2005
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!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
$?s
?B [±s 
? c<ls¨
8ù5
<ls￥?l#
l ??
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μ? uW
￥s
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 í? uW
 ￥s
?  ?  í?f
μ ? uW
￥
s
?  ? b
?l  ?B ?< l  s
!f
)(xf ),[ +∞a
=￥ ? iμK u W V 
i O K 
i
5?
∫
+∞→
A
a
A
dxxf )(lim )(xf  ),[ +∞a<ls
l ?
<ls1
∫∫
+∞→
∞+
=
A
a
A
a
dxxfdxxf )(lim)(  ??
l ?
5?<ls? ?b
?l  ?= ?<ls
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)(xf 
=￥ ?i μ K>0 u W V
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a
bB
dxxf )(lim i
5? )(xf 
￥<ls
l ?
<ls1),[ ba
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→
=
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a
bB
b
a
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]"á
ì V[?l 
<ls￥
l ??
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a
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b
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b
A
aA
b
a
dxxfdxxf )(lim)(

l ??￥
5
?B ?<ls
l ??￥
5
5  ? ?B ?< l  s
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dxxf )(
l ?
5
∫
∞+
a
dxxf )( B?
l ?
N
H
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l ?
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a
dxxf )(
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l ?7
∫
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a
dxxf )(
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∞+
a
dxxf )( Z? ?
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l ?
￡
!·° Iù
T? www.tsinghuatutor.com - 1 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
5 1?E
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),[),()(0 +∞∈≤≤ axxgxf
 ?
l ?
 B?
l ? ?
∫
∞+
a
dxxg )(
∫
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a
dxxf )(
∫
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a
dxxf )( ? ?

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∫
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a
dxxg )(
5 
! )(),( xgxf ),[ +∞a
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 O
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)(
)(
lim
xg
xf
x

5

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H
<ls D
∫
∞+
a
dxxf )(
∫
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a
dxxg )( μM]￥ ? ??

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H
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l ?5
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dxxg )(
∫
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dxxf )(
l ?

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H
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l ?5
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a
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∫
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a
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l ?
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x
a
p
∫
∞+
1
 ?
H
l ? ?)0( >a 1>p 1≤p
H? ? yN  ?

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50)(lim ≥=
+∞→
λxfx
p
x
1>p
∫
∞+
a
dxxf )(
l ?b
è 
 dx
x
xx
∫
∞+
+
1
5
1
ln
￥
l ??
3? 0
ln
lim
3
=
+∞→
x
x
x
i 0>X 
P¤? 0>> Xx
H
3
ln xx < 
1
6
7
,
11
ln
5
3
5
>=
+
<
+
p
x
xx
x
xx
?°¤1?E
l ?
è 
 dx
xxx
x
∫
∞+
++
1
2
1
arctan
￥
l ??
3D dx
x
∫
∞+
1
2
1
1??K1?E
l ?
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L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
è 

∫
∞+
e
p
xx
dx
2
ln
￥
l ??
3 ()+∞→→= x
xxx
x
p
p
0
ln
1
ln
22
yN
H1>p
∫
∞+
e
p
xx
dx
2
ln
l ?
1=p
H
∫
∞+
+∞→
=
=
e
B
B
e
xxx
dx
1
ln
1
lim
ln
2

1<p
HD
∫
∞+
e
x
dx
1?
V?
xnx
p
x
21
1
lim
+∞→
+∞= 
yNs?1
H
l ?1≥p 1<p
H? ?b

?= ?<ls
l ??￥
5
5  ??= ?< ls
∫
b
a
dxxf )(
l ?
B?
l ?
N
H?
'
l ? 
l ?7
∫
b
a
dxxf )(
∫
b
a
dxxf )(
∫
b
a
dxxf )(
∫
b
a
dxxf )( Z? ?
5?<l s
Hq
l ?
 5  1 ? E
d μf
),[),()(0 baxxgxf ∈≤≤
 ?
l ?
 B?
l ? ? ? ?

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∫
b
a
dxxg )(
∫
b
a
dxxf )(
∫
b
a
dxxf )(
∫
b
a
dxxg )(
5 f
 )(),( xgxf 
=￥ ?i uW
 V
),[ ba )(xg dμ
 O
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!·° Iù
T? www.tsinghuatutor.com - 3 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
λ=
→
)(
)(
lim
xg
xf
bx

5

? 0≠λ
H
<ls D μM]￥ ? ??
∫
b
a
dxxf )(
∫
b
a
dxxg )(

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H
<ls
l ?5
l ?
∫
b
a
dxxg )(
∫
b
a
dxxf )(

? ∞=λ
H
<ls
l ?5
l ?
∫
b
a
dxxf )(
∫
b
a
dxxg )(
 5  dx
bx
b
a
p
∫
)(
1
? 1<p
l ?
H? ? yN ?1≥p
0)()(lim ≥=?
→
λxfbx
p
bx

O 1<p
5
l ?b
∫
b
a
dxxf )(
è 
<ls dx
x
∫
π
0
sin
1
￥
l ??
3 dx
x
∫
π
0
sin
1

dx
x
dx
x
∫∫
+=
π
π
π
2
2
0
sin
1
sin
1

?B?sA ?
l ??=?s
7 dtdxtx ==?,π 
dx
x
dt
t
dx
x
∫∫∫
=?= 2
0
0
22
sin
1
sin
1
sin
1
π
π
π
π

l ?
è )
dx
x
x
p
∫
∞+
0
arctan
￥
l ??
3 dx
x
x
p
∫
∞+
0
arctan

￡
!·° Iù
T? www.tsinghuatutor.com - 4 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
dx
x
x
p
∫
=
1
0
arctan
dx
x
x
p
∫
∞+
+
1
arctan

?B?s
p
x
xarctan
D
1
1
p
x
?N 

l ?
0→x
2,11 <?<? pp
?=?s
p
x
xarctan
D
q
x
1
é?1¨
=
>
=
+∞→
qp
qp
x
x
qp
x
2
0
arctan
lim
π

yN?
H?=? s
l ?b8

s 1>≥ qp 21 << p
Hs
l ? b

è 9
∫
∞+
++
0
22
1)51(
1
dx
xx
b
3 |MD
2
1
,tan
t
dt
dxtx
+
== 5
∫
+
= 2
0
2
tan51
sec
π
dt
t
t
I
2arctan
2
1
)sin2arctan(
2
1
sin41
sin
2
0
2
0
2
==
+
=
∫
π
π
t
t
td
è 
!
è

?0>a
∫∫
∞+
+
=
+
a
a
dx
x
dx
x
2
0
2
1
1
1
1

5 =a b
￡
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T? www.tsinghuatutor.com - 5 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
3 aa arctan
2
arctan?=
π

1,
4
arctan == aa
π

è 9

∫
∞+
1
2
arctan
dx
x
x

3 )
1
(arctan
arctan
11
2
∫∫
∞+∞+
=
x
xddx
x
x

∫
∞+
∞+
+
+?=
1
21
)1(
11
dx
xx
nxarcta
x
dx
x
x
x
b
b
)
1
1
(lim
4
2
1
+
+=
∫
+∞→
π

]2ln
2
1
)1ln(
2
1
[lnlim
4
2
++?+=
+∞→
bb
b
π

2ln
2
1
4
+=
π
è 
∫
∞+
=
1
2
1xx
dx
b
∫∫∫
∞+
=
==
=
1
2
0
2
0
sec
2
2tansec
tansec
1
ππ
π
dtdt
tt
tt
xx
dx
tx

7
t
x
1
= ¨
X±sE
5
￡
!·° Iù
T? www.tsinghuatutor.com - 6 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
∫∫
=
∞+ 0
1
2
2
1
2
)
1
(
1
1
1
dt
t
t
t
xx
dx
∫
==
=
1
0
2
20
1
arcsin
1
1 π
tdt
t

è <ls =
∫
∞+
1
2
1
x
e
dx
s?
1
arccos
2
e
π

3 |MD 5te
x
sec=
tdttdxetx
x
tansec),ln(sec == 
11
2
arccos
arcsinarccos
2tan
tan
1

=?==
∫?
eedt
t
t
I
e
π
π
è 9
<ls bdxxx
n
∫
1
0
ln
<3>?¨s?s'μ
1
1
0
12
1
0
2
2
1
ln
2
1
ln
2
1
==
∫
n
nn
n
I
n
dx
x
xnxxxI
1
2
2
!)1(
2
1
2
+
==
=
n
n
n
n
I
nn
L

11
2
,
4
1
=?=
nn
I
n
II b
¨
ˉ8ù5 
?s?l Iù?￥?¨ ¨? p+?K
¨?s pKè¨
T1
∫
∑ =
+
=
∞→
b
a
n
k
dxxfk
n
ab
af
n
ab
)()(lim
1
n
b
￡
!·° Iù
T? www.tsinghuatutor.com - 7 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
? )(
k
fk
n
ab
ξ=

k
x
n
ab
=
b
è  pK
n
n
n
!
lim
n ∞→
bs?
e
1
b
bvD Iù￡??  <? è5
<3>:
n
n
y
n
n
!
= 5
nk
nn
n
y
n
k
n
n
lnln
1!
lnln
1
== ∑
=

:1
)lnln(
1
ln
1
nnk
n
y
n
k
n
= ∑
=
)ln(ln
1
1
nk
n
n
k
= ∑
=
∑
=
=
n
k
n
k
n
1
ln
1

K
n
n
ylnlim
∞→
∑
=
∞→
=
n
k
n
n
k
n
1
ln
1
lim ??<ls ￥′
∫
1
0
ln xdx
M??| uW sé?]1,0[ ),,2,1(],
1
[ nk
n
k
n
k
L=
￥s
T￥K
Os
T?￥
n
k
f
k
ln)( =ξ b
?i?<ls 1?= ?<lsi O
l ??
^ 
∫
1
0
ln xdx
n
n
ylnlim
∞→
∑
=
∞→
=
n
k
n
n
k
n
1
ln
1
lim
∫
=
1
0
ln xdx
￡
!·° Iù
T? www.tsinghuatutor.com - 8 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
1)ln(
1
0
=?= xxx 
[
n
n
y
∞→
lim ==
∞→
n
n
n
!
lim
n
e
1
b
?
ZE V[9

π
π
π
2
sin
1
sin
lim
1
0
1
==
+
∫
∑
=
∞→
xdx
k
n
n
k
n
k
n
b
?
n
k
k
x
kk
π
ξ sinsin,
1
==? b h A 
M I5


n
n
n
n
nn
222
1
2
1
1
1lnlim
+
+
+
∞→
L
??<#>
()
∫
Α
2
1
2
.ln xdx ( )
∫
Β
2
1
.ln2 xdx
() ( )
∫
+
2
1
.1ln2 dxxC ( ) ( )
∫
+
2
1
2
.1ln dxxD 
n
n
n
n
nn
222
1
2
1
1
1lnlim
+
+
+
∞→
L
+
+
+=
∞→
n
n
nnn
n
1
2
1
1
1ln
2
limL
∑
=
∞→
+=
n
k
n
n
k
n
1
1ln
1
lim2 
()
∫
+=
1
0
2
1ln2 dxx
∫
=
2
1
2
ln2 tdt #
b
42ln82ln4ln2
22
1
2
+?==
∫
tdt 
￡
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T? www.tsinghuatutor.com - 9 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
è
!
∫
+?
+=
)1(
0
1
1
2
3
nnnn
n
dxxxa 5 [ B ]b =
∞→
n
n
nalim
(A) b (B)1)1(
2/3
++e 1)
1
1(
2/3
+
e
b
(C) 1)
1
1(
2/3
++
e
b (D) b 1)1(
2/3
+e
3,s¤
)1()1(
1
2
3
21)1/(
0
nnnn
n
xdx
n
a ++?=
∫
+
=
1
0
2/3
)1(
1
+
+
n
n
n
x
n
= ]1))
1
(1[(
1
2/3
+
+
n
n
n
n
|K¤
1)
1
1(1])
1
(1[limlim
2/32/3
+=?
+
+=
∞→∞→
en
n
na
n
n
n
n
è X
?  O f(1)=0,5
xx
xeef
=
′
)(
()xf =
2
)(ln
2
1
x,
}s ~ 5 p ￥Vr
Ts' Vb
)
t
x
=
( xf ′
}3 ~
7 e 5 ?
^μ tx ln=
t
t
tf
ln
)( =
′
 ',
ln
)(
x
x
xf =
′
s¤ Cxdx
x
x
xf +==
∫
2
)(ln
2
1ln
)(, ?¨ 
SHq
,¤ C=0#
pf
1 () 01 =f
￡
!·° Iù
T? www.tsinghuatutor.com - 10 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
()xf =
2
)(ln
2
1
x,
è 
!
≥?
<≤?
=
2
1
,1
2
1
2
1
,
)(
2
x
xxe
xf
x

5
2
1
)1(
2
2
1
=?
∫
dxxf,
}s ~s
f
￥?s5 | uWMD x? 1 = t ?¨? uW
 
}f
￥s?éb
}3 ~
7 x? 1 = t
∫∫∫

==?
1
2
1
1
2
1
2
2
1
)()()1( dtxfdttfdxxf
2
1
)
2
1
(0)1(
1
2
1
2
1
2
1
2
=?+=?+=
∫∫
dxdxxe
x
,
è 
! SV
UC xàD wL y = F(x)-W￥
b 
? t > 0 V
U t ≤ x ≤ t0 ≤ y ≤ F(t)￥
, p
>
≤
=
0,
0,
)(
2
2
xe
xe
xF
x
x
)(
1
tS
(1) S(t) = S? ￥Vr
T )(
1
tS
(2) S(t)￥Kl′,
}s ~ wL y = F(x)1? yà? xàD wL y = F(x)??Bí? u×
[
 S V¨<lsV
U,
?'5?b
}3 ~(I) 12
0
2
0
2
=?==
∞+
∞+?
∫
xx
edxeS 
t ≤ x ≤ t0 ≤ y ≤ F(t)￥
1
t
tetS
2
1
2)(
= 
yN t ∈ (0,+∞),
t
tetS
2
21)(
=
￡
!·° Iù
T? www.tsinghuatutor.com - 11 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
(II) ??
t
ettS
2
)21(2)(
=
′

# S(t)￥·B?1
2
1
=t 
? 
t
ettS
2
)1(8)(
=
′′
0
4
)
2
1
( >=
′′
e
S 
[
e
S
1
1)
2
1
(?= 1l′
9
^Kl′,
è  p wL
∫
≤≤?
++
= 2
2
2
)22(
1sin2
cos
)(
π
π
xdt
txx
tx
xy D°L
?m? ?0,2,2 ==?= yxx xàè7?￥ ?88
3 
∫
++
= 2
2
2
1sin2
cos
)(
π
π
dt
txx
tx
xy 
( )
∫
++
++
= 2
2
2
2
1sin2
1sin2
2
1
π
π
txx
txxd

111sin2
2
2
2
+=++=
xxtxx
π
π
≥
<<?
≤?
=
12
112
12
x
xx
x

# 1 f
D°L)(xy 0,2,2 ==?= yxx 
?m? ? xàè7?￥ ?88??
}f
9
b
￡
!·° Iù
T? www.tsinghuatutor.com - 12 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
()πππ
3
32
2)2(22
2
1
21
0
22
0
2
=+==
∫∫∫
dxdxxdxyV è
!
)(xf 
 ?? O),( +∞?∞
 )()(,0)( tftfxf =?> 
∫
=
a
a
dttftxxF )(||)( 

￡
ü? ],[ aax?∈
H )(xF
′
??9 

 1′
Hx )(xF |Kl′ 

?ü )(xF ￥Kl′:1 a￥f
H
k p1)(
2
aaf )(xf 
3 
 ],[ aax?∈
∫∫
+?=
a
x
x
a
dttfxtdttftxxF )()()()()(

∫∫∫∫
+?=

a
x
a
x
x
a
x
a
dttfxdtttfdtttfdttfx )()()()(
)(xF
′
∫∫
∫∫
+=
++=
x
a
x
a
a
x
x
a
dttfdttf
dttfxxfxxfxxfdttfxxf
)()(
)()()()()()(
],[,0)(2)()()( aaxxfxfxfxF?∈>=+=
′′

# )(xF
′
??9

 
∫∫
+=
′
x
a
x
a
dttfdttfxF )()()(
∫∫
+=
x
a
x
a
dttfuduf )()()( 
∫∫∫
=+?=
x
x
x
a
x
a
dttfdttfdttf )()()( 
y1 0)( >xf  0)( =
′
xF μ·B3 0=x ? 0)0( >
′′
F ?
^0=x
)(xF ￥l′?
￡
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L 1101 è 62781785
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￡
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L 1101 è 62781785

 1)()(2)0(
2
0
==
∫
aafdtttfF
a
1)0( =f b p?
¤?B¨L?Z?
a
aafaaf 2)()(2?
′
= 
∫
=+=
1)2()(
2
22
a
CeCdaaeeaf
aa

? 2,1)0( == Cf ¤? b12)(
2
=
x
exf
è 
! )(xf 
=μ?l O),( ba ),(
0
bax ∈ ) V?b

@Hq
}{},{
nn
yx
00
limlim,xyxbyxxa
n
n
n
nnn
==<<<<
∞→∞→
b
k p
nn
nn
n
xy
xfyf
∞→
)()(
lim b
3 
ü à
Tí kl￥
?
à
QKDí kl￥1"
?  )￥ V±?i O)(xf
0
x
0
limlim xyx
n
n
n
n
==
∞→∞→
?
^
)())(()()(
0000
xxoxxxfxfxf
nnn
+?
′
+=,
)())(()()(
0000
xyoxyxfxfyf
nnn
+?
′
+=#
nn
nn
n
xy
xfyf
∞→
)()(
lim
+
′
=
∞→
nn
n
nn
n
n
xy
xxo
xy
xyo
xf
)()(
)(lim
00
0
,
?y1 byxxa
nn
<<<<
0

[¤?
0
00
)()(
0
xy
xyo
xy
xyo
n
n
nn
n
≤
≤
￡
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L 1101 è 62781785
2005
￡
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L 1101 è 62781785
0
00
)()(
0
xx
xxo
x
xy
xxo
n
n
nn
n
≤
≤ 
[ )(
)()(
lim
0
xf
xy
xfyf
nn
nn
n
′
=
∞→
b
è  pK
x
dttt
x
x
∫
+∞→
0
])[(
lim  ? V
U??V t￥Kv?
b][t
3 I
n sv￥ 1,+<< nxnx
Hμ
=?
∫
x
dttt
0
])[( +?
∫
1
0
])[( dttt +?
∫
2
1
])[( dttt
+?
∫
n
n
dttt
1
])[(L
∫
x
n
dttt ])[(
+?=
∫
x
n
dttt ])[( ∑
∫
=
n
k
k
k
dttt
1
1
])[( 
7 )1( +?= ktu  5μ dtdu =
∫
k
k
dttt
1
])[(
∫
+?=
1
0
)]1(1[ dukku
2
1
1
0
==
∫
udu
7
∫∫∫
=<=?
1
00
2
1
])[( uduududttt
nxx
n

yN
22
1
])[(
2
0
n
dttt
n
x
+<?<
∫

?
^ )
22
1
(
1
])[(
)1(2
0
n
nx
dttt
n
n
x
+<
<
+
∫

?C/? ?¤?
2
1
])[(
lim
0
=
∫
+∞→
x
dttt
x
x
b

è 
! ?? O)(xf 1)0(?=f 
￡
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L 1101 è 62781785
2005
￡
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L 1101 è 62781785
1cos)(cos2
0
=
∫
duuuf
π

5?
H0>x =
∫
])([
0
22
dt
x
tx
f
dx
d
x

C b
(A) 0b (B) xxf cos)(cos b
(C) 1b (D) -1b
3,
7,5μ uxt sin=
1cos)(cos)( 2
00
22
==
∫∫
π
uduxufdt
x
tx
f
x
s? 
Cb
è 
! 0>a )(xf  ],[ aa +?
μ= ¨ ???
 O 0)0( =f 

 ￥{ -BHSBOHF?[￥B¨
X ?

T
Tb)(xf

￡
ü ],[ aa +?
à
iB? η
P¤
∫
=
′′
a
a
dxxffa )(3)(
3
η b
<pSE>?
ü à
T¤
2
0
)("
2
1
)0(')( xxfxfxf += 
? ),(
0
aax?∈ b
HVa? ? s¤a
∫
=
a
a
xfadxxf )("
3
1
)(
0
3
b
<3>
 ],[ aax?∈? μ
22
!2
)(
)0(
!2
)(
)0()0()( x
f
xfx
f
xffxf
ξξ
′′
+
′
=
′′
+
′
+=
? ξ M

 -Wbx,0
 
?

T
H |s¤?
￡
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L 1101 è 62781785
2005
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L 1101 è 62781785
∫∫ ∫

′′
+=
a
a
a
a
a
a
dxf
x
xdxfdxxf )(
2
)0()(
2
ξ
∫
′′
=
a
a
dxfx )(
2
1
2
ξ 
?? )(xf
′′

 ??yN],[ aa +? )(xf
′′
 ],[ aa +?

iKvKl′ Mm,
P Mxfm ≤
′′
≤ )( b
?
^?s′? ? V¤?
∫∫
≤
aa
a
xdxxfdxxm
0
2
)(
∫∫
≤
′′
=
aa
a
dxxMdxfx
0
22
)(
2
1
ξ 
è  pK dx
x
nx
n
n
∫
+
∞→
1
0
1
sin1
lim 
<3>  dx
x
nx
n
∫
+
1
0
1
sin1

∫∫
+
+
+
=
+
=
1
0
2
1
0
1
0
)sin1(
cos
sin1sin1 x
xdxx
x
x
x
dx
nnn

: dx
x
xx
I
n
n
∫
+
=
1
0
2
)sin1(
cos
5
1
1
0
1
0
+
=≤<
∫
n
dxxI
n
n

yN
n
n
n
n
Idx
x
nx
∞→
∞→
+
+
=
+
∫
lim
1sin1
1
sin1
lim
1
0
1

1sin1
1
+
= b
￡
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L 1101 è 62781785
2005
￡
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L 1101 è 62781785
è  
! )(xf  ),( +∞?∞
 V? 50>a
=+
∫
→
dttftf
α
α
α
αα
α
)]()([
4
1
lim
2
0
<#>b
"
)2( αf
′
b #
)0(f
′
b
$
)(αf
′
b %
)0(
2
1
f
′

3?
ArE5
 dttftf
∫
→
+
α
α
α
αα
α
)]()([
4
1
lim
2
0

[]
α
α
α
α
α
8
)()(
lim
0
2
2
0
0
∫∫
→
=
dxxfdxxf
d
d
)0(
8
)2(2)2(2
lim
0
f
ff
′
=

=
→
α
αα
α



è ü
+
→ 0x
H￥í kl

dttdttdtt
xxx
∫∫∫
===
0
3
00
2
sin,tan,cos
2
γβα 
P
a
￥
^ -B?￥ú¨í kl5? ?￥
Q?
^
(A) γβα,,,(B) βγα,,,
(C) γαβ,,,(D) αγβ,,,[ B ]
}3 ~
0
cos
2tan
lim
cos
tan
limlim
2
0
0
2
0
00
2
=
==
+++
→→→
∫
∫
x
xx
dtt
dtt
x
x
x
xx
α
β
 V
" (C),(D)
ê[
6?
￡
!·° Iù
T? www.tsinghuatutor.com - 18 - bvD ? S
L 1101 è 62781785
2005
￡
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TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
xx
x
x
dtt
dtt
x
x
x
xx
tan2
2
1
sin
lim
tan
sin
limlim
2
3
0
0
0
3
00
2
==
+++
→→→
∫
∫
β
γ
∞==
+
→
2
0
lim
4
1
x
x
x
 Vn γ
^1 β?¨￥í kl
#?ê (B),
è 
! 
<?
=
>
=
0,1
0,0
0,1
)(
x
x
x
xf
∫
=
x
dttfxF
0
)()( 5
(A) F(x) x = 0?? ??,
(B) F(x) (?∞,+∞)
= ??? x = 0?? V?,
(C) F(x) (?∞,+∞)
= V? O
@ )()( xfxF =
′
,
(D) F(x) (?∞,+∞)
= V???B?
@ )()( xfxF =
′
,[ B ]
}ps ~5 ps
f
)(xf ￥MKs )
f
F(x)
￥ ???D V??' Vb
∫
=
x
dttfxF
0
)()(
}? ?s ~ )(xf V F(x) ?? )(xf μ?B ?W? F(x)? V?#ê (B),
}?31￥W3 ~
? x < 0
H  xdtxF
x
=?=
∫
0
)1()(
? x > 0
H ? x = 0
HF(0) = 0,' F(x) = |x| xdtxF
x
==
∫
0
1)(
A ? F(x) (?∞,+∞)
= ??? x = 0?? V?,#ê (B),
è 
! )(xf  ],0[ β
 ?? O??h
 ? βα <<0 ￡
ü
 b
∫∫
<
αβ
βα
00
)()( dxxfdxxf
<￡> 
ZE 
∫∫
αβ
βα
00
)()( dxxfdxxf 
∫∫∫
+=
αβ
α
α
βαα
00
)()()( dxxfdxxfdxxf
￡
!·° Iù
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L 1101 è 62781785
2005
￡
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L 1101 è 62781785
)()()()(
21
ξβααξβαα ff= 
0)]()()[(
21
<= ξξβαα ff 

ZE  ],[ βα∈?x 
7
0)(,)()()(
00
=?=
∫∫
αα
α
FdttfxdttfxF
x

∫
=
′
α
α
0
)()()( dttfxfxF
),(,0)()( αξξαα ofxf ∈<?= 

ZE 
∫
β
0
)( dxxf | uWMD xt
β
α
= 
5 dxdt
β
α
=  O 1>
α
β
?
^
∫∫
αβ
βα
00
)()( dxxfdxxf 
∫∫
=
αα
β
α
β
β
00
)()( dxxfdt
t
f 
∫∫
=
αα
β
α
β
β
00
)()( dxxfdt
t
f 
0))()((
0
<?=
∫
α
α
β
β dttf
t
f b
è 
! 0>a )(xf  ],[ aa +?
μ=¨ ???
 O 0)0( =f 

 )(xf ￥{ -BHSBOHF?[￥B¨
X ?

T
Tb

￡
ü
à
iB?],[ aa +? η
P¤
∫
=
′′
a
a
dxxffa )(3)(
3
η b
<3>
 ],[ aax?∈? μ
22
!2
)(
)0(
!2
)(
)0()0()( x
f
xfx
f
xffxf
ξξ
′′
+
′
=
′′
+
′
+=
? ξ -Wbx,0
￡
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L 1101 è 62781785
2005
￡
!·°
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ê?7<v0 1006 bvD ? S
L 1101 è 62781785

2?

T
H |s¤?
∫∫ ∫

′′
+=
a
a
a
a
a
a
dxf
x
xdxfdxxf )(
2
)0()(
2
ξ
∫
′′
=
a
a
dxfx )(
2
1
2
ξ 
?? )(xf
′′

 ??yN],[ aa +? )(xf
′′
 ],[ aa +?

iKvKl′ Mm,
P Mxfm ≤
′′
≤ )( b
?
^?s′? ? V¤?
∫∫
≤
aa
a
xdxxfdxxm
0
2
)(
∫∫
≤
′′
=
aa
a
dxxMdxfx
0
22
)(
2
1
ξ 
'μ Mdxxf
a
m
a
a
≤≤
∫
)(
3
3

 )(xf
′′

?¨ ??f
￥ o′? ? 5],[ aa +? ],[ aa +?
 à
i B
? η
P¤
∫
=
′′
a
a
dxxffa )(3)(
3
η 
?i ?/pSEb?
ü à
T¤
2
0
)("
2
1
)0(')( xxfxfxf +=  ? ],[
0
aax?∈ b
HV -a ? a s¤
∫
=
a
a
xfadxxf )("
3
1
)(
0
3
b
pey  -W OD μ1b
0
x x,0 x

￡
!·° Iù
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L 1101 è 62781785