
vaalisinte integralukal

ganithashaasthrathilum vishakalanathilum, kooduthal krithyamaayi paranjaal, joonn valis avatharippicha integralukalude oru kudumbamaanu valis integralukal.
nirvachanam, adisthaana gunangal.
valis integralukal" ennathu _ 1 enna krama soothravaakyathinte padangalaanu
allengil thulyamaayi,
eee shreniyude aadya kurachu padangal ivayaanu
sequence formula _ 4 kurayukayum positteev padangal undaakukayum cheiyunnu. vaasthavathil, ellah soothravaakyangalkkum _ 5
sequence formula _ 4 kurayukayum 0 kondu parimithappedutthukayum cheiyunnathinaal, athu oru noonn-negatteevu paridhiyilekku othucherunnu. theerchayaayum, paridhi poojyamaanu (chuvade kaanuka).
aavarthana bandham.
bhagangal cherthu samyojippikkunnathiloode oru reduction formula labikkum. aidantitti formula _ 9 upayogichu, namukk ellah formulakalkkum _ 10 undu,
randaamathe avibhaajyaghadakathe bhagangalaal samyojippikkuka, awah
namukk undu
eee phalam samavaakyam (1) aayi maattisthaapikkunnath nalkunnu
angane
ellah formulakalkkum _ 19
formula _ 21 nte adisthaanathil formula _ 20 nalkunna oru aavarthana bandhamaanith. ithu, formula _ 22, formula _ 23 ennivayude moolyangalumaayi chernnu formula _ 25 vichithramaano erattayaano ennathine aashrayich sequence formula _ 4 leeii padangalkku rund settu formulakal nalkunnu
vaalisinte integralukal vilayiruthunathinulla mattoru bandham.
euler integralukal upayogichu vaalisinte integralukal vilayiruthaam
beetta functionullil inipparayunna pakarakkaaran nammal cheiyukayaanenkil
soothravaakyam _ 30 <bii. aarr>
namukk labhikkunnatha
athinaal valis integralukal vilayiruthunnathin ithu inipparayunna bandham nalkunnu
athinaal, otta formula _ 25, formula _ 34 ezhuthunnathinu, namukk undu
athesamayam, formula _ 25-nuu, formula _ 37 ezhuthukayum aaa formula _ 38 ariyukayum cheythaal namukk labikkum
stirilinginte soothravaakyam nirnnayikkunnu.
namukk inipparayunna thulyathayundennu karuthuka (stirilinginte formula ennariyappedunnu):
nammal nirnnayikkaan aagrahikkunna chila sthiramaaya formula _ 54-nuu. mukalil ninnu, njangalkk
formula _ 56 vikasippikkukayum faaktoriyalukalkkaayi mukalilulla formula upayogikkukayum cheyyumbol namukk labikkum
(3), (4) ennivayil ninnu, naam traansittivitti vazhi nedunnu
formula _ 54-nuu pariharam nalkunnathu mattoru vidhathil paranjaal formula _ 60 nalkunnu,
eratta factorial anupatham kanakkakkunnu.
athupole, mukalil ninnu, namukk undu
formula _ 56 vikasippikkukayum eratta faaktoriyalukalkkaayi mukalilulla formula upayogikkukayum cheythaal namukk labikkum
lalithamaakkumbol namukk labhikkunnatha
allengil
gausiyan integral vilayiruthunnu.
vaalisinte integralukal upayogichu gausiyan integral vilayiruthaam.
thaazhepparayunna asamathvangal naam aadyam theliyikkunnu
vaasthavathil, formula _ 69 anuvadikkuka,
aadyathe asamathvam (athil soothravaakyam _ 70)
soothravaakyam _ 71 nuu thulyamaanu;
athesamayam randaamathe asamathvam kurayunnu
soothravaakyam _ 72,
athu formula _ 73 aayi maarunnu.
eee 2 avasaanathe asamathvangal eee asamathvathinte convexityil ninnaan pinthudarunnathu.
exponential function
(allengil function formula _ 74-nte vishakalanathil ninnu).
soothravaakyam _ 75 anuvadikkuka
anuchithamaya integralukalude adisthaana gunangal upayogappeduthuka
(integralukalude samyojanam vyakthamaan),
namukk asamathvangal labhikkunnu
soothravaakyam _ 76
sandwich sidhaanthathinoppam upayogikkunnathinaayi (soothravaakyam _ 77 aayi).
aadyathethum avasaanathethumaaya integralukal eluppathil vilayiruthaan kazhiyum
valis 'integralukal.
aadyathethinu, soothravaakyam _ 78 aakatte.
(t ennathu 0 muthal formula 79 vare vyathyaasappedunnu).
appol, integral formula _ 80 aayi maarunnu.
avasaanathe integralinu, soothravaakyam _ 81 aakatte.
(soothravaakyam _ 82 muthal soothravaakyam _ 79 vare vyathyaasappedunnu).
appol, athu formula _ 84 aayi maarunnu.
nammal neratthe kaanichathupole,
soothravaakyam _ 85. athinaal, athu pinthudarunnu
formula _ 86.
abhiprayam gausiyan integral vilayiruthunnathin mattu reethikalundu.
avayil chilath kooduthal nerittullathaanu.
kuripp.
athe gunangal valis uthppannathilekku nayikkunnu,
ithu soothravaakyam _ 87 prakadippikkunnu
ananthamaaya oru uthppannathinte roopathil.

