definite integration theory_e

8/12/2019 Definite Integration Theory_e

1/21

INTJ[

"inèsjauinrpjyseks.e`" >

@bwth` Cbeoètz ghriucn

Cbt dx

d(G(x)) 9 g(x)x(n, o). Tjb`

o

n

dx)x(g 9 oxcei G(x) nx

cei G(x).

@htb < >. Eg n 0 o, tjb` o

n

dx)x(g 9 oxcei G(x) nx

cei G(x).

?. Eg G(x) es kh`teùhus nt n n`d o, tjb` o

n

dx)x(g 9 G(o) G(n)

Bxnipcb # > Bvncuntb ?

>)?x)(>x(

dx

[hcuteh` )?x)(>x(

>

9 >x>

?x

>

(oy pnrtenc grnkteh`s)

?

>)?x)(>x(

dx9 ?>)?x(`)>x(

9 `; `4 `? + `; 9

:5`

[bcg prnktekb prhocbis ) ?

>

?

?

;x4x

x2dx

(?)

4

6

;? ?xxsbk? dx

(;)

;

6xsbk>

xdx

N`swbrs < (>) 2 ?

2

?

;`

4

2`5 (?)

>6?4

4+

?

+ ?

Dbgeètb e`tbmrncs


2/21

INTJ[

"in`esjauinrpjyseks.e`" ?

(;)>:

?

;;

+ ? `

;

?

Vrhpbrty (>) o

n

)x(g dx 9 o

n

)t(g dt

e.b. dbge etb e tbmrnc es e`dbpb`db t hg vnrenocb hg e tbmrnteh`.

Vrhpbrty (?) o

n

)x(g dx 9 n

o

)x(g dx

Vrhpbrty (;) o

n

)x(g dx 9 k

n

)x(g dx + o

k

)x(g dx, wjbrb k iny ceb e`sedb hr hutsedb tjb e tbrvnc Qn, oY.

Bxnipcb # ? Eg g(x) 9

;xx;

;xx;( dx 9

;

?

?

x;?

x

+ 2;; xx

9?

45+ ; (; ?) + 2; ;; + 2 ; 9

?

?>>

Bxnipcb # ; Bvncuntb :

?

|2x| dx.

[hcuteh` :

?

|2x| dx 9 2

?

)2x( dx + :

2

)2x( dx 9 5

Bxnipcb # 4 [jhw tjnt ?

6)>x?( dx 9

2

6

)>x?( +

?

2

)>x?(

[hcuteh` C.J.[. 9 x? + x Y?6

9 4 + ? 9 7

_.J.[. 9 ?2 + 2 6 + (4 + ?) (?2 + 2) 9 7

C.J.[. 9 _.J.[

[bcg prnktekb prhocbis ;

Vrhpbrty (4)

n

n

)x(g dx 9 n

6

))x(g)x(g( dx 9

hddes)x(g.b.e)x(g)x(geg,6

bvb`es)x(g.b.e)x(g)x(geg,dxxg?

n

6

Bxnipcb # 2 Bvncuntb

>

>

x

xx

b>

bbdx

[hcuteh`

>

>

x

xx

b>

bbdx 9

>

6

x

xx

x

xx

b>

bb

b>

bbdx

9

>

6

x

xxx

x

xx

>b

)bb(b

b>

bbdx 9

>

6

xx )bb( dx 9 b > +>

)>b( >

9b

>b?


?

?

xkhs dx.

[hcuteh`

?

?

xkhs dx 9 ?

?

6

xkhs dx 9 ? (khs x es bvb` gu`kteh`)

Bxnipcb # = Bvncuntb

>

>

bx?

x?chm dx.

[hcuteh` Cbt g(x) 9 chmb

x?

x?

g(x) 9 chmb

x?

x?9 chm

b

x?

x?9 g(x)

e.b. g(x) es hdd gu`kteh`

>

>

bx?

x?chm dx 9 6

[bcg prnktekb prhocbis

>

|x| dx.

(:)

?

?

= xse` dx.

(5)

?

?

xb>

xkhsdx.

N`swbrs < (=) > (:) 6 (5) >

Vrhpbrty (2) o

n

)x(g dx 9 o

n

)xon(g dx

Gurtjbr n

6

)x(g dx 9 n

6

)xn(g dx

Bxnipcb # : Vrhvb tjnt

?

6)x(khsm)x(se`m

)x(se`mdx 9

?

6)x(khsm)x(se`m

)x(khsmdx 9

4

.

[hcuteh` Cbt 9

?

6 )x(khsm)x(se`m

)x(se`mdx

9

?

6 x?

khsmx?

se`m

x?

se`m

9

?

6 )x(se`m)x(khsm

)x(khsmdx

h` ndde m, wb hotne`

? 9

?

6)x(se`m)x(khsm

)x(khsm

)x(khsm)x(se`m

)x(se`mdx 9

?

6

dx 94

[bcg prnktekb prhocbis6)

6

xse`>

xdx.


5/21

INTJ[

"in`esjauinrpjyseks.e`" 2

(>>)

?

6xkhsxse`

xdx.

(>?)

?

6

44 xkhsxse`

xkhsxse`xdx.

(>;)

;

7

xtn`>

dx

N`swbrs < (>6) (>>)??

chm

b ?> (>?) >7?

(>;)>?

Vrhpbrty (7) n?

6

)x(g dx 9 n

6

))xn?(g)x(g( dx 9

)x(g)xn?(geg,6

)x(g)xn?(geg,dxxg?

n

6


6

;; xkhsxse` dx.

[hcuteh` Cbt g(x) 9 se`;x khs;x g( x) 9 g(x)

6

;; xkhsxse` dx 9 6

Bxnipcb # >6 Bvncuntb

6

? xse`?>

dxdx.

[hcuteh` Cbt g(x) 9xse`?>

>?

g( x) 9 g(x)

6?

xse`?>

dx9 ?

?

6

?

xse`?>

dx9 ?

?

6

??

?

xtn`?xtn`>

dxxsbk

9 ?

?

6

?

?

xtn`;>

dxxsbk9

;

?

?6> xtn`;tn`

tn`?

es u`dbge`bd, wb tnab ceiet

9;

?

6tn`;tn`xtn`;tn`Ct >>

?x


6/21

INTJ[


9;

?

?

9

;

Nctbr`ntevbcy <

6

? xse`?>

dx9

6

?

?

?xbkkhs

xbkkhsdx 9

6

?

?

;xkht

dxxbkkhs

Hosbrvb tjnt wb nrb `ht kh`vbrte`m e` tbris hg tn` x ns et es `ht kh`te`uhus e` (6, )

9 ;

>

6

>

;xkhttn` 9

;>

;xkhttn`Ct

;xkhttn`Ct >

6x

>

x

9 ;

>

?? 9

;

Bxnipcb # >> Vrhvb tjnt

?

6

xse`` dx 9

?

6

xkhs` dx 9

?

6

)x?(se`` dx 9 ?

` ? .

[hcuteh` Cbt 9

?

6

xse`` dx ..........(e)

9

?

6

x?

se`` dx (oy prhpbrty V 2)

9

?

6

)x(khs` dx ..........(ee)

Ndde`m (e) n`d (ee)

? 9 ?

6

)xkhs.x(se`` dx 9

?

6?

x?se`` dx

? 9

?

6

)x?(se`` dx

?

6

?` dx

? 9 >

?

` ? ..........(eee)

wjbrb 9

?

6

)x?(se`` dx

put ?x 9 t dx 9?

>dt

C . C < x 9 6 t 9 6

R . C < x 9?

t 9

>

9

6

)t(se`` ?

>dt

9?

> ?

?

6

)t(se`` dt (oy use`m prhpbrty V 7)


7/21

INTJ[

"in`esjauinrpjyseks.e`" =

>

9 (eee) mevbs 9 ?

` ?

[bcg prnktekb prhocbis 4)

6

?x>

x

>

x`dx.

(>2) >

6

>

x

xse`dx.

(>7)

6

xse``x dx.

N`swbrs < (>4) ?` (>2)? ?` (>7)

?

?

?`

Vrhpbrty (=) Eg g(x) es n pbrehdek gu`kteh` wetj pbrehd T, tjb`

(e) `T

6

)x(g dx 9 T

6

)x(g dx, ` z

(ee) `Tn

n

)x(g dx 9 ` T

6

)x(g dx, ` z, n _

(eee) `T

iT

)x(g dx 9 (` i) T

6

)x(g dx, i, ` z

(ev) `Tn

`T

)x(g dx 9 n

6

)x(g dx, ` z, n _

(v)

`To

`Tn

)x(g dx 9 o

n

)x(g dx, z, n, o _

Bxnipcb # >? Bvncuntb

?

>

}x{b dx.

[hcuteh`

?

>

}x{b dx 9

;>

>

}x{b dx 9 ; >

6

}x{b dx 9 ; >

6

}x{b dx 9 ;(b >)

Bxnipcb # >; Bvncuntb v`

6

|xkhs| dx ,?

3 v 3n`d ` z.


8/21

INTJ[

"in`esjauinrpjyseks.e`" :

[hcuteh` v`

6

|xkhs| dx 9 v

6

|xkhs| dx + v`

v

|xkhs| dx

9

?

6

xkhs

v

?/

xkhs dx + `

6

|xkhs| dx

9 (> 6) (se` v >) + ?`

?

6

xkhs dx

9 ? se` v + ?` (> 6) 9 ?` + ? se` v

[bcg prnktekb prhocbis =)

?

>

}x;{b dx.

(>:)

?666

6

xse`b>

dxdx.

(>5)

4

2

44 xkhsxse`

x?se`dx.

N`swbrs < (>=) ; (b >) (>:) >666 (>5)4

Vrhpbrty (:) Eg(x)g(x) (x) ghr nxo, tjb`

o

n

)x( dx o

n

)x(g dx o

n

)x( dx

Vrhpbrty (5) Eg ig(x)I ghr nxo, tjb` i (o n) o

n

)x(g dxI (o n)

Gurtjbr eg g(x) es ih`hth eknccydbkrbnse m e` (n, o), tjb` g(o) (o n) 3

o

n

)x(g dx 3 g(n) (o n) n`d eg g(x)

es ih`hth`eknccy e`krbnse m e` (n, o), tjb` g(n) (o n) 3 o

n

)x(g dx 3 g(o) (o n)

Vrhpbrty (>6) o

n

dx)x(g o

n

dx)x(g


9/21

INTJ[


Vrhpbrty (>>) Eg g(x)6 h` Qn, oY, tjb` o

n

dx)x(g 6

Bxnipcb # >4 Ghr x (6, >) nrrn`mb g>(x) 9

?x4

>

, g

?(x) 9

?x?4

>

n`d g

;(x) 9

;? xx4

>

e` nskb`de`m

hrdbr n`d jb`kb prhvb tjnt7

3

>

6;? xx4

dx3

?4

.

[hcuteh` 6 3 x; 3 x? x? 3 x? + x; 3 ?x? ?x? 3 x? x; 3 x?

4 ?x? 3 4 x? x; 3 4 x?

?x?4 3 ;? xx4 3 ?x4 g

>(x) 3 g

;(x) 3 g

?(x) ghr x(6, >)

>

6

> )x(g dx 3 >

6

; )x(g dx 3 >

6

? )x(g dx

se`>>

6?

x

3

>

6;? xx4

dx3

?

>se`>

>

6?

x

7

3

>

6;? xx4

dx3

?4

Bxnipcb # >2 Bsteintb tjb vncub hg

?

6x

xse`dx.

[hcuteh` Cbt g(x) 9x

xse`

g(x) 9 ?x

xse`xkhsx 9 ?x

)xtn`x)(x(khs 3 6

g(x) es ih`hth eknccy dbkrbnse`m gu`kteh`.g(6) es ht dbge bd, sh wb bvncuntb

6xCt g(x) 9 6x

Ctx

xse`9 >. Tnab g(6) 9 6x

Ct g(x) 9 >

g

?

9

?

?

.

6

? 3

?

6x

xse`dx 3 > .

6

?

> 3

?

6x

xse`dx 3

?

Bxnipcb # >7 Bsteintb tjb vncub hg

>

6

x

dxb

?

use`m (e) rbktn`mcb, (ee) tren`mcb.

[hcuteh` (e) Oy use`m rbktn`mcb


10/21

INTJ[

"in`esjauinrpjyseks.e`" >6

Nrbn HNBD 3 >

6

x dxb?

3 Nrbn HNOK

> 3 >

6

x dxb?

3 > . b

> 3 >

6

x dxb?

3 b

(ee) Oy use`m tren`mcb

Nrbn HNBD 3 >

6

x dxb?

3 Nrbn HNBD + Nrbn hg tren`mcb DBO

> 3>

6

x dxb?

3 > +?

>. >. (b >) > 3

>

6

x dxb?

3?

>b

Bxnipcb # >= Bsteintb tjb vncub hg >

6

x dxb?

oy use`m >

6

xdxb .

[hcuteh` Ghr x (6, >), ?xb 3 bx

> > 3 >

6

x dxb?

3 >

6

xdxb

> 3 >

6

x dxb?

3 b >

[bcg prnktekb prhocbis

6

?x xkhsb dx 3 >

6

?x xkhsb?

dx

(?>) Vrhvb tjb ghcchwe`m < 6 3

?

6

>` xse` dx 3

?

6

? xse` dx , ` 0 >

(??) Vrhvb tjb ghcchwe m < 4>

b

3

>

6

xx?b dx 3 >


11/21

INTJ[

"in`esjauinrpjyseks.e`" >>

(?;) Vrhvb tjb ghcchwe`m < ?

>

>

6

?

;

x?

xkhsxdx 3

?

>

(?4) Vrhvb tjb ghcchwe`m < > 3

?

6

xse` dx 3?

(?2) Vrhvb tjb ghcchwe`m < 6 3 ?

6

;x>7

dxx3

7

>

Cbeo`etz Tjbhrbi < Eg G(x) 9 )x(j

)x(m

dt)t(g , tjb`

dx

)x(dG

9 j(x) g(j(x)) m(x) g(m(x))

Vrhhg < Cbt V(t) 9 dt)t(g

G(x) 9 )x(j

)x(m

dt)t(g 9 V(j(x)) V(m(x))

dx

)x(dG9 V(j(x)) j(x) V(m(x)) m(x)

9 g(j(x)) j( x ) g (m(x)) m(x)

Bxnipcb # >: Eg G(x) 9 dttse`

?x

x

, tjb` ge`d G(x).

[hcuteh` G(x) 9 ?x . ?xse` > . xse`

Bxnipcb # >5 E g G(x) 9 dttchm

tx;

x?

b

b b

, tjb` ge`d gerst n`d sbkh`d dbrevntevb hg G(x) wetj rbspbkt th ` x

nt x 9 ` ?.

[hcuteh` x`d)x(dG

9

dx

)x(dG

)x`(d

dx

9

x?

x?x?

x;

x;x;

b`

bb.?

b`

b.b.;

x 9 b7x b4x.

?

?

)x`(d

)x(Gd

9

)x`(d

d

(b7x b4x) 9

dx

d(b7x b4x)

dx/`xd

>

9 (7 b7x 4 b4x) x

Gerst dbrevntevb hg G(x) nt x 9 ` ? (e.b. bx 9 ?) es ?7 ?4 9 4:

[bkh`d dbrevntevb hg G(x) nt x 9 ` ? (e.b. bx 9 ?) es (7 . ?7 4 . ?4) . ` ? 9 2 . ?7 . ` ?.


12/21

INTJ[

"in`esjauinrpjyseks.e`" >?

Bxnipcb # ?6 Bvncuntb xCt

x

6

t?

?x

6

t

dtb

dtb

?

?

.

[hcuteh` xCt

x

6

t?

?x

6

t

dtb

dtb

?

?

ghri

Nppcye`m CJhspetnc rucb

9 xCt

?

??

x?

x

6

xt

b.>

b.dtb.?

9 xCt

?

?

x

x

6

t

b

dtb.? 9 x

Ct?

?

x

x

b.x?

b.?9 6

Bxnipcb # ?> Eg g(x) 9 x

chmxbtx

dt

, tjb` ge`d g(x).

[hcuteh` g(x) 9

x

x`

? dt

tx

>

+ > .x?

>

x

>

x`x>

9

x

x`)tx(

>

+ x?>

x`xx>

9x?

>

x`x

>

+ x?>

x`xx>

9 x>

x`xx>x

9 x`xx>x`

Nctbr`ntevbcy < g(x) 9

x

x`

x

x`

)tx(`tx

dt

(trbnte`m t ns kh`stn`t)

g(x) 9 ` ?x ` (x + ` x)

g(x) 9x

> x`x

>

x

>> 9 x`xx

>x`

Bxnipcb # ?? Bvncuntb

>

6

o

`x

>x

, o obe m pnrnibtbr..

[hcuteh` Cbt(o) 9

>

6

o

`x

>x

dx

do

)o(d9

>

6

o

`x

`xx

dx + 6 6


13/21

INTJ[

"in`esjauinrpjyseks.e`" >;

(use m ihdegebd Cbeo etz Tjbhrbi)

9 >

6

o dxx 9

>

6

>o

>o

x

9>o

>

(o) 9 ` (o + >) + ko 9 6 (6) 9 6

k 9 6 (o) 9 ` (o+>)

Bxnipcb # ?; Bvncuntb

>

6?

>

x>x

)nx(tn`dx , n obe`m pnrnibtbr..

[hcuteh` Cbt(n) 9

>

6?

>

x>x

)nx(tn`dx

dn

)n(d9

>

6

?? )xn>(

x

?x>x

>

dx 9

>

6??? x>)xn>(

dx

Vut x 9 se` t dx 9 khs t dtC.C. < x 9 6 t 9 6

R.C. < x 9 > t 9?

dn

)n(d9

?

6

?? tse`n>

>

tkhs

>khs t dt 9

?

6

?? tse`n>

dt

9

?

6

??

?

ttn`)n>(>

dttsbk9

?

n>

>

tn`> ?

6

? ttn`n>

9?n>

>

.

?

(n) 9?

`

?n>n + k

Out(6) 9 6 k 9 6

(n) 9?

`

?n>n

[bcg Vrnktekb Vrhocbis

tdt, tjb` ge`d tjb vncub hg g(?).

(?:) Eg x 9

y

6?t4>

dtn`d ?

?

dx

yd9 _y, tjb` ge`d _


14/21

INTJ[


(?5) Eg g(x) 9 ?x

x

? dttse`x , tjb` ge`d g(x).

(;6) Eg (x) 9 khs x x

6

)t()tx( dt, tjb` ge`d tjb vncub hg (x) +(x).

(;>) Ge`d tjb vncub hg tjb gu`kteh` g(x) 9 > + x + x

>

? t`?)t`( dt, wjbrb g(x) vnèsjbs.

(;?) Bvncuntb6x

Ct xse`x

dttkhs

?x

6

?.

(;;) Bvncuntb

6

)xkhso>(` dx, o obe m pnrnibtbr..

N`swbrs < (?7) ;x? ;xkhs (?=) >=

?(?:) 4

(?5) x? (?x se` x? se` x) + (khs x khs x?) ?x (;6) khs x

(;>) > +b

?(;?) > (;;)

`

?

o>> ?

Dbgeètb E`tbmrnc ns n ceiet hg suiCbt g(x) ob n kh`teùhus rbnc vncubd gu`kteh` dbge bd h` tjb kchsbd e`tbrvnc Qn, oY wjekj es devedbd e th ` pnrts

ns sjhw` e` gemurb.

Tjb phe`t hg deveseh` h` x-nxes nrb n, n + j, n + ?j ..........n + (` >)j, n + `j, wjbrb`

no 9 j.

Cbt [`db`htbs tjb nrbn hg tjbsb ` rbktn`mcbs.

Tjb`, [`9 jg(n) + jg(n + j) + jg(n + ?j) + ........+jg(n + (` >)j)

Kcbnrcy, [`

es nrbn vbry kchsb th tjb nrbn hg tjb rbmeh` ohu`dbd oy kurvb y 9 g(x), xnxes n`d tjb hrde`ntbs


15/21

INTJ[


x 9 n, x 9 o.

Jb`kb o

n

dx)x(g 9`

Ct [`

o

n

dx)x(g 9`

Ct

>`

6r

)rjn(gj 9`

Ct

>`

6r `

nog

`

r)no(n

@htb . ]b kn` ncsh wretb

[`9 jg(n + j) + jg (n + ?j) + .........+ jg(n + `j) n`d

o

n

dx)x(g 9`

Ct

`

>r `

nog

r

`

non

?. Eg n 9 6, o 9 >, >

6

dx)x(g 9 `Ct

>`

6r `

rg

`

>

[tbps th bxprbss tjb ceiet hg sui ns dbge`tb e`tbmrnc ._bpcnkb

`

roy x,

`

>oy dx n`d `

Ct oy

[tbp ?.Bvncuntb `Ct

`

roy putte`m cbnst n`d mrbntbst vncubs hg r ns chwbr n`d uppbr ceiets rbspbktevbcy..

Ghr bxnipcb `Ct

p`

>r `

rg

`

>9

p

6

dx)x(g (>r

`

`

rCt

9 6,

`pr`

`

rCt

9 p)

Bxnipcb # ?4 < Bvncuntb

`Ct

`?>

.........`;

>

`?

>

`>

>

[hcuteh` `Ct

`?>

.........`;

>

`?

>

`>

>

9 ` Ct `

>r `r

>

9 `Ct

`

>r `

>

>`

r

>

9 >

6>x

dx9 >6)>x(` 9 ` ?

Bxnipcb # ?2< Bvncuntb `Ct

`2

;.........

;`

;`

?`

?`

>`

>`?????? .


16/21

INTJ[


[hcuteh` `Ct

`?

>r?? r`

r`9 `

Ct

`?

>r `

>?

`

r>

`

r>

`Ct

`

r9 6, wjb` r 9 >, chwbr ceiet 9 6

n`d `Ct

`

r9 `

Ct

`

`?9 ?, wjb` r 9 ?`, uppbr ceiet 9 ?

?

6

?x>

x>dx 9

?

6

?x>

>dx +

?

> ?

6

?x>

x?dx

9 Q tn`>x Y?6

+

?

6

?b )x>(chm

?

>

9 tn`> ? +?

>` 2

Bxnipcb # ?7 < Bvncuntb

`Ct `

>

``

!

.

[hcuteh` Cbt y 9 `Ct `

>

``

!

` y 9 `Ct

`

>

`

``

!

9 `Ct

`

>`

``

`........;.?.>

9 `Ct

`

>

`

``.....

`

;`

`

?`

`

>`

9 `Ct

`

>

`

>r `

r`

9

>

6

>

6

xx`xdx`x

9 (6 >) 6x

Ct x ` x + 6

9 > 6 9 >

y 9b

>

[bcg Vrnktekb Vrhocbis =

(;4) `Ct

????? ``

>....

`?`

>

``

>

`

>

(;2) `Ct

`2>

.......`;

>

`?

>

`>

>

(;7) `Ct

?`

>

`4

`se``........

`4

;se`;

`4

?se`?

`4se` ;;;;

(;=) `Ct

>`

6r?? r`

>

(;:) `Ct

`

;

)>`(;`

`......

5`

`

7`

`

;`

`>

N`swbrs < (;4) ? >? (;2) ` 2 (;7) ?5?

(2? >2)

(;=)?

(;:) ?

_bdukteh` ghriucnb e` dbg e`etb E`tbmrncs. Eg`9

?

6

` xse` dx , tjb` sjhw tjnt`9

`

>`

`?

Vrhhg < `

9

?

6

` xse` dx

`

9 ?6

>` xkhsxse`

+

?

6

??` dxxkhs.xse`)>`(

9 (` >)

?

6

??` dx)xse`>(.xse`

9 (` >)

?

6

?` )>`(dxxse`

?

6

` dxxse`

`+ (` >)

`9 (` >)

`?

`9

`

>`

`?


18/21

INTJ[

"inèsjauinrpjyseks.e`" >:

@htb < >.

?

6

` dxxse` 9

?

6

` dxxkhs

?. `

9

`

>`

?`

;`

4`

2`.....

6hr

>

nkkhrde`m ns ` es bvb` hr hdd.69 ?,>9 >

Jb`kb`9

hddesèg,>.;

?........

4`

2`

?`

;`

`

>`

bvbèsèg,?

.?

>........

4`

2`

?`

;`

`

>`

?. Eg`9

4

6

` dxxtn` , tjb` sjhw tjnt`+

`?9

>`

>

[hcuteh` `9

4

6

?`)x(tn` . tn`?x dx

9

4

6

?`)x(tn` (sbk?x >) dx

9

4

6

?`)x(tn` sbk?x dx

4

6

?`)x(tn` dx

94

6

>`

>`

)x(tn`

`?

`9 >`

>

`?

`+

`?9

>`

>

;. Egi,`

9

?

6

ì dxxkhs.xse` , tjb` sjhw tjnti,`

9ì

>i

i?, `


19/21

INTJ[


[hcuteh` i,`

9

?

6

`>i )xkhsx(se`xse` dx

9

?

6

>`>i

>`

xkhs.xse`

+

?

6

>`

>`

xkhs

(i >) se`

i?

x khs x dx

9

>`

>i

?

6

?`?i xkhs.xkhs.xse` dx

9

>`

>i

?

6

ì`?i xkhs.xse`xkhs.xse` dx

9

>`

>i

i?,`

>`

>i

i,`

>`

>i>

i,`9

>`

>i

i?,`

i,`

9

ì

>i

i?,`

@htb < >. i,`

9

ì

>i

?ì

;i

4ì

2i........

6,`hr

>,`nkkhrde`m ns i es bvb` hr hdd.

6,`

9

?

6

` dxxkhs n`d>,`

9

?

6

` dxxkhs.xse` 9>`

>

?. ]ncces Ghriucn

i,`

9

htjbrwesb)........4ì()?ì()ì(

).......2`();`()>`.........()2i();i()>i(

bvb`nrb`,iohtjwjb`

?)........4ì()?ì()ì(

).......2`();`()>`.........()2i();i()>i(

Bxnipcb # ?= < Bvncuntb

?

?

?? dx)xkhsx(se`xkhsxse` .

[hcuteh` Mevb` e`tbmrnc 9

?

?

?; dxxkhsxse` +

?

?

;? dxxkhsxse`


20/21

INTJ[

"in`esjauinrpjyseks.e`" ?6

9 6 + ?

?

6

;? dxxkhsxse` (se`;x khs?x es hdd n`d se`?x khs;x es bvb`)

9 ?.>.;.2

?.>9

>2

4

Bxnipcb # ?: < Bvncuntb

6

72 dxxkhsxse`x .

[hcuteh` Cbt 9

6

72 dxxkhsxse`x

9

6

72 dx)x(khs)x(se`)x(

9

6

72 dxxkhs.se`

6

72 dxxkhs.xse`x

? 9 . ?

?

6

72 dxxkhs.xse`

9>.;.2.=.5.>>

>.;.2.?.4

9 75;

:

Bxnipcb # ?5 < Bvncuntb >

6

2; dx)x>(x .

[hcuteh` Vut x 9 se`? dx 9 ? se`khsdC . C < x 9 6 9 6

R.C. < x 9 > 9?

>

6

2; dx)x>(x 9

?

6

2?7 )(khsse` ? . se . khs d

9 ? .

?

6

>>= dkhsse`

9 ? .?.4.7.:.>6.>?.>4.>7.>:

?.4.7.:.>6.?.4.79

264

>


21/21

INTJ[

[bcg Vrnktekb Vrhocbis) >

6

>7 dxxse`x .

(4?) n

6

?

=?? dxxnx .

(4;) .dxx?x

?

6

?/;

N`swbrs < (;5)>2

:(46)

;>2

:(4>)

>4

?42

>7(4?)

5

n5

(4;)?

definite integration theory_e

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