~ ?yvD ?Dy
? 8 ?
<ls
~ ?yvD ?Dy
è
{
xx
x
y
ü
u×¥


??¥# p wL 1,
1
2
==
ABCD
b
SS
+∞→
= lim
dx
x
b
b

+∞→
=
1
2
1
lim
)
1
1(lim
b
b
=
+∞→
1=
· u×1í? u×
?m
Baí kK¥<ls
~ ?yvD ?Dy
?l  
!f
)(xf  uW ),[ +∞a
 ?? |
ab>  ?TK

+∞→
b
a
b
dxxf )(lim i5?N
K1f
)(xf í k uW ),[ +∞a
¥<l
s:T

∞+
a
dxxf )( 

∞+
a
dxxf )(

+∞→
=
b
a
b
dxxf )(lim
?Ki
H?<ls
l ? ?K?i
H?<ls? ? 

∞+
=
1
2
1
1 dx
x
S? è
~ ?yvD ?Dy
?
1
!f
)(xf  uW ],( b?∞
 ?? |
ba <  ?TK

∞→
b
a
a
dxxf )(lim i5?N
K1f
)(xf í k uW ],( b?∞
¥<l
s:T

∞?
b
dxxf )( 

∞?
b
dxxf )(

∞→
=
b
a
a
dxxf )(lim
?Ki
H?<ls
l ? ?K?i
H?<ls? ? 
~ ?yvD ?Dy
!f
)(xf  uW ),( +∞?∞
 ??
?T
<ls

∞?
0
)( dxxf

∞+
0
)( dxxf ?
l ?5
?

<ls-1f
)(xf í k uW
),( +∞?∞
¥<ls:T

∞+
∞?
dxxf )( 

∞+
∞?
dxxf )(

∞?
=
0
)( dxxf

∞+
+
0
)( dxxf

∞→
=
0
)(lim
a
a
dxxf

+∞→
+
b
b
dxxf
0
)(lim
Ki?<ls
l ? ?5?<ls? ? 
~ ?yvD ?Dy
?T

+∞→
b
b
b
dxxf )(lim
i

∞+
∞?
dx)x(f
^?
l ?$
è ?

∞+
∞?
xdx

+
+∞→
=
b
b
b
xdxlim
b
b
b
x
+
+∞→
2
2
1
lim
0)0(lim ==
+∞→b
1aí kK¥<ls
Lé

^ ?iμK
uW
?s¥Kb

∞+
∞?
dxxf )(2a
l ??N?

∞?
b
dxxf )(

∞+
b
dxxf )(
D
]
H
l ?b
?
?
~ ?yvD ?Dy
dxx
b
b

+
+∞→
0
lim
b
b
x
+
+∞→
=
0
2
2
1
lim
7
=
+∞→
2
2
1
lim b
b
dxx

∞+

0
? ?
dxx

∞+
∞?
? ?b
yN
+∞=
~ ?yvD ?Dy
è 9
<ls,
1
arctan
0
2
dx
x
x

∞?
+
dx
x
x
a
a

+
=
∞→
0
2
1
arctan
lim

∞→
=
0
arctanarctanlim
a
a
xxd
0
2
)(arctan
2
1
lim
a
a
x
∞→
=
2
)(arctan
2
1
lim a
a?∞→
=
8
2
π
=
3 e
T
~ ?yvD ?Dy
è 9
<ls
).0(
0
>

∞+
pdtte
pt
3

b
pt
dtte
0

=
b
pt
)e
p
(td
0
1


+
=
b
pt
b
pt
dte
p
e
p
t
0
0
1
b
pt
b
pt
e
pp
e
p
t
0
0
11
+
=

pbpb
e
pp
e
p
b

+
=
22
11

∞+
0
dtte
pt

+∞→
=
b
pt
b
dtte
0
lim
)
11
(lim
22
pbpb
b
e
pp
e
p
b

+∞→
+
=,
1
2
p
=
~ ?yvD ?Dy
è 9
<ls
.
1
2

∞+
∞?
+ x
dx
3

∞+
∞?
+
2
1 x
dx

∞?
+
=
0
2
1 x
dx

∞+
+
+
0
2
1 x
dx

+
=
∞→
0
2
1
1
lim
a
a
dx
x

+
+
+∞→
b
b
dx
x
0
2
1
1
lim
[]
0
arctanlim
a
a
x
∞→
=
[ ]
b
b
x
0
arctanlim
+∞→
+
a
a
arctanlim
∞→
= b
b
arctanlim
+∞→
+
.
22
π=
π
+
π
=
~ ?yvD ?Dy

∞+
∞?
+
2
1 x
dx
[ ]
∞+
∞?
= arctan x
)
2
(
2
ππ
=
π=
63
~ ?yvD ?Dy
è 9
<ls
3
.
1
sin
1
2 2∫
∞+
π
dx
xx

∞+
π
2
1
sin
1
2
dx
xx

∞+
π
=
2
11
sin
x
d
x

π
=
+∞→
b
b
x
d
x
2
11
sinlim
b
b
x
π
=
+∞→
2
1
coslim
=
+∞→
2
cos
1
coslim
π
b
b
.1=
~ ?yvD ?Dy
è X?
3
>
≤<

=
2,1
20,
2
1
0,0
)(
x
xx
x
xf
dttf
x

∞?
)(
dt
x
a
a
0lim

∞→
=
k¨s
f
V
U,
,)( dttf
x

∞?
H? 01
0
≤x
0=
dttf
x

∞?
)(
dttfdttf
x
∫∫
+=
∞? 0
0
)()(
H? 202
0
≤< x
tdtdt
x
a
a
∫∫
+=
∞→
0
0
2
1
0lim
x
t
0
2
4
1
0+=
2
4
1
x=
~ ?yvD ?Dy
dttf
x

∞?
)( dttfdttfdttf
∫∫∫
∞+
∞?
++=
2
2
0
0
)()()(
H? 23
0
>x
dttdtdt
x
a
a
1
2
1
0lim
2
2
0
0
∫∫∫
++=
∞→
x
tt
2
2
0
2
4
1
0 ++= 1?= x
,
2,1
20,
4
1
0,0
)(
2
>?
≤<

=∴

∞?
xx
xx
x
dttf
x
~ ?yvD ?Dy
è £
ü<ls

∞+
1
1
dx
x
p
? 1>p
H
l ?
? 1≤p
H? ?,
£
,1)1( =p

∞+
1
1
dx
x
p

∞+
=
1
1
dx
x
[ ]
∞+
=
1
ln x
,+∞=
,1)2( ≠p

∞+
1
1
dx
x
p
∞+
=
1
1
1 p
x
p
>
<∞+
=
1,
1
1
1,
p
p
p
yN? 1>p
H<ls
l ? ′1
1
1
p

? 1≤p
H<ls? ?,
~ ?yvD ?Dy
è £
ü<ls

∞+
a
px
dxe ? 0>p
H
l ?
? 0<p
H? ?,
£

∞+
a
px
dxe

+∞→
=
b
a
px
b
dxelim
b
a
px
b
p
e
=
+∞→
lim
=

+∞→
p
e
p
e
pbpa
b
lim
<∞
>
=
0,
0,
p
p
p
e
ap
'? 0>p
H
l ?? 0<p
H? ?,
~ ?yvD ?Dy
?l  
!f
)(xf  uW ],( ba
 ??7
? a¥·
#×
=í? | 0>ε  ?TK

+
+→
b
a
dxxf
ε
ε
)(lim
0
i5?NK1f
)(xf
 uW ],( ba
¥<ls:T

b
a
dxxf )( 

b
a
dxxf )(

+
+→
=
b
a
dxxf
ε
ε
)(lim
0
?Ki
H?<ls
l ? ?K?i
H?<ls? ?
=aí?f
¥<ls
~ ?yvD ?Dy
?
1
!f
)(xf  uW ),[ ba
 ??
7? b¥P
#×
=í? | 0>ε  ?TK

+→
ε
ε
b
a
dxxf )(lim
0
i5?NK1f
)(xf
 uW ),[ ba
¥<ls
:T

b
a
dxxf )(

+→
=
ε
ε
b
a
dxxf )(lim
0

?Ki
H?<ls
l ? ?K?i
H?<ls? ?
~ ?yvD ?Dy
!f
)(xf  uW ],[ ba
"? )( bcac << ? ?
?7? c¥
#×
=í? ?T
?<ls

c
a
dxxf )(

b
c
dxxf )( ?
l ?5?l

b
a
dxxf )(

=
c
a
dxxf )(

+
b
c
dxxf )(

+→
=
ε
ε
c
a
dxxf )(lim
0


+
+→

+
b
c
dxxf
ε
ε
)(lim
0
?5ü?<ls

b
a
dxxf )( ? ?
?l? C1 <? [
s?1 <s,
~ ?yvD ?Dy
è 9
<ls
3
).0(
0
22
>

a
xa
dx
a
,
1
lim
22
0
+∞=

xa
ax
Q
ax =∴ 1$f
¥í kW?,

a
xa
dx
0
22

+→
=
ε
ε
a
xa
dx
0
22
0
lim
ε
ε
+→
=
a
a
x
0
0
arcsinlim
=
+→
0arcsinlim
0
a
a ε
ε
.
2
π
=
22
1
xa
y
=
~ ?yvD ?Dy
è £
ü<ls

1
0
1
dx
x
q
? 1<q
H
l ??
1≥q
H? ?,
£
,1)1( =q

=
1
0
1
dx
x
[ ]
1
0
ln x=
,+∞=
,1)2( ≠q

1
0
1
dx
x
q
1
0
1
1
=
q
x
q
<
>∞+
=
1,
1
1
1,
q
q
q
yN? 1<q
H<ls
l ? ′1
q?1
1

? 1≥q
H<ls? ?,

1
0
1
dx
x
q
~ ?yvD ?Dy
è 9
<ls
3
.
ln
2
1

xx
dx

2
1
ln xx
dx

+
+→
=
2
1
0
ln
lim
ε
ε
xx
dx

+
+→
=
2
1
0
ln
)(ln
lim
ε
ε
x
xd
[ ]
2
1
0
)ln(lnlim
ε
ε
+
+→
= x
[ ]))1ln(ln()2ln(lnlim
0
ε
ε
+?=
+→
.∞=
#e<ls? ?,
~ ?yvD ?Dy
è 9
<ls
3
.
)1(
3
0 3
2

x
dx
1=x
<?

3
0
3
2
)1(x
dx
∫∫
+=
1
0
3
1 3
2
)1(
)(
x
dx

1
0
3
2
)1(x
dx

+→
=
ε
ε
1
0
0 3
2
)1(
lim
x
dx
3=

3
1
3
2
)1(x
dx

+
+→
=
3
1
0 3
2
)1(
lim
ε
ε
x
dx
,23
3
=


3
0
3
2
)1(x
dx
).21(3
3
+=
~ ?yvD ?Dy
è 9
<ls,
1
1

x
dx
0=x <?
3 e
T
1
1
||ln
= x 000 =?=
∫∫
+→
=
1
0
1
0
lim
ε
ε x
dx
x
dx
Q
3
1
0
lnlim
ε
ε
x
+→
=
ε
ε
lnlim
0+→
=
+∞=
.
1
0
? ?


x
dx
.
1
1
? ?


x
dx
~ ?yvD ?Dy
í?f
¥<ls
<s 
í kK¥<ls

∞+
∞?
dxxf )(

∞?
b
dxxf )(

∞+
a
dxxf )(
∫ ∫∫
+=
c
a
b
c
b
a
dxxfdxxfdxxf )()()(

?i ?
-
{
=?¥<?

b
a
dxxf )(
tal2
~ ?yvD ?Dy
± I5
s ¥<?
^
'+?$

1
0
1
ln
dx
x
x
~ ?yvD ?Dy
± I53s
s V
¥<?
^

1
0
1
ln
dx
x
x
1,0 == xx
1
ln
lim
1

x
x
x
Q
,1
1
lim
1
==

x
x
1=∴ x
?
^<?,


1
0
1
ln
dx
x
x
¥<?
^
.0=x
~ ?yvD ?Dy
Ba A b5
a<ls

∞+
1
p
x
dx
?@@@@@@@
H
l ? ?@@@@@@
H
? ?
a<ls

1
0
q
x
dx
?@@@@@@@
H
l ? ?@@@@@@@
H?
?
a<ls

∞+
2
)(ln
k
xx
dx
@@@@@@
H
l ?  @@@@@@@

H? ?
a<ls

∞+
∞?
+
dx
x
x
2
1
@@@@
5
~ ?yvD ?Dy
a<ls =

1
0
2
1 x
xdx
@@@@@@@@
a<ls

∞?
x
dttf )( ¥+il
^@@@@@@@@@@@@@@
@@@@@@@@@@@@@@@@@@@@@@@@
=a
Y/
ò<ls¥
l ?? ?T
l ?5 9
<ls¥′
a

∞+
0
coshtdte
pt
 )1( >p  a

∞+
∞?
++ 22
2
xx
dx

a

∞+
0
dxex
xn

11 ?
n  a

2
0
2
)1( x
dx

~ ?yvD ?Dy
a

2
1
1x
xdx
 a

∞+
+
0
22
)1(
ln
dx
x
xx

a

1
0
ln xdx
n

?a p? 1′
Hk <ls )(
)(
ab
ax
dx
b
a
k
>

l ?$? 1′
Hk ?<ls? ?$
1aX?
<
≤<
≤<∞?
=
x
xx
x
xf
2,1
20,
2
1
0,0
)( 
k¨s
f
V
U


∞?
x
dttf )( 
~ ?yvD ?Dy
Baa 1,1 ≤> pp  a 1,1 ≥< qq  a 1,1 ≤> kk 
a? ? a aV? à
ü?? yx ¥°
LPH
wL )(xfy = à x
?m?¥

=aa
1
2
p
p
 a π a !n  a? ?
a
3
2
2  a a !)1( n
n

?a? 1<k
H
l ??
k
ab
k
1
)(
1
1
 ? 1≥k
H? ?
1a
<?
≤<
≤<∞?
=

∞?
xx
xx
x
dttf
x
2,1
20,
4
1
0,0
)(
2

5s?