b .5apule2y

Se a < b ⇒ / fcxidx .oe

ee

Se a=b ⇒ ) fkiok = 0a

Se as b ⇒ fabfcxidx = - fhfaddx

fabfkidx ta,belk

Ifq b) = interim di estonia ebPROPRIETA ' ( seuneohiis

. )1) Saw f , g : Iconb ) R ieutegcabnh

.

so I ( ab ) .I

Allow 42, B e R srme 2ft pg : I Cqb ) R

uitegaebile so Icqb ) e si he

±€µ,

kfcxstfgsxs )d× = ¥,pwdx+ PfgcxkdxIcqb )

Cpcopcieta di lnieooatoi )2) ( monotone ) Sano f , g : End - AIR integrals . etali che ftx eggs txe[qb] . Allot

§ Lksdx ± { gcxsdx .Tu partisan se fCx)=o txeEeuD , alloy } gcxidx 20 .

2

3) ( disaguoyoanhefmdarueutale )

Son f : Eu b) R integrator n Egd . Allow If 16 anooae uitegcabrle W Ea ,b] e

/ } fcxidx / ± } lfanldxSe f : Ica , b) 112 allow lfl e- one coca integrated so

Icoub ) e lee ohsuguageauhefoudewoutale ohoente

¥ * *lttkbpexndxl3

4) ( adokkvila aspeth all'

iutowallo so Cai integrand

I c- R interval e srauo a ,b , a EI ed f : I 112

ntegaewee so I . Allow

{ bfandr. -fdfcndx t{faad*SIGNIFICATO GEOMETRKO

B Se fcxszo GXEEUD =D { fcxidx 20

If# . S sottogeaficodifAreas = {faaolx1 to > ,on

4

2) Se fly EO =D - fcx ) 20 ⇒ £h fcxidx 20D

.fu ⇒tfjlb• < {f*' dx 20 (per euearila )#j: #s fabftxldxeoAreas ' = Areas per costume

Areas' =€ . fcndx = - € fxidx = Areas

3) 5 = SZUSZSs # ' Areas =Auaszt Arose#¥¥ ={ fxidx - {bfkldx52

J

TEOREMA DELA MEDIA INTEGRALE

Sia f : Fa ,b] R iutegubrle au Ea ,b] . Allow

nif f E ¥ } flxldx < aupf .laid Lend- = M

media uitegcole difso Eub ]

Se f ni agguinte ie continue so Eeub] allow Fxoe Eidtale che me = fcxo ) .

din :ttxefoub] a he sniff < fcxs egg¥ =p ( pecMomotmi )

Loud

{bEa¥§ &×s{E*d×±{ sage o.es ⇒6

[wgjfgf. ( b- a) c- {bfxldx I[§f .G- a ) as

Egf ± ¥ fitandxeeaggf .

Se f Econtinue

,peril teorewadiWeierstrass si wa che ewstovo

£bdgf . dudxf =D peril teoaeuedcinlociuetoomedi/ LeadF xoe Eeub] t.ci fko ) = m esseudo nee fuuuf ,woof ]

BUD Could ↳

:-

KPROBLGTA DELLA PA MITIVA : sia JER un iutecrallo e sre

f : J R . Determined ,se ewste , and Lemmon decwebrle

di ai f e los dourota .

DEI : srdeaohe F : J -3113 e- awe pcimutwre per f se

F'

k ) = fcx ) the End .proposition : F

,G : F - > IR pcimuhoedif =D F- G Eastone

deter : §, ( FCA - Gas ) = Fkxl- G ' G) = fcx ) - fix ) = 0 the Eeab]

=D FG ) - Gtx ) = C,CER

. Ee

FQ ) = GQ ) tc tcelr ⇒ le piwntwe,

seewstaeo.

soo

infinite .8

ESEMPW ( feeuioue che won ouamette and pzivntwo )Caesrdeciauuo lee fiuioume oegwo :

1 se x so

Fa - sgmx = { Issexgzg f : R £ li 1 }Supp per oessecdo che 2 F Wcinutwadif so R .In pacticoeare

¥6) = Leggio F 't ) =×Q,ui+fs= 1

xso-

'" * ' ' ' fsegifa, = . <

} ⇒ 'HOH Eco )Fl ( o ) = lain ASSURDO

=D fan ÷ sgux won eeumette one pcivnture so tahr R .

&

FED fY× ) fcx ) Fcx )

K,KER 0 0 K

,KER

× 1 t X

M . 1×?neH ux xn NI netNTI

logx In§ loglxl

ex exex ex

seux eosx

semi - eoox

coax - senxcost senx

£#÷= I¥=×n ; adobe.ua#oEtFtI?*c.n..g a

DEF : She f : Eub ] R uitegeobrle so Eub ] . 8 defines a

Leeuoeome integral oh . f ( respeto ol panto a) lee feunome

Fakl = FG ) := / fktdt ttxe Eu D .Proposing . . FCA hai roppoati incremental funk (edicouxgueuveFG ? sang continue ni Ea D ) .

ehien X .xoeEb] :

÷¥⇐ , {e⇒dt - { Eat flyetsdtitatksdexxo

= - = =Ao Xtxo

= £¥4# passado al modulo who#

11

15,58kt out 15,5 lets out1¥ITI= - ± - ±K . xol ( txol[§H Hahdxl sopf.kxI< - = # txyoe ED1×+61 ¥01 UE

TEORETLA FONDA MENTHE DEL CALCOLO INTEGRA LE

Sra f una fuuowme intgcable so feub ] e meppmraeuo che

f she continue in xoe feud.

Allow si he do

FCXI ={×fHdt E decioebneenixo esr he Fkxo ) =fKo ) .

Se f e continue with Eeub ],

allow Fed =€fk)dt e-continued con derivate prime continue in tatro Eaid esr he

F ' CA = fcx ) the Eoub ] . 12

did : dale

ipotesiolicoutimntedifwixoseguedrelfgixfK7-fkda-bV@Eo.toleckeVtEEiDioult.xo

/ < of risotto foot E ⇐ f# e- fcxdt E

integrand (INDI{ Cfao) -e) out e § fktdt e §forte ) out( fed -e) ( x* ) < £tCHdt < ( fed te ) . € xo )§ fktdt = fakefki - E E --

- ,×Ii.tt#F@akbaueorealiroatofufE,tEt6=f*

. ) # Fkxo ) =f(xD:FEB

FORMULA FONDAMENTALE DEL CALCDWINTEGRALE

Se f : Eeub] ( K e continued nituth [oub] , allow/

bb[ fkidx = FC b) - Ffa ) = [7×7] oppue FG ) /be

on

dove F E

uvapcinmtwredif.

old : deal teamprecedent

If auunutte minute e lluisreeue

delle piwukke : FK)=€fk) .dtt C , CERb

Se x = b =D Fcb ) = Sfktdttc.

be

Sex = a =D Faa ) = £f#tc = Otc ⇒ C= Fca )⇒ FED = Sba fat t Foal .

14

fDEF : llieweeme oh

. tattle premature di and funnomef pause'd move oh ' uitegcole indefinite dif , in sumboli

S fax .

Se f E continue in Eu b) =D F FG ) pcimutwed . fcx )

⇒ Sfxidx = Fast c,

⇐ IR.

ESEMPI.

D ) sent at = - eosx + c , celr

2) ) ×2 ok = xI÷ + c = §+c , cells

fbxzdx = §3§ = § - o = ¥ = aggregate, ,