Definite Integration Theory_e
Transcript of Definite Integration Theory_e
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INTJ[
"in`esjauinrpjyseks.e`" >
@bwth` Cbeo`etz 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`te`uhus 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+
?
+ ?
Dbge`etb e`tbmrncs
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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?
Bxnipcb # 7 Bvncuntb
?
?
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.
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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
Bxnipcb # 5 Bvncuntb
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
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INTJ[
"in`esjauinrpjyseks.e`" 7
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)
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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.
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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
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INTJ[
"in`esjauinrpjyseks.e`" 5
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
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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 >
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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 . ` ?.
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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
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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 _
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INTJ[
"in`esjauinrpjyseks.e`" >4
(?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) vn`esjbs.
(;?) 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>> ?
Dbge`etb E`tbmrnc ns n ceiet hg suiCbt g(x) ob n kh`te`uhus 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
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INTJ[
"in`esjauinrpjyseks.e`" >2
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
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roy x,
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Ct oy
[tbp ?.Bvncuntb `Ct
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roy putte`m cbnst n`d mrbntbst vncubs hg r ns chwbr n`d uppbr ceiets rbspbktevbcy..
Ghr bxnipcb `Ct
p`
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p
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dx)x(g (>r
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Bxnipcb # ?4 < Bvncuntb
`Ct
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.........`;
>
`?
>
`>
>
[hcuteh` `Ct
`?>
.........`;
>
`?
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9 ` Ct `
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>
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dx9 >6)>x(` 9 ` ?
Bxnipcb # ?2< Bvncuntb `Ct
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?`
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>`?????? .
-
8/12/2019 Definite Integration Theory_e
16/21
INTJ[
"in`esjauinrpjyseks.e`" >7
[hcuteh` `Ct
`?
>r?? r`
r`9 `
Ct
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>?
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r>
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r>
`Ct
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r9 6, wjb` r 9 >, chwbr ceiet 9 6
n`d `Ct
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r9 `
Ct
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?
6
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x>dx 9
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?
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6
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x?dx
9 Q tn`>x Y?6
+
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?
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9 tn`> ? +?
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Bxnipcb # ?7 < Bvncuntb
`Ct `
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y 9b
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9
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` xse` dx
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9 (` >)
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-
8/12/2019 Definite Integration Theory_e
18/21
INTJ[
"in`esjauinrpjyseks.e`" >:
@htb < >.
?
6
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?
6
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?. `
9
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nkkhrde`m ns ` es bvb` hr hdd.69 ?,>9 >
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4
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9
4
6
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9
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6
?`)x(tn` sbk?x dx
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94
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i?, `
-
8/12/2019 Definite Integration Theory_e
19/21
INTJ[
"in`esjauinrpjyseks.e`" >5
[hcuteh` i,`
9
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6
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9
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?
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?
?
;? dxxkhsxse`
-
8/12/2019 Definite Integration Theory_e
20/21
INTJ[
"in`esjauinrpjyseks.e`" ?6
9 6 + ?
?
6
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9 ?.>.;.2
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4
Bxnipcb # ?: < Bvncuntb
6
72 dxxkhsxse`x .
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6
72 dxxkhsxse`x
9
6
72 dx)x(khs)x(se`)x(
9
6
72 dxxkhs.se`
6
72 dxxkhs.xse`x
? 9 . ?
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6
72 dxxkhs.xse`
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9 75;
:
Bxnipcb # ?5 < Bvncuntb >
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9 ? .?.4.7.:.>6.>?.>4.>7.>:
?.4.7.:.>6.?.4.79
264
>
-
8/12/2019 Definite Integration Theory_e
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>)
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?42
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5
n5
(4;)?