2008
￡
!·° Iù￡?$? bê
ê]Z<? BV 609 è 62701055
? ?<ls ¨
8ù5
<l  s￥?l #
l ??
?sù?￥ù5μ ?f
μ? uW
￥s
<l s ù?￥ù 5μ ?f
 í? uW
 ￥s 
?  ?  í?f
μ? u W
￥ s 
?  ? b
?l  ?B ?<ls
!f
)(xf ),[ +∞a
=￥ ?i μ K uW V 
i OK i
5? 
∫
+∞→
A
aA
dxxf )(lim )(xf ),[ +∞a<ls
l ?

<ls 1  ??
l ?
5?<l  s? ?b 
∫∫
+∞→
+∞
=
A
aAa
dxxfdxxf )(lim)(
?l  ? = ?<l  s
!f

=￥ ?i μ K>0 u W V

i O K  i
5 ? 
￥<ls
l ?
<l s1
)(xf ),[ ba
∫?
→
B
a
bB
dxxf )(lim )(xf ),[ ba
∫∫?
→
=
B
a
bB
b
a
dxxfdxxf )(lim)( b
]"á
ì V [?l 
<ls ￥
l ?? 
∫∫
∞→∞?
=
a
AA
a
dxxfdxxf )(lim)(
 b
∫∫ +
→
=
b
A
aA
b
a
dxxfdxxf )(lim)(

l ??￥
 5
?B ?<ls
l ??￥
5
5  ?? B ?<l s
∫
+∞
a
dxxf )(
l ?
5 B ?
l ?

N
H? '
l ?
∫
+∞
a
dxxf )(
∫
+∞
a
dxxf )(
?
l ?7
∫
+∞
a
dxxf )(
∫
+∞
a
dxxf )( Z? ?
H
?<l sHq
l ?
5  ? MKs ? ?μ? 5  B?
l
?b+Y dμf

μ?5 B?
l ? b
),[ +∞a
∫
x
a
dttf )(
∫
+∞
a
dxxf )(
)(xf ),[ +∞a
∫
+∞
a
dxxf )(
5  °¤1? E
dμf
),[),()(0 +∞∈≤≤ axxgxf
 ?
l ?
 B?
l ? ? ? ?

B?? ? 
∫
+∞
a
dxxg )(
∫
+∞
a
dxxf )(
∫
+∞
a
dxxf )(
∫
+∞
a
dxxg )(
> ?
 
6; I
￡
!·° Iù
T? www.tsinghuatutor.com è 82378805 1
2008
￡
!·° Iù￡?$? bê
ê]Z<? BV 609 è 62701055
5  K1? E

! )(),( xgxf ),[ +∞a
=￥ ?iμ K uW V

dμ
 O)(xg λ=
+∞→
)(
)(
lim
xg
xf
x

5

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

? 0=λ
H
<ls
∫
l ?5
l ?
+∞
a
dxxg )(
∫
+∞
a
dxxf )(

? ∞=λ
H
<ls
l ?5
l ?
∫
+∞
a
dxxf )(
∫
+∞
a
dxxg )(
5 
E dx
x
a
p∫
∞+
1
 ?
H
l ??)0( >a 1>p 1≤p
H? ?
yN ?
O
5 
l ?b0)(lim ≥=
+∞→
λxfx
p
x
1>p
∫
+∞
a
dxxf )(
è 
 dx
x
xx
∫
∞+
+
1 5
1
ln
￥
l ?? 
}3~ ? 0
ln
lim
3
=
+∞→
x
x
x
i  
P¤ ?
H0>X 0>> Xx
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 ?? 
}3~D dx
x
∫
∞+
1
2
1
1 ?? K 1?E dx
xxx
x
∫
∞+
++
1 2
1
arctan
l ?
è 

∫
∞+
e
p
xx
dx
2
ln
￥
l ?? 
}3~  ()+∞→→= x
xxx
x
p
p
0
ln
1
ln
22
y N
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 ?
H? ?b1≥p 1<p
?= ?<ls
l ??￥
5
> ?
 
6; I
￡
!·° Iù
T? www.tsinghuatutor.com è 82378805 2
2008
￡
!·° Iù￡?$? bê
ê]Z<? BV 609 è 62701055
5  ? ?= ?< l  s
∫
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 
sHq
l ?
5  °¤1?E
dμf
),[),()(0 baxxgxf ∈≤≤
 ?
l ?
 B?
l ?  ? ? ?
 B
?? ? 
∫
b
a
dxxg )(
∫
b
a
dxxf )(
∫
b
a
dxxf )(
∫
b
a
dxxg )(
5 f
 K1? E
 
=￥ ?i uW
 V

dμ
 O
)(),( xgxf ),[ ba
)(xg λ=
→
)(
)(
lim
xg
xf
bx

5

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

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

? ∞=λ
H
<ls
l ?5
l ?
∫
b
a
dxxf )(
∫
b
a
dxxg )(
5 
 E  dx
bx
b
a
p∫
)(
1
? 1<p
l ?
H? ? y N  ?

O
5 
∫
l ?b
1≥p
0)()(lim ≥=?
→
λxfbx
p
bx
1<p
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? s A ?
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 9
∫
∞+
++
0 22
1)51(
1
dx
xx
b
}3~ | MD
2
1
,tan
t
dt
dxtx
+
== 5
> ?
 
6; I
￡
!·° Iù
T? www.tsinghuatutor.com è 82378805 3
2008
￡
!·° Iù￡?$? bê
ê]Z<? BV 609 è 62701055
∫
+
=
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
}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
}3~
∫∫∫
∞+
=
==
=
1
2
0
2
0
sec
2
2tansec
tansec
1
ππ
π
dtdt
tt
tt
xx
dx
tx

7
t
x
1
= 
¨X±s E 5
∫∫
=
∞+ 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
> ?
 
6; I
￡
!·° Iù
T? www.tsinghuatutor.com è 82378805 4
2008
￡
!·° Iù￡?$? bê
ê]Z<? BV 609 è 62701055
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
? 1,
2007-2-18
'5
s 11
!
^ê? wLD )0,1(
2
+∞≤≤>=
xaaxy
a
x
/Za
à
Z￥í? u×b x

ú p u× ? àèB?
?è8￥8  D x )(aV

?? 1′
H Kl$i pNKl′b a )(aV
}3~ 
ú
a
ax
dxe
a
a
dxxaaV
a
a
x
a
x
ln
ln
)(
0
ln
0
∫∫
∞+?∞+?
== ππ
a
a
x
dex
a
a
ln
0
ln
∞+
∫
=π
∫
∞+?
∞+
+?=
0
ln
2
2
0
ln
ln
lnln
x
a
a
de
a
a
e
a
ax
a
a
x
a
a
x
ππ
a
a
e
a
a
a
a
x
2
2
0
ln
2
2
lnln
0
π
π =?=
+∞

? )(aV′
a
a
a
a
32
ln
2
ln
2 ππ
= 
7 0)( =′ aV ¤·B? ea =  
§M
| O15?aμ
[ 1Kl′b
)(aV ea =
π
2
)( eeV =
¨
ˉ 8ù 5 
?s? l  Iù? ￥ ? ¨ ¨? p +?K
¨? s pKè ¨
T1 
∫
∑
=
+
=
∞→
b
a
n
k
dxxfk
n
ab
af
n
ab
)()(lim
1
n
b
? )(
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

> ?
 
6; I
￡
!·° Iù
T? www.tsinghuatutor.com è 82378805 5
2008
￡
!·° Iù￡?$? bê
ê]Z<? BV 609 è 62701055
K 
n
n
ylnlim
∞→
∑
=
∞→
=
n
k
n
n
k
n
1
ln
1
lim ? ?<ls ￥ ′
∫
1
0
ln xdx
M??| u W sé? ]1,0[ ),,2,1(],
1
[ nk
n
k
n
k
L=
￥ s
T ￥ K
O  s
T? ￥
n
k
f
k
ln)( =ξ b
? i ?<l s 1?= ?<l s i O
l ? ?
^  
∫
1
0
ln xdx
n
n
ylnlim
∞→
∑
=
∞→
=
n
k
n
n
k
n
1
ln
1
lim
∫
=
1
0
ln xdx 1)ln(
1
0
=?= xxx 
[ 
n
n
y
∞→
lim ==
∞→
n
n
n
!
lim
n
e
1
b
?
ZE V [9

π
π
π 2
sinsin
1`
lim
1
0
1
==
∫
∑
=
∞→
xdx
n
k
n
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
}3~
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
1ln2 dxx
∫
=
2
1
ln2 tdt #
b
22ln4)ln(2ln2
2
1
2
1
=?==
∫
ttttdt 
è
!
∫
+
+=
)1(
0
1
1
2
3
nn
nn
n
dxxxa 5 =
∞→
n
n
nalim [ B ]b
(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
nn
nn
n
xdx
n
a ++?=
∫
+
> ?
 
6; I
￡
!·° Iù
T? www.tsinghuatutor.com è 82378805 6
2008
￡
!·° Iù￡?$? bê
ê]Z<? BV 609 è 62701055
=
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
xx
xeef
=′ )( 0)1( =f,5 ( )xf =
2
)(ln
2
1
x,
}s ~ 5 p  )(xf ′ ￥Vr
Ts' Vb
}3 ~
7 5 ?
^μ te
x
= tx ln=
t
t
tf
ln
)( =′  ',
ln
)(
x
x
xf =′
s¤ Cxdx
x
x
xf +==
∫
2
)(ln
2
1ln
)(, ? ¨
SHq ( ) 01 =f,¤ C=0#
p
f
1 = ()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
￥?s 5 | 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 ~ w L y = F(x)1? yà? xàD w L y = F(x)??Bí? u×
[
 S V¨<lsV
U,
? '5?b
> ?
 
6; I
￡
!·° Iù
T? www.tsinghuatutor.com è 82378805 7
2008
￡
!·° Iù￡?$? bê
ê]Z<? BV 609 è 62701055
}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)(
=
(II) ?? # S(t)￥·B ?1
t
ettS
2
)21(2)(
=′
2
1
=t 
? 
t
ettS
2
)1(8)(
=′′ 0
4
)
2
1
( >=′′
e
S 
[
e
S
1
1)
2
1
(?= 1l′
9
^Kl′,
è 
! )(xf ),( +∞?∞
 ?? O )()(,0)( tftfxf =?> 
∫
=
a
a
dttftxxF )(||)( 

￡
ü?
H ??9 ],[ aax?∈ )(xF′

 1′
H |Kl′ x )(xF

?ü ￥Kl′:1 ￥f
H
k p )(xF a 1)(
2
aaf )(xf
}3~ 
 ],[ aax?∈
∫∫
+?=
a
x
x
a
dttfxtdttftxxF )()()()()( 
)(xF′
∫∫
∫∫
+=
++=
x
a
x
a
a
x
x
a
dttfdttf
dttfxxfxxfxxfdttfxxf
)()(
)()()()()()(
],[,0)(2)()()( aaxxfxfxfxF?∈>=+=′′ 
# ??9)(xF′


∫∫
+=′
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
^ ￥l′
?
)(xF
> ?
 
6; I
￡
!·° Iù
T? www.tsinghuatutor.com è 82378805 8
2008
￡
!·° Iù￡?$? bê
ê]Z<? BV 609 è 62701055

 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

? ¤? b2,1)0( == Cf 12)(
2
=
x
exf
è 
! 
=μ?l O)(xf ),( ba ),(
0
bax ∈ ) V?b

@ H
q
}{},{
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
≤
≤
0
00
)()(
0
xx
xxo
x
xy
xxo
n
n
nn
n
≤
≤ 
[ )(
)()(
lim
0
xf
xy
xfyf
nn
nn
n
′=
∞→
b
è  p K
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  5μ )1(= ktu dtdu =
> ?
 
6; I
￡
!·° Iù
T? www.tsinghuatutor.com è 82378805 9
2008
￡
!·° Iù￡?$? bê
ê]Z<? BV 609 è 62701055
∫
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 1cos)(cos
2
0
=
∫
duuuf
π

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

C b
(A) 0b (B) b (C) 1b (D) -1b xxf cos)(cos
}3~
7,5μuxt sin= 1cos)(cos)(
2
00
22
==
∫∫
π
uduufdt
x
tx
f
x
s? 
Cb
è 
!  
μ= ¨ ???
 O0>a )(xf ],[ aa +? 0)0( =f 

  ￥{ -BHSBOHF?[￥B¨
X ?

T
Tb)(xf

 ￡
ü
 à
iB ?],[ aa +? η 
P¤ b
∫
=′′
a
a
dxxffa )(3)(
3
η
}3~ 
 μ],[ aax?∈?
22
!2
)(
)0(
!2
)(
)0()0()( x
f
xfx
f
xffxf
ξξ ′′
+′=
′′
+′+= ? ξ M

 -Wbx,0
 
 ?

T
H | s ¤?
∫∫∫

′′+=
a
a
a
a
a
a
dxf
x
xdxfdxxf )(
2
)0()(
2
ξ
∫
′′=
a
a
dxfx )(
2
1
2
ξ 
?? )(xf ′′ ],[ aa +?
 ? ?yN )(xf ′′  ],[ aa +?

i  K vKl ′ 
PMm,Mxfm ≤′′≤ )( b
?
^? s′? ? V ¤?
∫∫
≤
aa
a
xdxxfdxxm
0
2
)(
∫∫
≤′′=
aa
a
dxxMdxfx
0
22
)(
2
1
ξ
> ?
 
6; I
￡
!·° Iù
T? www.tsinghuatutor.com è 82378805 10
2008
￡
!·° Iù￡?$? bê
ê]Z<? BV 609 è 62701055
'μ  Mdxxf
a
m
a
a
≤≤
∫
)(
3
3

 
 ?¨ ? ?f
￥o ′? ?5 )(xf ′′ ],[ aa +? ],[ aa +?
 à
i B ?
η
P¤ 
∫
=′′
a
a
dxxffa )(3)(
3
η
?i ?/ p SEb ?
ü à
T ¤
2
0
)("
2
1
)0(')( xxfxfxf +=  ? ],[
0
aax?∈ b
HV a? ? s¤a
∫
=
a
a
xfadxxf )("
3
1
)(
0
3
b
pey  -W OD μ1b
0
x x,0 x
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
è 
! 
 V?  5)(xf ),( +∞?∞ 0>a
=+
∫
→
dttftf
α
αα
αα
α
)]()([
4
1
lim
2
0
<#>b
"
)2( αf ′ b #
b $
)0(f ′ )(αf ′ b %
)0(
2
1
f ′ 
}3~?
ArE5
 dttftf
∫
→
+
α
αα
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
H￥í kl

+
→ 0x
dttdttdtt
xxx
∫∫∫
===
0
3
00
2
sin,tan,cos
2
γβα
?
P
a
￥
^ -B?￥ú
¨í kl5? ?￥
Q?
^ [ B ]
(A) γβα,,,(B) βγα,,,(C) γαβ,,,(D) αγβ,,,
> ?
 
6; I
￡
!·° Iù
T? www.tsinghuatutor.com è 82378805 11
2008
￡
!·° Iù￡?$? bê
ê]Z<? BV 609 è 62701055
}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 ?
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
 V n γ
^1 β?
¨￥í kl
#?ê (B),
è 
!  5 [ B ]
<?
=
>
=
0,1
0,0
0,1
)(
x
x
x
xf
∫
=
x
dttfxF
0
)()(
(A) F(x) x = 0?? ??,
(B) F(x) (?∞,+∞)
= ??? x = 0?? V?,
(C) F(x) (?∞,+∞)
= V? O
@ )()( xfxF =′,
(D) F(x) (?∞,+∞)
= V???B?
@ )()( xfxF =′,
}ps ~5 ps 
f
￥MK s )
f
F(x)
￥ ???D V??' Vb
)(xf
∫
=
x
dttfxF
0
)()(
}? ?s ~ V F(x) ?? μ?B ?W? F(x)? V?#ê (B),)(xf )(xf
}?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 )()()()(
21
ξβααξβαα ff= 
0)]()()[(
21
<= ξξβαα ff 

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

∫
=′
α
α
0
)()()( dttfxfxF ),(,0)()( αξξαα ofxf ∈<?= 
> ?
 
6; I
￡
!·° Iù
T? www.tsinghuatutor.com è 82378805 12
2008
￡
!·° Iù￡?$? bê
ê]Z<? BV 609 è 62701055

ZE   | uWMD
∫
β
0
)( dxxf 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
> ?
 
6; I
￡
!·° Iù
T? www.tsinghuatutor.com è 82378805 13