~ ?yvD ?Dy
? 2 ?
±s¥'
T
D'? ?
~ ?yvD ?Dy
M
°L??ê?f
D
f
¥ ó"
M
°L??
^?1

2
1
)(
T
T
dttv
!
t8T°L?X?
)(tvv =
^
H
WW? ],[
21
TT
 t¥B? ??f
 O
0)( ≥tv  pt8?
HW
=
üV¥
^?,
6BZ
?
^? VV
U1 )()(
12
TsTs?
Baù5¥4
).()()(
12
2
1
TsTsdttv
T
T
=∴

).()( tvts =

?
~ ?yvD ?Dy
=a±s'
T
?l 1
.)( )(
)()(
)()(
)(
)(
? uW
=¥ef
ü?1
*
1f

μ
B?
P¤? uW
=¥ ? ?Tif
=¥f

^B??l
B uWX?
xfxF
dxxfxdF
xfxF
xF
xf
=
=

~ ?yvD ?Dy
? ? 
±s'
T
?T )(xF
^ ??f
)(xf  uW ],[ ba

¥B?ef
5 )()()( aFbFdxxf
b
a
=


?s?
=? ? uW 1],[?nba
£
bxxxxa
nn
=<<<<=
110
L
),,2,1](,[],[
1
nixxnba
ii
L=
?l uW$sé1
*
1
?′? ?μ?
! Lagrangexxx
iii
,
1?
=?
iiii
xFxFxF?ξ )()()(
1

=?
),(
1 iii
xx
∈ξ
~ ?yvD ?Dy
[

=
=?
n
i
ii
xFxFaFbF
1
1
)]()([)()(
i
n
i
i
xF?ξ?

=

=1
)(

=
=
n
i
ii
xf
1
)(?ξ
'¤

T?
7,0},,2,1|max{ →== nix
i
L?λ
.],[)(
 ??# V?? baxf
.)()()( dxxfaFbF
b
a

=?
——
d — ??
DG
T
~ ?yvD ?Dy
)()()( aFbFdxxf
b
a
=

±s'
TV
ü
[]
b
a
xF )(=
B? ??f
 uW ],[ ba
¥?s??
¥ ?iB?ef
 uW ],[ ba
¥9
,
?i ? ba >
H )()()( aFbFdxxf
b
a
=

ˉ? ?,
p?sù51 pef
¥ù5,
~ ?yvD ?Dy
è p,)1sincos2(
2
0

π
+ dxxx
e
T [ ]
2
0
cossin2
π
xxx=,
2
3
π
=
è
!, p,
≤<
≤≤
=
215
102
)(
x
xx
xf

2
0
)( dxxf
3
3
∫ ∫∫
+=
1
0
2
1
2
0
)()()( dxxfdxxfdxxf
 ]2,1[
?? 1=x
H 5)( =xf,
∫ ∫
+=
1
0
2
1
52 dxxdxe
T
.6=
x
y
o 1 2
~ ?yvD ?Dy
è p
.},max{
2
2
2

dxxx
3
?m? V?
},max{)(
2
xxxf =
,
21
10
02
2
2
≤≤
≤≤
≤≤?
=
xx
xx
xx
∫∫∫
++=∴
2
1
2
1
0
0
2
2
dxxxdxdxxe
T
.
2
11
=
x
y
o
2
xy =
xy =
1
2
2?
~ ?yvD ?Dy
è p
3
.
1
1
2
dx
x

? 0<x
H
x
1
¥B?ef
^ ||ln x,
dx
x

1
2
1
[ ]
1
2
||ln
= x
.2ln2ln1ln?=?=
è 9
wL xy sin=  ],0[ π
D xà
?
?¥
ü
m?¥
,
3

x
y
o
π

π
=
0
sin xdxA
[ ]
π
=
0
cos x
.2=
~ ?yvD ?Dy
!f
)(xf  uW ],[ ba
 ?? i O
! x1
],[ ba
¥B?

x
a
dxxf )(
I3?s

=
x
a
dttf )(
:
.)()(


x
a
dttfx s
Kf
?T
K x uW ],[ ba
 ?iM?5
?
B? |?¥ x′?sμB??′
[
 ],[ ba
?l
B?f

?as
Kf
# ?
~ ?yvD ?Dy
a
b
x
y
o
? ?  ?T )(xf  ],[ ba
 ??5s
K¥f
dttfx
x
a

=Φ )()(  ],[ ba
 μ?
 O
¥?
^ )()()( xfdttf
dx
d
x
x
a
==Φ


 )( bxa ≤≤ 
s
Kf
¥?é
xx?+
£
dttfxx
xx
a

+
=?+Φ )()(
)()( xxx Φ+Φ=?Φ
dttfdttf
x
a
xx
a
∫∫
=
+
)()(
)(xΦ
x
~ ?yvD ?Dy
ξ
dttfdttfdttf
x
a
xx
x
x
a
∫∫∫
+=
+
)()()(
,)(

+
=
xx
x
dttf
?s?′? ?¤
xf?=?Φ )(ξ
],,[ xxx?+∈ξ
xx →→? ξ,0
),(ξf
x
=
Φ
)(limlim
00
ξf
x
xx →?→?
=
Φ
).()( xfx =Φ


a
b
x
y
o
xx?+
)( xΦ
x
~ ?yvD ?Dy
 ?T )(tf ?? )(xa a )(xb V?
5 dttfxF
xb
xa

=
)(
)(
)()( ¥?
)(xF

1
?
[ ] [ ] )()()()( xaxafxbxbf


=
£
dttfxF
xa
xb
)()(
0
)(
)(
0
+=
∫∫
dttf
xb

=
)(
0
)(
,)(
)(
0
dttf
xa

[ ] [ ] )()()()()( xaxafxbxbfxF


=


=

)(
)(
)()(
xb
xa
dttf
dx
d
xF
~ ?yvD ?Dy
è p,lim
2
1
cos
0
2
x
dte
x
t
x


3

1
cos
2
x
t
dte
dx
d
,
cos
1
2

=
x
t
dte
dx
d
)(cos
2
cos

=
xe
x
,sin
2
cos x
ex
=
2
1
cos
0
2
lim
x
dte
x
t
x


x
ex
x
x
2
sin
lim
2
cos
0

=
.
2
1
e
=
0
0
s ?
^ ???
T?¨
ArE5,
~ ?yvD ?Dy
è
! )(xf  ),( +∞?∞
= ?? O 0)( >xf,
£
üf


=
x
x
dttf
dtttf
xF
0
0
)(
)(
)(  ),0( +∞
=1??9
Ff
,
£

x
dtttf
dx
d
0
)( ),(xxf=

x
dttf
dx
d
0
)(
),(xf=
2
0
00
)(
)()()()(
)(
=


∫∫
x
xx
dttf
dtttfxfdttfxxf
xF
~ ?yvD ?Dy
()
,
)(
)()()(
)(
2
0
0


=

x
x
dttf
dttftxxf
xF
)0(,0)( >> xxfQ,0)(
0

>∴
x
dttf
,0)()( >? tftxQ
,0)()(
0

>?∴
x
dttftx
).0(0)( >>

∴ xxF
# )(xF  ),0( +∞
=1??9Ff
,
~ ?yvD ?Dy
è
! )(xf  ]1,0[
 ?? O 1)( <xf,£
ü
1)(2
0
=?

dttfx
x
 ]1,0[
oμB?3,
£,1)(2)(
0
=

dttfxxF
x
,0)(2)( >?=

∴ xfxF
,1)( <xfQ
)(xF  ]1,0[
1??9Ff
,
,01)0( <?=F

=
1
0
)(1)1( dttfF

=
1
0
)](1[ dttf,0>
[ 0)( =xF 'eZ? ]1,0[
oμB?3,
7
~ ?yvD ?Dy
? ? 
ef
i? ?
 ?T )(xf  ],[ ba
 ??5s
K¥f
dttfx
x
a

=Φ )()( ü
^ )(xf  ],[ ba
¥B?
ef

? ?¥×1il

1 \?
 ??f
¥ef
^i¥,

2?£
U
sD?¥?sDef
-
W¥ ó",
~ ?yvD ?Dy
1.±s'
T
2.s
Kf


x
a
dttfx )()(
3.s
Kf
¥?
)()( xfx =Φ

)()()( aFbFdxxf
b
a
=

1al2
d ??
DG
TY
±sDDsD-
W¥1"
~ ?yvD ?Dy
± I5
! )(xf  ],[ ba
 ??5 dttf
x
a

)( D
duuf
b
x

)(
^ x¥f

^ tD u¥f
$
ì
¥?
i
$ ?i??
I
1$
~ ?yvD ?Dy
± I53s
dttf
x
a

)( D duuf
b
x

)( ?
^ x¥f
)()( xfdttf
dx
d
x
a
=

)()( xfduuf
dx
d
b
x
=

~ ?yvD ?Dy
BaA b5
a

b
a
x
dxe
dx
d
2
2
@@@@@@@
a

=
x
a
dxxf
dx
d
))(( @@@@@@@@@@
a =+

2
2
3
)1ln(
x
dttt
dx
d
@@@@@@@
a

=
2
0
)( dxxf @@@@ ?
<<?
≤≤
=
21,2
10,
)(
2
xx
xx
xf 
a
!

π
π?
=,coscos
1
nxdxmxI 
 dxnxmxI

=
π
π
sinsin
2

5
~ ?yvD ?Dy

a? nm =
H
1
I @@
2
I @@@@@

a? nm ≠
H
1
I @@@
2
I @@@@@
a
!,sincos
3

=
π
π
nxdxmxI 

a? nm =
H
3
I @@@@


a? nm ≠
H
3
I @@@@@
a =+

9
4
)1( dxxx @@@@@
a =
+

3
3
1
2
1 x
dx
@@@@@
a =


x
dtt
x
x
0
2
0
cos
lim @@@@@@@@
~ ?yvD ?Dy
?z  p?

a
!f
)(xyy = ?Z? 0cos
00
=+
∫∫
xy
t
tdtdte
?
? p
dx
dy
 
a
!
=
=


1
2
1
2
2
,ln
,ln
t
t
uduuy
uduux
)1( >t
p
2
2
dx
yd
 
a

π
x
x
dtt
dx
d
cos
sin
2
)cos(  
a
!

+
=
2
0
3
1
)(
x
x
dx
xg  p )1(g
′′


~ ?yvD ?Dy
?a 9
/
ò?s
a

+
2
1
2
2
)
1
( dx
x
x a

2
1
2
1
2
1 x
dx

a

+
++
0
1
2
24
1
133
dx
x
xx
a

π2
0
sin dxx 
1a p/
K
1a


+∞→
x
t
x
t
x
dte
dte
0
2
2
0
2
2
)(
lim ; 2a
2
5
0
2
0
2
1
)cos1(
lim
x
dtt
x
x

+→
.
~ ?yvD ?Dy
?a
! )(xf 1 ??f
£
ü

∫∫∫
=?
xxt
dtduufdttxtf
000
))(())(( 
Ba pf

+?
+
=
x
dt
tt
t
xf
0
2
1
13
)(  uW []1,0
¥K
v′DKl′
ta
!
><
≤≤
=
H ?
H ?
π
π
xx
xx
xf
00
0,sin
2
1
)(
 p

=
x
dttfx
0
)()
 ),( ∞+?∞
=¥Vr
T
~ ?yvD ?Dy
?a
! [ ]baxf,)( 
 ?? O,0)( >xf
∫∫
+=
x
a
x
b
tf
dt
dttfxF
)(
)()( 
£
ü

a 2)(
'
≥xF 

aZ? 0)( =xF  ),( ba
=μ O?μB??
~ ?yvD ?Dy
Baa a )()( afxf?  a )1ln(
2
3
+? xx 
a
6
5
 a 
ππ,  


a ;
6
1
45 a
6
π
 a
=aa
1sin
cos
x
x
 a
tt ln2
1
2

a )sincos()cos(sin
2
xxx π  a 2? 
?aa
8
5
2  a
3
π
 a 1
4
+
π
 a
5s?
~ ?yvD ?Dy
1aa a
10
1

Ba
33



ta
π>
π≤≤?
<

x
xx
x
x
,1
0,)cos1(
2
1
0,0
)(