清华名师：微积分课程各章节知识点题解：第6章定积分的概念与计算

分类：数学格式：pdf 日期：2005年11月22日

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dttfxF )()( ],[ ba
())()( xfdttf
dx
d
x
a
=
∫
b
?i B?
^)(xF )(xf

￥B?ef
b
],[ ba
￡
ü
],,[ bax∈?
∫
=
x
a
dttfxF )()( 5
∫
+
=?
xx
x
dttfxF )()(
y1 )(xf
V
y N
μ? '],[ ba )(xf ],[ ba ],,[ bax∈? i
0>M
P¤ Mxf ≤)( ?
^
xMdttfxF
xx
x
≤≤?≤
∫
+
)()(0
? C/? ? V¤ 0)(lim
0
=?
→?
xF
x
yN
∫
=
x
a
dttfxF )()(
??],[ ba
￡
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L 1101 è 62781785
2005
￡
!·°
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L 1101 è 62781785

()
∫
x
a
dttf
dx
d
)(
[]
∫∫
=
+
→?
x
a
xx
a
x
dttfdttf
x
)()(
1
lim
0
[]
∫
+
→?
=
xx
x
x
dttf
x
)(
1
lim
0

∫
+
→?
=
xx
x
x
dt
x
f )(
lim
0
ξ
)()(lim
0
xff
x
==
→?
ξ

T? ξ1 xD xx?+ -W￥B?

￡
ü¨?s?′? ? b
è
! ?dttfxF
x
∫
=
0
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≤<?
≤≤?
=
.20,2
,01,
)(
2
2
xx
xe
xf
x

5
=<%>b)(xF ]2,1[?
"
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#
V?
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? ?? %
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3 )(xf
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μ?B ?W?￥ Vf

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b]2,1[?
M/Ks μ]"￥?é
∫
b
x
dttf )(
? ?
? )(xf
V
5M /Ks],[ ba
∫
=
b
x
dttftF )()(
^
??f

? )(xf
? ?
5 M/K s
^ V ?f
b
O
],[ ba
∫
b
x
dttf )(
￡
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L 1101 è 62781785
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￡
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L 1101 è 62781785
())()( xfdttf
dx
d
b
x
=
∫

ˉMKs
∫
)(
)(
)(
x
x
dttf
β
α

? ?
? )(xf
V
],[ ba )(),( xx βα
??5
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],[ ba
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)(
)(
)(
x
x
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α
^ ??f
b

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?? ],[ ba )(),( xx βα
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5
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∫
)(
)(
)(
x
x
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β
α
^ V?f

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1
() )())(()())(()(
)(
)(
xxfxxfdttf
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x
x
ααββ
β
α
′
′
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1?
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^
0)(
0
=
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dxxf ? 0>T 1?
ùb
2
μ?B ?W?￥f
àμef
b
2
μ?= ?W?￥f
V[μef
b
2
MKsV
U￥f
?B?
^ef
b

è￡
ü ??
}f
-ef
A1f
Dè
-b
<￡>:
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=
x
a
dttfxF )()(
1￡if
P¤)(xG CxGxF += )()(
? )()( xfxF =
′
C1è
b
￡
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L 1101 è 62781785
2005
￡
!·°
TDn bê
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L 1101 è 62781785
! 1￡
∫
=
x
dttfxG
0
)()( )(xG
@
)()( xGxG?=?
A? CxGxF += )()( ? )()( xfxf =? yN
)( xG?
∫
=
x
dttf
0
)(
7 tu?= ?
^¤?
∫
=?
x
udufxG
0
)()()(
∫
=
x
uduf
0
)()( )(xG?= b
è
! )(xf
^ ? ?￥?
ù1 T ￥f

￡
ü
∫∫
=
+ TaT
a
dxxfdxxf
0
)()(
￡
ü￡EBy1
^ ??f

| T1M

)(xf a
0)()()( =?+=
∫
+
afaTfdxxf
da
d
aT
a

∫
+aT
a
dxxf )(1?M
1è

7?a 0=a
H

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TaT
a
==
∫∫
+
0
)()(

yN Cdxxfdxxf
TaT
a
==
∫∫
+
0
)()( ?i￥ (? ?a
￡E=

∫
+aT
a
dxxf )(
∫∫∫
+
++=
aT
T
T
a
dxxfdxxfdxxf )()()(
0
0

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∫∫∫
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++=
T
aT
a
dxxf
dxTxfdxxfdxxf
0
00
0
)(
)()()(

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^ ??f

￡E=?1p )(xf
^ Vf

è
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^[ T 1?
ù￥ ??f

￡
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A1?
ùf
DL?f
bax+ -O?
ù?Mb

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ùf
)(xf -ef
1?
ùf
￥ 1Hq
^
0)(
0
=
∫
T
dxxf ? 0>T 1?
ùb
￡

! )()( xfTxf =+
￡
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L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
CdttfxF
x
+=
∫
0
)()(
o3￡i Rba ∈,D[ T 1?
ù￥ ??f
)(xg
P¤
baxxgxF ++= )()(
CaxaxdttfxF
x
+?+=
∫
0
)()(
Caxdtatf
x
++?=
∫
0
))((
| Cb=
∫
=
x
dtatfxg
0
))(()(
baxxgxF ++= )()( b
o3
7 )()( xgTxg =+
p Ra ∈,' Vb
∫
+
=+
Tx
dtatfTxg
0
))(()(
∫∫
+
+?=
Tx
x
x
dtatfdtatf ))(())((
0

aTdttfxg
T
+=
∫
0
)()(
7 3¤0)(
0
=?
∫
aTdttf
T
Rdttf
T
a
T
∈=
∫
0
)(
1
b

?X?Hq¤? b[
￡
üV? V
Ib0=a

è
! dttxf
x 2cos1
0
sin)(
∫
=
65
)(
65
xx
xg += 5?
H0→x )(xf
^
)(xg ￥
b
"
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b #
ú¨íkl
b
$
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b %
]¨?d?Níkl

s? #

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F
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L 1101 è 62781785
2005
￡
!·°
TDn bê
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L 1101 è 62781785
5
dttf
dttgtf
x
x
x
)(
)()(
)(
0
0
∫
∫
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),0[ +∞ ),0[ +∞
??h

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??h
b
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2
0
00
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)()()()()()(
)(
dttf
dttgtfxfdttfxgxf
x
x
xx
∫
∫∫
=
′

[]
,0
)(
)]()()[()(
2
0
0
>
=
∫
∫
dttf
dttgxgtfxf
x
x

[ )(x?
??9F
s? "
),0[ +∞
è
!,0)( >xf [ ]ba,
??
￡
üf
[]
∫∫
+=
x
b
x
a
dttfdttfxF
1
)()()(
[
μM?μB?
,?b]ba,
￡
ü,0)( >xf [ ]ba,
??
5 )(xF [ ]ba,
??O V?
0
)(
1
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′
xf
xfxF
)(xF [ ]ba,
??9?y1
0)()(,0
)(
1
)( >=<=
∫∫
b
a
a
b
dttfbFdt
tf
aF
? ??f
,?? ? V?
)(xF [ ]ba,
μO?μB?
,?b
è ??è
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Pcba,,
￡
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L 1101 è 62781785
2005
￡
!·°
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L 1101 è 62781785
0
)1ln(
sin
lim
3
0
≠=
+
∫
→
c
dt
t
t
xax
x
b
x

<3>
n5?s
K1
,Aμ
0
)1ln(
lim
3
0
=
+
∫
→
x
b
x
dt
t
t
b?
^ 0=b b
Q
∫
+
→
x
b
x
dt
t
t
xax
)1ln(
sin
lim
3
0

)1ln(
)cos(
lim
)1ln(
cos
lim
3
0
3
0
x
xax
x
x
xa
xx
+
=
+
=
→→
0
cos
lim
2
0
≠=
=
→
c
x
xa
x

Aμ
2
1
,1 =?= ca b
è
! )(xf
V?),0[ ∞+ 0)0( =f Qf
1 )(xg
p
xxf
exdttg
2)(
0
)(
∫
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<3>??
T
§1?
∫
=
)(
0
2
)(
xf x
exduug
xp?
?i? xxfg =))(( ¤?
,2)(
2 xx
exxexfx +=
′
K? 0≠x 5μ

xx
xeexf +=
′
2)(
s¤? C)1()( ++=
x
exxf ? 0)0( =f #
)0(1)(lim
0
fCxf
x
=+=
+
→
3 1?=C ?
^
b1-)1()(
x
exxf +=
è
! )(xf 1 ??dμf

μv? ￥è
C? #bx≤≤1
￡
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L 1101 è 62781785
2005
￡
!·°
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L 1101 è 62781785
)(0 xfy ≤≤ ??u×￥
1 21
2
+b p )(xf b
<3>? 21)(
2
1
+=
∫
bdxxf
b
¤
21)(
2
1
+=
∫
xdttf
x

|

T
H p?¤x
1
)(
2
+
=
x
x
xf b
è X ? )(xf ),( +∞?∞
1
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O
dttfttxF
x
∫
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0
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^<">)(xF
"a
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#af
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ùf
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￥s
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[y0?
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￥y0M?s|?

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￥?sM?s|?
5??M
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è =?
∫
x
dttx
dx
d
0
2
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<3>
7 dudtutx?==?,2 5
∫∫
=?=
x
x
x
x
duuduuxI
2 2
2
2
coscos)( b
=
′
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22
cos)2cos(2 xx? b
è
! )(xf ??X? 1)1( =f O
2
0
arctan
2
1
)2( xdttxtf
x
=?
∫

p b
∫
2
1
)( dttf
<3>
n5) ??

7 dudttux?==?,2 ?
^
∫
x
dttxtf
0
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=
x
x
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2
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∫
=
x
x
duufx
2
)(2
∫
x
x
duuuf
2
)(
2
arctan
2
1
x=
￡
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L 1101 è 62781785
2005
￡
!·°
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L 1101 è 62781785

p?
¤
)(2)2(4)(2
2
xxfxxfduuf
x
x
+
∫

4
1
)()2(4
x
x
xxfxxf
+
=+?
'μ
4
2
1
)()(2
x
x
xxfduuf
x
x
+
+=
∫

7 ?i?1=x 1)1( =f ¤?
4
3
)(
2
1
=
∫
dttf b
è
! )(xf ),( +∞?∞
??:
∫
=
x
dttftxxF
0
)()2()(
k￡

? )(xf 1
}f
5 9
^
}f
)(xF

? )(xf ???95 ???hb)(xF
<￡>
! )(xf 1
}f
I
n
∫
=?
x
dttftxxF
0
)()2()(
7 dudtut?=?=,5μ
∫
=+=?
x
xFduufuxxF
0
)()()2()( # 1
}f

)(xF

! )(xf ???9 I
n
[]
′
=
′
∫∫
xx
dtttfdttfxxF
00
)(2)()(
)(2)()(
0
xxfxxfdttf
x
∫
+=

∫
=
x
xxfdttf
0
)()(
)]()([ xffx?= ξ
? ξ D -W?
^0 x
?
H0>x 0)(,0)()( ≥
′
≥? xFxff ξ
?
H0<x 0)(,0)()( ≥
′
≤? xFxff ξ
?
H yN0=x 0)0( =
′
F ),( +∞?∞
μ 0)( ≥
′
xF b
[ ???hb)(xF
￡
!·° Iù
T? www.tsinghuatutor.com - 20 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
è X ? )(xf ??O A
x
xf
x
=
→
)(
lim
0

!
p
∫
=
1
0
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′
i)
)(xF
′
￥ ???b
<3>
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Q
7 0,,≠== x
x
du
dtxtu 5μ

=
≠
=
∫
00
0)(
1
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0
x
xduuf
x
xF
x

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∫
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′
x
duuf
xx
xf
xF
0
2
)(
1)(
)(
1 ??f
b

22
)(
lim
)(
lim)0(
0
2
0
0
A
x
xf
x
duuf
F
x
x
x
===
′
→→
∫

=
≠?
=
′
∫
0
2
0,)(
1)(
)(
0
2
x
A
xduuf
xx
xf
xF
x

uduf
xx
xf
xF
x
xxx
∫
→→→
=
′
0
2
000
)(
1
lim
)(
lim)(lim
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22
F
AA
A
′
==?=
yN )) ??b )(xF
′
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è
!
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1X? V?f

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1
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￥Qf

5
=?
∫
)(
)(
xfx
x
dtxtxg
dx
d
"
￡
!·° Iù
T? www.tsinghuatutor.com - 21 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785

" )()(
2)(
0
xfxdttg
xf
′
+
∫

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2)(
0
xfxdttg
xf
′
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0
xfxdttg
xf
′
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0
xfxdttg
xf
′
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xfx
x
xfx
x
dtxtgxdtxtxgxI

T
?￥s
7 dudtuxt ==?,
∫
=
)(
0
)()(
xf
duugxxI
))())((()()(
)(
0
xfxfxgduugxI
xf
′
+=
′
∫

))())((()(
)(
0
xfxfxgduug
xf
′
+=
∫

)()(
2)(
0
xfxduug
xf
′
+=
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"

è =
+
∫
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x
x
t
x
dt
e
t
2
1
ln
lim b
s? b0 0>?X
P? 0>> Xx
H
x
x
x
t
e
xx
dt
e
t
+
≤
+
≤
∫
1
2ln
1
ln
0
2

?¨C/? ?¤? 0
1
ln
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2
=
+
∫
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t
x
dt
e
t
b
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c
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dxxaf
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0
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￡
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T? www.tsinghuatutor.com - 22 - bvD ? S
L 1101 è 62781785
2005
￡
!·°
TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
<3>?+il V¤?ê %
wL )(xfy =1?? ?b)0()0,( ≠aa
[/
^+?¨?uWMD￥ è5b
è X?
￥ ??wL1?°L],[ ba )(xfy =
2
ba
x
+
= ?

￡
ü
∫∫
+
= 2 )(2)(
ba
a
b
a
dxxfdxxf
<￡>
∫∫∫
+
+
+=
b
ba
ba
a
b
a
dxxfdxxfdxxf
2
2 )()()(
??1?°L)(xfy =
2
ba
x
+
= ?
aBsμ
∫
+
b
ba
dxxf
2
)(
∫
+
+=
b
ba
dxxbaf
2
)(
7 xbat?+= 5 dtdx?= ?
^¤?
∫∫
++
=?+
a
ba
b
ba
dttfdxxbaf
22
)()(
∫
+
=
2
ba
a
dttf )(
[
∫∫
+
= 2 )(2)(
ba
a
b
a
dxxfdxxf b
è p ￥KvKl′bdtetxf
x t
∫
=
2
0
)2()(
<3> )(xf 1
}f
o3p ),[ +∞0
￥KvKl′b
7 2,0)2(2)(
2
2
==?=
′
xexxxf
x
1·B?O
? 20 << x
H,0)( >
′
xf
? 2>x
H,0)( <
′
xf yN 2=x 1v′?'Kv′?b
Kv′1
22
0
1)2()2(

+=?=
∫
edtetf
t
b
=
+∞→
)(lim xf
x
dtet
t
∫
∞+?
0
)2(
112)2(
00
=?=+=
∞+
∞+
tt
eet
? 0)0( =f yN 1Kl′?Kl′1 b0=x

è
! )(xf
V±],[ ba )(xf
′
dh￡
ü
￡
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￡
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L 1101 è 62781785
)]()([
2
)( bfaf
ab
dxxf
b
a
+
≤
∫
b
￡
!],[ bax∈?
∫
+
=
x
a
dttfxfaf
ax
xF )()]()([
2
)(

0)( =aF
5o?ù? ￥???
￡
ü b)(xF 0)()( =≥ aFxF
)()(
2
)]()([
2
1
)( xfxf
ax
xfafxF?
′
++=
′
)()(
2
)]()([
2
1
xfxf
ax
xfaf?
′
+?=
)(
2
)(
2
)(
xf
ax
f
xa
′
+
′
= ξ
)]()([
2
ξfxf
ax
′
′
=
),( xa∈ξ
)(xf
′
dh
? 0)( ≥
′
xF

0)()( =≥ aFxF b
è￡
ü
∫∫
+
≤
+
2
0
2
0
22
1
cos
1
sin
ππ
dx
x
x
dx
x
x

<￡>
ZE
∫∫
+
+
+
= 2
4
2
4
0
2
1
cossin
1
cossin
π
π
π
dx
x
xx
dx
x
xx
I

?=?s
7 xt?=
2
π
5μ
￡
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L 1101 è 62781785
∫
+
= 2
0
2
1
cossin
π
dx
x
xx
I
∫∫
+
+
+
= 4
0
2
4
0
2
)
2
(1
sincos
1
cossin
ππ
π
dt
t
tt
dx
x
xx

∫
+
+
= 4
0
2
2
]
)
2
(1
1
1
1
)[cos(sin
π
π
dt
t
t
tt
0
)
2
(1)1(
)
4
)(cos(sin
4
0
22
2
<
++

=
∫
π
π
π
π
dt
tt
ttt

ZE
∫
+
= 2
0
2
1
cossin
π
dx
x
xx
I
∫∫
+
+
+
= 2
4
2
4
0
2
1
cossin
1
cossin
π
π
π
dx
x
xx
dx
x
xx
I
∫
+
= 4
0
2
1
)cos(sin
1
1
π
ξ
dxxx
∫
+
+ 2
4
2
2
)cos(sin
1
1
π
π
ξ
dxxx
? )
2
,
4
(),
4
,0(
21
ππ
ξ
π
ξ ∈∈

?= ?s
7 xt?=
2
π
5
μ
￡
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L 1101 è 62781785
∫
+
+
= 4
0
2
2
2
1
)cos(sin)
1
1
1
1
(
π
ξξ
dxxxI
0)21)(
1
1
1
1
(
2
2
2
1
<?
+
+
=
ξξ

è Z? 01
0
cos0
2
2
=++
∫∫
x
tx
dtedtt ￥?￥?
1 #
b
"
#
b $
b %
b
3A
! dtedttxf
t
x
x
2
0
2
cos
2
0
1)(
∫∫
++=
π

μ 0)0(
0
1
2
<=
∫
dtef
t
01)1(
1
0
2
>+=
∫
dttf
[ )1,0(∈?ξ
P 0)( =ξf
? 0
2
sin
2
1)(
2
cos
2
2
>?++=
′
x
e
x
xxf
π
ππ

[μO ?μ
B??b
è
! 1>k
f ]1,0[
??
= V?O
@)1,0(
0)()1(
1
0
1
==
∫
dxxfxekf
k
x

￡
üà
iB? )1,0(∈ξ
P¤ )()1()(
1
ξξξ ff
=
′
b
<￡>?s?′? ?¤ )()1(
1
ξξ
ξ
fef
= )
1
,0(
ξ
ξ∈
! )()(
1
xfxex
x?
=? 5 )(x? ]1,[
1
ξ
?? )1,(
1
ξ
= V?
O )1()1()(
1
f==?ξ?
yNi )1,0(),1(
1
∈ ξξ
P¤
￡
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L 1101 è 62781785
0)()()()(
111
=
′
+?=
′

ξξξξξξ?
ξξξ
fefefe
?
^0
1
≠
ξ
e 0)()()( =
′
+? ξξξξξ fff
[ )()1()(
1
ξξξ ff
=
′
b

è
! )(xf ]
2
,0[
π
??O
@
0)(cos
2
0
2
=?
∫
dxxfx
π

￡
ü

? )(xf )
2
,0(
π
= V?5i )
2
,0(
π
ξ∈
P¤
ξξξ tan)(2)( ff =
′

? )(xf
′
)
2
,0(
π
= ?? 5i )
2
,0(
π
ξ∈ D )
2
,0(
π
η∈
sY
P¤
ξξξ tan)(2)( ff =
′
D ηηη tan)()( ff =
′

<￡>

n5? 0)(cos2
0
2
=?
∫
dxxfx
π
5 )
2
,0(
0
π
∈?x
P¤
?0)(cos
00
2
=xfx 0)(,0cos
00
=?≠ xfx b
6??s?′? ?¤?|￡ùf

)(cos)(
2
xxfx =? 5 )(x? ]
2
,0[
π
? ? )
2
,0(
π
= V
?O 0)
2
(,0)(
0
==
π
x
yN )
2
,0()
2
,(
0
ππ
ξ?∈? x
P¤
￡
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L 1101 è 62781785
0)(cos)(cossin2)(
2
=+?=
′
ξξξξξξ? ff
'μ ξξξ tan)(2)( ff =
′

n5?s?s
=?
∫
dxxfx )(cos2
0
2
π
0))(cos)(sincos2(
2
2
0
=
′
+
∫
dxxfxxfxxx
π
?$f
￥ ???5i )
2
,0(
π
ξ∈
P¤
0))(cos)(sincos2(
2
=
′
+ ξξξξξξ ff Q
,0cos ≠ξξ Aμ
0))(cos)(sin2 =
′
ξξξξ ff
'μ ξξξ tan)(2)( ff =
′
b
6?s?s
=?
∫
dxxfx )(cos2
0
2
π
xdxfx sin)(cos2
0
∫
π

dxxxfxfxx ))(sin)((cossin2
0
′
=
∫
π

0sin))(sin)((cos 2
0
=
′
=
∫
dxxff
π
ηηηη ?
)
2
,0(
π
η∈ b
6?,01sin2
0
≠=?
∫
dxx
π
yN
0))(sin)(cos =
′
ηηηη ff
'μ ηηη tan)()( ff =
′
b
￡
!·° Iù
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L 1101 è 62781785
2005
￡
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TDn bê
ê?7<v0 1006 bvD ? S
L 1101 è 62781785
è
! ()
∫
+
= 2,sin
π
x
x
dttxf

￡
ü ( )xf
^[ π1?
ù￥?
ùf

p ( )xf ￥′×b
3
()
∫
+
+
=+
π
π
π 2
3
,sin
x
x
dttxf
! π+=ut 5μ
()
∫
+
+=+ 2 )sin(
π
ππ
x
x
duuxf
∫
+
== 2 )(sin
π
x
x
xfduu
# ( )xf
^[ π1?
ù￥?
ùf
b

y1 xsin ( )+∞∞?,
?? ? i? ( )xf ￥?
ù 1 π# o3
[ ]π,0
)
′×by1
() )sin()cos()sin()
2
sin( xxxxxf?=?+=
′
π
7
( ) 0=
′
xf ¤,
4
3
,
4
21
ππ
== xx O
∫
==
4
3
4
2sin
4
π
π
π
tdtf
∫∫∫
=?==
π
π
π
π
π
π
π
4
3
4
5
4
5
4
3
,22sinsinsin
4
3
tdttdtdttf
? () () ( )
∫∫
=?=== 2
0
2
3
,1sin,1sin0
π π
π
π dttftdtf y7
( )xf ￥Kl′
^ 22? Kv′
^ 2 # ( )xf ￥′×
^
[ ].2,22? b
￡
!·° Iù
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L 1101 è 62781785

课件简介

课件名称：	清华名师：微积分课程各章节知识点题解
课件分类：	数学
课件类型：	电子教案
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