

Cx # 2.: ]sct # 99(i) U

Y?) ] U= ;

_klutnkg4-

] U=

Lct,

]

Hnjjcrcgtnitnga kg `ktb snhc w.r.t “x― .

]

] Jkr “ y

0 “ 4-







Viac 28-KJ-99?

]

] ]

Jkr Pihnoil ] U4-

] U=

] U=

] U=

] U=

Xbc i`kvc anvcg ngtcaril `cokec„s.

]

U=

] U=

] U=

] U=

] U=

----------

]

] ] ] U=

] U=

] U=

] U=

] U=

]

U=

-------------------------







Viac 23-KJ-99?

] U=

] U=Igswcr.

[-----------------Jngnsbch-bcrc-----------

_ct 99 (i)4

Y1) ] U= ;

_klutnkg4-

] U=

] U=

]

U=

] U=

] U=

Lct,

] U=

Hnjjcrcgtnga wntb rcspcot tk “y―.

] U=

] U=

] U=

Jkr (x+0) 4-

] U=

] U=

Jkr Pihnoil4-

]

U=

] U=

Xbc i`kvc ngtcaril `cokec

] U=







Viac 2:-KJ-99?

] U=

] U=

] U=

] U= Igswcr.

-------------------

] U=

] U=

---------------------------

-------------------Jngnsbch--------------

Cx # 2.: , sct # 99(i)4

Y2) ] U= ;

_klutnkg4-

] U=

] U=

] U=

] U=

Lct,

] U=

Hnjjcr…………….

] U=

] U=

JKP PIHNOIL4-

] U=

] U=

] U=

] U=

] U=







Viac 21-KJ-99?

] U=

Xbc i`kvc ngtcaril `cokecs .

]

U=

] U=

] U=

] U=

---------------------------

Jkr “y―4-

] U=

] U=

-----------------

] U=

] U=Igswcr.

-------------------Jngnsbch----------

Cx # 2.: sct #99(i)

Y> ) ] U= ;

_klutnkg4-

] U=

] U=

] U=

] U=

] U=

] U=

lct,

] ----------------hk nt ̀ y ykur sclj----







Viac 22-KJ-99?

.

.

-------------Jngnsbch----------

_ct # 90 (i) ]stirt-bcrcU ]Ngtcaritnkg @y VirtsU4-

NGXCAPIXNKG-@^-VIPX_4-

JKPESLI4-

] U=

--------------

_ct # 90 (i)

Y9) ]

U= ;

_klutnkg4-

JKPESLI4-

] U=

Gkw,

] U=

Lct,

] U= ] U=

Hnjjcrcgtnga kg `ktb snhc w.r.t “x―.

] U=

] U=

] U=

Ngtcaritnga kg `ktb snhc w.r.t “x―

] U=] U=

] U= ] U=

JKPESLI4-

] U=

]

U=

Ipplnga cq-O ng cq-@ , tbcg cq-@ `cokecs,

_k,

] U=

{=. ] U= }

] U=







Viac 2>-KJ-99?

] U=

] U=

]

U=

] U= Igswcr.

-------------Jngnsbch---------

_ct # 90 (i)

Y0) ] U= ;

_klutnkg 4-

] U=

Lct,

] U= ] U=

Hnjjcrcgtnitnga kg `ktb snhc w.r.t “x―

] U=

] U=

Ngtcaritnga w.r.t “x―

] U=

Lct,

] U=

] U=

] U=

] U=

] ‟ U=

] U=

] U=

]

U=

] U=

] U=

] U= ] U=

] U=

] ‟ U=

] U=

] U=

] U=

] U=







Viac >6-KJ-99?

] U=

] U=

] U=] U=

] U=

OKG_NHCP ] U4-

] U=

Lct,

]

U=

] U=

] U=

] U=

] U=

] U=

]

U=

] U=] U=

Vlionga tbc viluc kj cq-aa ng cq-C wc act .

] U=

] U=

{4. ] U= }

] U=Igswcr.

---------------Jngnsbch -------------

Y?) ] U= ;

_klutnkg4-

] U=







Viac >9-KJ-99?

] U=

] U ] U

Hnjjcrcgtnga w.r.t “x―

]

U

] U

Ngtcaritnga w.r.t “x―

]

U

Lct ,] U=

Hnjjcrcgtnitnga kg `ktb snh…―x―

] U=

] U=

] U=

] U

] U

] U

] U

] U

] U

] U ] U


] U=

] U=

] U=

] U=

OKG_NHCP ] U4-

] U=

Lct,

] U=

Hnjjcrcgtnitnga kg `ktb snhc w.r.t “x―

] U=

] U=

] U=

_k,







Viac >0-KJ-99?

] U=

] U=

] U=

] U=] U=

Gkw `y kenttnga tbc gcaitnvc snag wc hk lnmc tbit , kc Ipplynga lka prkpcrtnty.

] U=] U=] U=Gkw `y plionga tbc viluc kj cq-A ng cq-J,sk, cq-J `cokecs.

_k,

] U=

] U=

] U= Igswcr.

--------------Jngnsbch--------------

_ct # 90(i)

Y8) ] U= ;

_klutnkg4-

] U=

Lct,

] U=

Hnjjcrcgtnitnga kg `ktb snhc w.r.t “x―.

] U=

] U=

] U=

Jkr hx 4-







Viac >?-KJ-99?

] U=

] U=

] U=


] U=

] U=

] U=

Gkw `y usnga ngtcaritnkg `y pirt kg “cq-@― , wc act.

JKPESLI-KJ-NGXCAPIXNKG-@^-VIPX4-

] U=

Gkw , `y ipplynga cq-I ng cq-@,sk cq-@ ,`cokecs.

] U=

Lct,

] U= ] U=

Hnjjcrcgtnitnga kg ciob snhc w.r.t “s―

] U=

] U=

]

U=

Ngtcaritnga kg `ktb snhc w.r.t “s―.

] U=

]

U=

] U=] U=

] U= ] U=

Vut tbc viluc ng cq-@ igh act tbc igswcr.

] U=

]

U=

] U=

] U=

] U=

] U=







Viac >8-KJ-99?

] U=

{4. ] U= }

]

U= Igswcr.

--------------Jngnsbch---------

Y3) ] U= ;

_klutnkg4-

JKPESLI4-

] U=

] U=

] U=

Lct,

] U=

Hnjjcrcgtnitnga kg `ktb snhc wntb rcspcot tk “x―.

] U=

] U=

]

U=

JKP “ x0 “ 4-

] U=


] U=

] U=

] U=

Lct,

Hnjjcrcgtnga kg `ktb snhc w.r.t “u―

Ngtcaritnga kg `ktb snhc w.r.t “u―







Viac >3-KJ-99?

S_NGA JKPESLI4-

] U=

Ipplynga tbc cq-@ ng cq-H,cq-H `cokecs.

] U=

] U=

] U=

]

U=

] U=

{4. }

] U=

{4. ] U= }

] U= Igswcr.

------------jngnsbch--------

Y:) ]

U= ;

_klutnkg4-

] U=

Lct,

Hnjjcrcgtnga kg `ktb snhc

Ngtcaritnga kg `ktb snhc







Viac >:-KJ-99?

Ssnga Jkreuli4-

] U=

Ssnga cq-H ng cq-I,

] U=

] U=

] U=

]

U=

] U=

] U=

] U=

Hk nt littcr…..

--------------------Jngnsbch---------------

Y1) ] U= ;

_klutnkg4-

{4. }

] U=

Lct,

Hnjjcrcgtnga

Ngtcaritnga







Viac >1-KJ-99?

Ssnga Jkreuli4-

] U=

Gkw,

] U=

] U=

] U=

] U=

] U=

] U=

Okgsnhcr 4-

] U=

Lct,

Hnjjcrcgtnitnga

Ngtcaritnga

Lct,







Viac >2-KJ-99?

_k,

] U=

] U=

] U=

] U=

{4.

}

] U=

] U=

] U=

Vlioch cq-H ng cq-@ ,cq-@ `cokecs.

] U=

] U=

] U=

]

U= Igswcr.

-------------------------Jngnsbch-----------------

--------------_ct # 90(i) ]Jngnsbch-bcrcU----------







Viac >>-KJ-99?

_ct # 90 (`) ]_tirt-bcrcU Ngtcaritnkg @y Virts4

Jkreuli4-

] U=

Y9) ] U= ;

_klutnkg4-

] U=

N hnh tbns qucstnkg ng Y3 ng sct # 90(i).

--------------jngnsbch-----------

--------------sct # 90(`) JNGN_BCH-BCPC--------------

_ct # 90 (o) ]_tirt-BcrcU

JKPESLI4-

] U=

---------------------

Y9) ] U= ;

_klutnkg4-

] U=

Lct,

Hnjjcrcgtnil

Ngtcaritnkg







Viac 966-KJ-99?

Ssnga Jkreuli4-

]

Ipplynga cq-H kg i`kvc ngtcaril.

] U=

]

U=

]

U=

] U=

] U=

]

U=

] U= Igswcr.

-----------------Jngnsbch----------

Y0) ] U= ;

_klutnkg4-

] U=

Lct=

Hnjjcrcgtnitnga

Ngtcaritnga







Viac 969-KJ-99?

S_NGA JKPESLI4-

] U=

Ipplynga cq-H jkr i`kvc ngtcaril

] U=

] U=

]

U=

]

U=

]

U=

] U=

]

U=

]

U=Igswcr.

-------------Jngnsbch------------

--------------_ct # 90 (o ) Jngnsbch-bcrc--------------

_ct # 90 (h) ]_tirtU Ngtcaritnkg @y Virts4-

Jkreuli4-

] U=

------------------

Cx # 2.1

Y9 ,Y?(nx) ) ] U= ;

_klutnkg4-

]

U=

Lct,

Hnjjcrcgtnitnga

Ngtcaritnga







Viac 960-KJ-99?

{4. }

Ssnga Ngtcaritnkg @y Virts Jkreuli4-

] U=

Ipplnch cq-H jkr i`kvc ngtcaril sk,

] U=

] U=

] U=

] U=

Okgsnhcr ] U4-

]

U=

Lct,

] U=

Hnjjcrcgtnitnga

] U=

] U=

Jkr x? 4-

] U=

]

U=

Jkr Pihnoil4-

] U=

] U=

@y plionga i`kvc viluc ng cq-J,







Viac 96?-KJ-99?

]

U=

]

U=

] U=

]

U=

]

U=

]

U=

OKG_NHCP 4-

] U=

LCX,

] Hnjjcrcgtnitnga

]

] ] U=

]

U=

] U=]

U={4. ] }] U=Vlioc cq-pt ng cq-A

]

U=] U=

]

U=

]

U=

{4. }







Viac 968-KJ-99?

{4. }

]

U=

] U=

]

U=

]

U=

Gkw plionga tbc i`kvc cq-B ng cq-C, cq-C `cokecs

] U=

] U=

] U= Igswcr.

------------------Jngnsbch-------------

Y0) ] U= ;

_klutnkg4-

] U=

Lct,

Hnjjcrcgtnitnga

Ngtcaritnga

S_NGA JKPESLI KJ NGXCAPIXNKG @^ VIPX_4-

] U=

@y plionga tbc i`kvc ngtcaril ng cq-I„, sk tbc i`kvc ngtcaril `cokecs.

] U=

] U=







Viac 963-KJ-99?

]

U=

]

U=

Okgsnhcr ] U4-

]

U=

Lct.

] U=

Ssnga hnjjcrcgtnitnga kg `ktb snhc w.r.t “y―

] U=

]

U=

]

U=

Jkr x8 4-

] U=

] U=

] U=

Jkr Pihnoil ] U4-

]

U=

] U=

] U= nts kur Pihnoil.

Vlionga tbc i`kvc viluc ng cq-O tbcg cq-O `cokecs.

]

U=

] U=

] U=

] U=

] U=

Okgsnhcr ] U4-

] U=

] U=







Viac 96:-KJ-99?

] U=

] U=

]

U=

] U=

] U=

] U=

] U=

]

U=

] U=

] U=

] U=

] U=

Okgsnhcr 4-

] U=

Lct,

] U=

] U=

] U=

_k,

] U=

] U=

] U=

] U=

] U=

@y plionga tbc viluc kj cq-A, ng cq-J, sk cq-J `cokecs.

] U=







Viac 961-KJ-99?

] U=

] U=

] U=

] U=

] U=

] U=

] U=

]

U=

Gkw `y plionga tbc viluc kj cq-L, ng cq-@,sk cq-@ `cokecs.

]

U=

] U=

] U=

{4. ] U= } {4. ] U= }

] U=

] U= Igswcr.

-----------------Jngnsbch-------------

-----------_ct # 90 (h ) ] Jngnsbch-bcrcU-----------

_ct # 9? (i) ]_tirt-bcrcU “IPCI-SGHCP-XBC-OSPZC― 4-

“Xbc eidkr okgocpt kj ngtcaril (Igtn-Hcrnvitnvc) kj i jugotnkg “j(x)― wis hcvclkpch “Ng krhcr tk jngh i ocrting Irci―=

CRIEVLC-KJ-IPCI-SGHCP-XBC-OSPZC4-

Hniarie ]Jna # 0 U4-







Viac 962-KJ-99?

Nj “i― igh “`― irc twk “rcil-gue`cr― i,` } sk, ( )= wc hcjngc tbc quigtnty=

{4. }

Ilsk

] U=

] U=

] U=

{4. }

Cx # 2.1 , IPCI-SGHCP ‟XBC-OSPZC4-

Okgsnhcr tbc Aripb *`clkw+ ng wbnob tbc bnablnabtch irci “I― wbnob lncs, ughcr tbc pksntnvc x-ixns `ctwccg tbc krhngitcs ]x5iU igh ]x5`U .

] U=

Aripb ]Jna # ? U 4-







Viac 96>-KJ-99?

] U=

] U=

] U= nts ig Irci ughcr tbc anvcg ourvc i`kvc tbc pksntnvc x-ixns igh `ctwccg tbc

krhngitcs ]x5iU igh ]x5`U .

---------------Jngnsbch---------

Cxieplc # 94-

Jngh irci ]IU i`kvc tbc pksntnvc x-ixns ughcr tbc ourvc ] U wbnob ns i oup-up piri`kli `ctwccg tbc krhngitcs ]x59U igh ]x5?U ;

_klutnkg4-

Jna # 8







Viac 996-KJ-99?

S_NGA JKPESLI KJ IPCI SGHCP OSPZC4-

] U=

Jkr Irci4-

] U=

] U=

]

U=

]

U=

] U= Igswcr.] nts rcqunrch Irci.U

------------------Jngnsbch----------

Cxieplc # 0 4- Jngh irci i`kvc x-ixns ughcr tbc anvcg onrolc wbksc ocgtcr it krnang ] U `ctwccg krhngitcs it ] U= ;







Viac 990-KJ-99?

] U=

Ipplynga cq-@ jkr cq‘O, sk cq-O `cokecs.

]

U=

] U=

] U=

Ssnga Jkreuli4-

] U=

Ipplynga cq-H jkr i`kvc ngtcaril sk, tbc i`kvc ngtcaril `cokecs.

]

U=

] U=

] U=

……

] U= Igswcr.

----------------Jngnsbch------------

Y9 ) Jngh Xbc irci i`kvc tbc x-ixns ughcr tbc ourvc, `ctwccg tbc anvcg krhngitcs ;

Ourvc 4 piri`kli57 ] U, Nt„s tbc cquitnkg kj ourvc wbksc vcrtcx it krnakg igh nts ixns ilkga tbc pksntnvc x-ixns.

Krhngitcs 57 ] U=

Jna # :







Viac 99?-KJ-99?

Ssnga Jkreuli4-

] U=

Ipplynga tbc cq-H jkr i`kvc oup-rnabt Viri`kli ] U,

Jkr y4-

] U=

] U=

Qc timc Vksntnvc ]kr uppcr bilj kj anvcg piri`kli ng jna # :U.

Xbcrcjkrc,

] U=

Ipplynga cq-H kg cq-n wc act,

] U=

] U= Igswcr , nt ns tbc rcqunrch irci .







Viac 998-KJ-99?

---------------------Jngnsbch-bcrc-------------

-----------------_ct 9? (i) Jngnsbch-bcrc-----------------

Ob-: ]HnjjcrcgtnilU ]Jnrst-Vrngonplc U 4-Y9 ) ] U= ;

Ssnga Jnrst Vrngonplc

] U=

] U=

{4. }

] U=

] U=

Ipplynga lnent.

] U=

] U=

] U= Igswcr.

-----------Jngnsbch----------

Y9) ] U= ;

Ssnga Jnrst Vrngonplc4-

] U= Igswcr.

-------Jngnsbch------

------------Ob-: Jngnsbch-bcrc----------

------------Ob-2 + Ob-: Jngnsbch-Bcrc--------

REKBIEEIH-_NMIGHIP-MBIG-LKHBN

KQGCP-KJ-E^-VCP_KGIL-GKXC_Qchgcshiy, Duly 96, 069?







Viac 993-KJ-99?







Viac 991-KJ-99?







Viac 992-KJ-99?







Viac 99>-KJ-99?







Viac 906-KJ-99?







Viac 909-KJ-99?







Viac 900-KJ-99?







Viac 90?-KJ-99?







Viac 908-KJ-99?

Ch-8, Integrations or Anti-Derivatives [ ch-8 ] + differentiations [ ch-6 ].

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Transcript of Ch-8, Integrations or Anti-Derivatives [ ch-8 ] + differentiations [ ch-6 ].