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6; I
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4
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6; I
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T? www.tsinghuatutor.com è 82378805
5
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!·° Iù￡?$? bê
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6; I
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T? www.tsinghuatutor.com è 82378805
6
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6; I
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7
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!·° Iù￡?$? bê
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6; I
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!·° Iù￡?$? bê
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6; I
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9
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!·° Iù￡?$? bê
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fxf
x
xf
xx
′==
=
→→

[s?1
$ b

? cef
D ??s 
??sDef

? ?sDef
￥?l
?l 
^?l  uW)(xf RI ∈
￥f

 ? i ?l I
 ￥ V?f


P¤
5? 1 
)(xF
IxxfxF ∈?=′ ),()( )(xF )(xf I
 ￥B?ef
b
 ? 1 )(xF )(xf I
￥B?ef

 CxF +)( 91 )(xf I
￥ef


? 1 ?iè
]" V[ ￡
ü
 ￥ ? i
?ef
￥μ1è
C )(xf
?l ? ￥
μef
 ?￥")(xf { }CxF +)( 1 ￥ ??s
:T

)(xf
CxFdxxf +=
∫
)()(
? 1 )(xF )(xf I
￥B ?ef
 1 ?iè
bC
??  si￥ sHqA1Hq
? ?  ? ?f
B?i??sb

Y
L

 ??f
￥M
Ks )(xf
∫
x
a
dttf )( ü
^ ￥ B?ef

y N )(xf
 bCdxxfdxxf
x
a
+=
∫∫
)()(
> ?
 
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 è ? ? ? ￥f
9 V
μef
??s
 è ?f


=
≠?
=
00
0
1
cos
1
sin2
)(
x
x
xx
x
xf 
μef

=
≠
=
00
0
1
sin
)(
2
x
x
x
x
xF b
?
^ ￥ ?= ?W ?′¤ ·￥
^ 0=x )(xf
? ?  ?f
 uW
=μ? B ? W?
5  u W
=
à
μef
b
)(xf ),( ba )(xf ),( ba
è 
! 
<?
≥
=
.0,/1
,0,cos
)(
xxae
xx
xf
x
 ? )(xf R
μef
5 =a 

"
 b #
b $
b
%
?ib
1?
e a
}3 ~ A? ? ??')(xf 0=x 1)(lim
0
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→
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x
b
n 5,1)(lim
0
=
+
→
xf
x
Q? μ
x
ae
xf
x
xx
1
lim)(lim
00
=

→→
1
1
1
lim
1
1
1
lim
00
=
=
+
=

→→
x
a
aa
x
a
e
a
x
x
x

[
 2To? 1=a
H ? ?b
è 
!
<+
≥+
=
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2
1
2
1;0,1
)(
xe
xx
xf
x

5 ￥B?ef
1
#bf

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<?
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=
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1;0,
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2
xe
xxx
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x

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22
1;0,
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1
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1
)(
2
x
x
e
xxx
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x


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1;0,
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1
)(
2
xe
xxx
xF
x

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22
1;0,1
2
1
)(
2
x
x
e
xxx
xF
x

?s 
f
￥? ? ss
9
31- %bef
1 p 
n5 ?? ??
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6; I
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11
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s 
?  Q B?
^ V±f
b
è 
!
>+
≤
+
=
01
2
1
0
1
1
)(
2
xx
x
x
xf
k ?? ￥ B?ef

P)(xf
4
)0(
π
=F 
s? 
>++
≤+
=
0
4
0
4
arctan
)(
2
xxx
xx
xF
π
π

è 
≥
<
=
0
01
)(
xx
x
xf ),( +∞?∞∈x bíef

y 0=t 1? B ?W? 

??s￥9
ZE
Dí E 

?BD íE
X±sZE
?T1'ZE <T

Cxfdxxf +=′
∫
)()(  bCxfdxxxf +=′?′
∫
))(()())((
è 9

∫
+ xbxa
dx
2222
cossin
b
}3 ~?i ? 
C
a
x
aa
x
d
a
x
a
dx
xa
+=
+
=
+
∫∫
arctan
1
)(
)(1
111
2
22 ∫
+ xbxa
dx
2222
cossin

Cx
b
a
abbxa
xad
a
+
=
+
=
∫
tanarctan
1
tan
)tan(1
222
è 
∫
dx
x
x
cos
tan

∫
=
xx
xd
coscos
)(cos
C
x
+=
cos
2
b
è 
∫∫
+
=
dx
axaxaax
dx 11
2
1
22

+
+

=
∫
)(
1
)(
1
2
1
axd
ax
axd
axa
C
ax
ax
a
+
+
= ||ln
2
1

> ?
 
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￡
!·° Iù￡?$? bê
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è 9
∫
)ln1(ln xxx
dx

?i ? Cxdx
x
+=
∫
arcsin
1
1
2

C
a
x
a
x
d
a
x
dx
xa
+=
=
∫∫
arcsin)(
)(1
11
2
22

dx
xx
∫
)1(
1
Cx
x
xd
+=
=
∫
arcsin2
1
2 
?
^
∫
)ln1(ln xxx
dx

∫
=
)ln1(ln
)(ln
xx
xd
∫
=
x
xd
ln1
ln
2 Cx += lnarcsin2 b
? ? dx
xx
x
∫
)1(
arcsin
)(arcsinarcsin2 xdx
∫
= Cx +=
2
)(arcsin 
è 
∫
+ xx
dx
sin)cos2(


∫
+
xx
xd
2
sin)cos2(
)(cos

 )
cos1
)(cos)cos2(
)cos2(
)(cos
(
3
1
2∫∫
+
+
x
xdx
x
xd

 |cos1|ln
2
1
|cos2|ln
3
1
xx +?+  Cx +?+ |cos1|ln
6
1
b
è 
∫
+
+
dx
xx
x
)1)(1(
22
2

∫
+
= dx
x
x
x
)
)1(1
1
(2
2
" C
x
x
+
+
1
)1(
ln
2
2
b
è  dx
x
xxx
∫
2
sincos
C
x
x
+=
sin

> ?
 
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￡
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?=DíE
?
 OCtFdtttf +=′?′
∫
)()())(( )(tx?= μQf

 5

)(
1
xt
=?
CxFdxxf +=
∫
))(()(
1
è 
∫
dxxa
22
)0( >a 
}3 ~
7 )
2
0(,sin
π
≤≤= ttax
tdtadx cos= 
∫∫
=? dtatadxxa coscos
22

∫
= tdta
22
cos 
C
t
t
a
dt
t
a +
+=
+
=
∫
2
2sin
22
2cos1
2
2
Cxax
a
xa
+?+=
22
2
2
1
arcsin
2

μ
∫
+ dxxa
22
 
7)0( >a tax tan= b

∫
dxax
22

7 tax sec= b
è 9
??s
∫

2
1 xx
dx

}3 ~ Z E 
7
tdtdxtx cos,sin ==
dt
tt
t
xx
dx
∫∫
=

cossin
cos
1
2

Cxxx
dt
tt
tt
dt
tt
tt
+=
+
+
=
∫∫
arcsin
2
1
1ln
2
1
)
cossin
sincos
cossin
sincos
(
2
1
2

ZE 

dt
tt
t
I
∫
=
cossin
cos
 dt
tt
ttttt
∫
+?+
=
cossin
coscossinsincos

CxIxx += arcsin1ln
2

íBE 3 I ' V
b
è  p
∫
42
2
xx
dx

}3 ~
7 tx =?42
5
> ?
 
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￡
!·° Iù￡?$? bê
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∫∫
+
=
22
2
)4(
4
42
t
dt
xx
dx

C
t
t
t
++
+
=
2
arctan
4
1
)4(2
2
C
x
xx
+
+
=
4
42
2
42
arctan
4
1

[
ZE? 1μ ? 
s?  sE
∫∫
= )()()()()()( xduxvxvxuxdvxu 
s? sE￥? ¨
è  )(cossin
22
xdxxdxx
∫∫
=
∫
∫
+?=
+?=
xdxxxx
xxdxx
cos2cos
)(coscos
2
22

Cxxxxx +++?= cossin2cos
2

è 
Cxxxdx
x
xxxxdx +?==
∫∫
ln
1
lnln 
?
￥ è5 μ
? 1 1 ?


∫
xdxx
nm
ln nm,
∫
xdxarcsin

∫
xdxarctan


∫
bxdxe
ax
sin
è 
!
x
x
xf
)1ln(
)(ln
+
= 9
∫
dxxf )( b
}3 ~

! 5,ln tx =
t
t
t
e
e
tfex
)1ln(
)(,
+
== 5
∫
dxxf )( 
∫∫
+
++?=+?=

x
xxxx
e
dx
eedee
1
)1ln()1ln( Cexee
xxx
++?++=
)1ln()1ln(
> ?
 
6; I
￡
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T? www.tsinghuatutor.com è 82378805
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2008
￡
!·° Iù￡?$? bê
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dx
e
e
ee
x
x
xx
)
1
1()1ln(
∫
+
++?=

Ceex
xx
+++?=
)1ln()1( b
è 
! B ? ￥ ef
1)(xf
x
xsin
 p dxxfx
∫
′
π
π
2
)( b
}3 ~ )(xf )sincos(
1
)
sin
(
2
xxx
xx
x
=′= 
2
4
)
2
(,
1
)(
π
π
π
π?=?= ff 
dxxfxI
∫
′=
π
π
2
)( 
π
π
π
π
2
2
)()( xxfxxdf ==
∫ ∫
π
π
2
)( dxxf 
π
π
2
)
sin
)((
x
x
xxf?= 1
42
)
4
(
2
1
2
=+=
πππ
π
b
è 9

∫
+
dx
e
xe
x
x
1
b
}3 ~
7 te
x
=+1
 dt
t
t
dxtx
1
2
),1ln(
2
2
=?=


∫∫
=
+
dttdx
e
xe
x
x
)1ln(2
1
2
∫
= dt
t
t
tt
1
2
2)1ln(2
2
2
2
∫
+= dt
t
tt
1
2
22)1ln(2
2
2
∫
+
= dt
tt
ttt
1
1
1
1
24)1ln(2
2
C
t
t
ttt +
+
+=
1
1
ln24)1ln(2
2
C
e
e
eex
x
x
xx
+
+
++
++?+=
11
11
ln21412 
B? V [¨?f
V
U￥??s
> ?
 
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￡
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ê]Z<? B V 609 è 62701055
??s? B? V[¨ ?f
V
U
 è ?
á
ì V[￡
ü
∫
dxe
x
2
B ?? V[
¨ ?f
V
U B?7y
 1￡
üB ???s ? V[¨ ?f
V
U
^Bqd
è ?
4￥
Y?á
ì V[￡
ü μt?? sB? V[ ¨?f
V
U

μ ?s
TsB? V[¨?f
V
U
μ ?s
T
)(
)(
)(
xQ
xP
xR =
 ?
^ [
T)(xP )(xQ
)(
)(
)()(
2
1
xQ
xP
xPxR += b
?
^ [
T

^ Q
?? ￥ [
T μ ?s
T)(
1
xP )(
2
xP )(xQ
)(
)(
2
xQ
xP
V [V
U
?B tKeμ ? s
T￥ e?1 μ ?s
T￥Kes 3


 ￥ BQy 0
P¤μ ?s
T)(xQ )( ax?
)(
)(
2
xQ
xP
s3acμ
ax
A
?￥ Keμ ?
s
T 

 ￥ ×y0
P¤μ ?s
T)(xQ k
k
ax )(?
)(
)(
2
xQ
xP
cμ
() ()
k
k
ax
A
ax
A
ax
A

,,,
2
21
L?￥ Keμ ?s
T

 ￥=Qy0 ?
P¤ μ ?s
T)(xQ qpxx ++
2
04
2
<? qp
)(
)(
2
xQ
xP
μ
qpxx
NMx
++
+
2
?￥ Keμ ?s
T

 ￥ ×=Qy0 () ?
P¤μ ?s
T)(xQ k
k
qpxx ++
2
04
2
<? qp
)(
)(
2
xQ
xP
μ
()()
k
kk
qpxx
NxM
qpxx
NxM
qpxx
NxM
++
+
++
+
++
+
2
2
2
22
2
11
,,,L? ￥Keμ ? s
T
è 
I
1Hq/
dx
xx
pbxax
∫
++
23
2
)1(
^μ ?f

}3 ~y1 
2
21
3
3
2
21
23
2
)1(1)1(?
+
+++=
++
x
B
x
B
x
A
x
A
x
A
xx
pbxax
[? 0
11
== BA
H
> ?
 
6; I
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17
2008
￡
!·° Iù￡?$? bê
ê]Z<? B V 609 è 62701055
dx
xx
pbxax
∫
++
23
2
)1(

^μ ?f
?
H 
23
3
2
2
3
2
2
23
2
)1(
)1()1(
)1(?
+?+?
=
++
xx
xBxAxxA
xx
pbxax
?1 ?"
¤ 
aAA =+?
32
2  bAA =?
32
2 pA =
3

h ? ¤? 
32
,AA 032 =++ pba 
è  p
∫
++
dx
xx
x
22
)1)(1(
2

}3 ~
22222
)1(11)1)(1(
2
+
+
+
+
+
+
+
=
++ x
EDx
x
CBx
x
A
xx
x
??? "
E¤? 
,1,
2
1
,
2
1
===?== EDBCA ?
^
22222
)1(
1
1
1
2
1
1
1
2
1
)1)(1(
2
+
+
+
+
+
+
=
++ x
x
x
x
xxx
x

∫
=
++
dx
xx
x
22
)1)(1(
2
dx
x
x
x
x
x
)
)1(
1
1
1
2
1
1
1
2
1
(
222∫
+
+
+
+
+
+

()
C
x
x
x
x
dx
x
x
dx
x
x
x
dx
+
+
+
+
+
=
+
+
+
+
+
+
=
∫∫∫
)1(2
1
)1(
1
ln
4
1
1
1
1
1
2
1
12
1
22
2
2
2
2

?? μ ?s
T sB? V[ ¨?f
V
U 
?? μ ?s
T  ?
^μ ?s
T),cos,(sin xxR ),( vuR

?? ?D:
2
tan
x
t =
5μ
2
1
2
sin
t
t
x
+
= 
2
2
1
1
cos
t
t
x
+
=  dt
t
dt
2
1
2
+
= ?
^ V | ??μ ? s
T s
11? t￥μ ? s
Tsb 
∫
dxxxR )cos,(sin
> ?
 
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￡
!·° Iù￡?$? bê
ê]Z<? B V 609 è 62701055
è 9
∫
+
+
dx
x
x
cos1
sin1
b
}3 ~:
2
tan
x
t =
5μ
∫∫
+
++
=
+
+
dt
t
tt
dx
x
x
2
2
1
2
2
21
cos1
sin1
C
xx
+?=
2
cosln2
2
tan 
? ?
i?
^
μ￥ ??μ ?s
T￥ s?1¨

ZE
 

 ?? ?D 
? 
)cos,sin( xxR? )cos,(sin xxR?
5 | xt cos= e?Bt b

x
dt
dx
sin
=  ?ec ￥  Qy0|M 1
}Qy0 V7
P
¤
)cos,(sin xxR xsin
xx
2
cos1sin?= d L?1"¤ [3"b 
 ? 
5MD)cos,(sin xxR? )cos,(sin xxR? xt sin=e?B t
 ? 
5MD)cos,sin( xxR )cos,(sin xxR xt tan=e?B t
B? μ )cos,(sin xxR
))cos,sin()cos,(sin[(
2
1
xxRxxR= 
))cos,sin()cos,sin(( xxRxxR+ ))]cos,(sin)cos,sin(( xxRxxR ++ 
è 
∫
+ xx
dx
sin)cos2(

}3 ~ V
7 xt cos= 
∫
+ xx
dx
sin)cos2(
∫
+
=
xt
dt
2
sin)2(

∫
+
=
)1)(2(
2
tt
dt

 è 9

∫
dx
x
xx
232
)1(
arcsin

> ?
 
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￡
!·° Iù￡?$? bê
ê]Z<? B V 609 è 62701055
∫
= )
1
1
(arcsin
2
x
xd 
∫
2
2
1
1
arcsin
x
dx
x
x
 C
x
x
x
x
+
+
1
1
ln
2
1
1
arcsin
2

è 9

∫
+ xx
dx
sin22sin

}3 ~
∫
+ xx
dx
sin22sin
∫
+
=
)cos1(sin2
1
xx
dx


ZE 
7 tx=cos 
x
dt
dx
sin
= 
∫
+ )cos1(sin2
1
xx
dx
∫
+?
=
)1)(1(2
1
2
tt
dt

∫
+?
=
2
)1)(1(2
1
tt
dt
Ct
t
t +?+
+
++?= 1ln
8
1
1
1
4
1
1ln
8
1

dt
ttt
)
1
1
4
1
)1(
1
2
1
1
1
4
1
(
2
1
2
+
+
+
+
=
∫
C
x
x
x
+
+
+
+
=
cos1
cos1
ln
8
1
)cos1(4
1
b

ZE 
∫
+ xx
dx
sin22sin
∫
+
=
)cos1(sin2
1
xx
dx
∫
=
2
cos2
2
cos
2
sin
4
1
2
xxx
dx
∫
=
2
cos
2
tan
2
tan
4
1
2
xx
x
d

2
tan
2
tan
2
tan1
4
1
2
x
d
x
x
∫
+
= C
xx
++=
2
tan
8
1
2
tanln
4
1
2
b
è 5.27 p? ?s
∫
+
dx
x
x
1
arcsin b
}3 ~
∫∫
′
+
+
=
+
dx
x
x
x
x
x
xdx
x
x
1
arcsin
1
arcsin
1
arcsin
∫
+
+
= dx
x
x
x
x
x
)1(21
arcsin
?? CxxCttdt
t
t
dx
x
x
tx
+?=+?=
+
=
+
∫∫
=
arctanarctan
1)1(2
2
2

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∫
+
dx
x
x
1
arcsin?
+
=
x
x
x
1
arcsin Cxx ++arctan


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!·° Iù
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21