~ ?yvD ?Dy
? 5 ?
??sDíE=
~ ?yvD ?Dy
ù5?1
25
=?

dxxx
3 %ZE ?M?WM
¥
!?ZE,
V?
7
tx sin=,costdtdx =?
=?

dxxx
25
1
tdttt cossin1)(sin
25

tdtt
25
cossin

= LL=

?¨oX±sp' V p2T
?= ?DíE
~ ?yvD ?Dy
? )(xψ
^ )(tx ψ= ¥Qf

£
! 1 ¥ef
,)(tΦ
)()]([ ttf ψψ

7
)]([)( xxF ψΦ=
5
dx
dt
dt
d
xF?
Φ
=

)(
)()]([ ttf ψψ

=,
)(
1


! )(tx ψ=
^??¥a V?¥f

[]
)(
)()]([)(
xt
dtttfdxxf
ψ
ψψ
=
∫∫

=5μDí
T
i O 0)( ≠

tψ 
?
! )()]([ ttf ψψ

μef

? ?
~ ?yvD ?Dy
?= ?sDí
T

+=∴ CxFdxxf )()(
,)]([ Cx +ψΦ=
[]
)(
)()]([)(
xt
dtttfdxxf
ψ
ψψ
=
∫∫

=
)]([ tf ψ=
).(xf=
a
ü )(xF 1 )(xf ¥ef
,
~ ?yvD ?Dy
è p
3
).0(
1
22
>
+

adx
ax
7 tax tan=
tdtadx
2
sec=?
=
+

dx
ax
22
1
tdta
ta
2
sec
sec
1


= tdtsec Ctt ++= )tanln(sec
t
a
x
22
ax +
.ln
22
C
a
ax
a
x
+
+
+=
ππ

2
,
2
t
~ ?yvD ?Dy
è p
3
.4
23
dxxx

7 tx sin2=
tdtdx cos2=
ππ

2
,
2
t
dxxx

23
4
( ) tdttt cos2sin44sin2
2
3
=

tdtt
23
cossin32

= tdttt
22
cos)cos1(sin32

=
tdtt cos)cos(cos32
42
=

Ctt += )cos
5
1
cos
3
1
(32
53
t
2
x
2
4 x?
()( ),4
5
1
4
3
4
5
2
3
2
Cxx +?+=
~ ?yvD ?Dy
è p
3
).0(
1
22
>

adx
ax
7 tax sec=

2
,0
π
t
tdttadx tansec=
=

dx
ax
22
1
dt
ta
tta

tan
tansec

= tdtsec Ctt ++= )tanln(sec
t
a
x
22
ax?
.ln
22
C
a
ax
a
x
+
+=
~ ?yvD ?Dy
a
ü 
[
+ è
P¨¥ (1 ??}D,
??}D¥
"¥
^??
T,
B?
p ?/?$f
?cμ
22
)1( xa?
V
7;sintax =
22
)2( xa +
V
7;tantax =
22
)3( ax?
V
7
.sectax =
~ ?yvD ?Dy
s?1
??
T
^?B??¨
??}Di?
^ '¥3? $f
¥
f ? ??,
a
ü 
è p
dx
x
x

+
2
5
1

??}DN

2
1 xt +=
7,1
22
=? tx,tdtxdx =
dx
x
x

+
2
5
1
( )
tdt
t
t

=
2
2
1
( )dttt

+?= 12
24
Cttt ++?=
35
3
2
5
1
.1)348(
15
1
242
Cxxx +++?=
3
~ ?yvD ?Dy
è p
3
.
1
1
dx
e
x∫
+
x
et += 1
7
,1
2
=? te
x
,
1
2
2
dt
t
t
dx
=
dx
e
x

+1
1
dt
t

=
1
2
2
dt
tt

+
=
1
1
1
1
C
t
t
+
+
=
1
1
ln
( ),11ln2 Cxe
x
++=
( ),1ln
2
= tx
~ ?yvD ?Dy
a
ü 
?s
¥¨?ú
H, V?¨ ?}D
.
1
t
x =
è p
dx
xx

+ )2(
1
7
7
t
x
1
=,
1
2
dt
t
dx?=?
dx
xx

+ )2(
1
7
dt
t
t
t

+
=
∫ 27
1
2
1

+
= dt
t
t
7
6
21
Ct ++?= |21|ln
14
1
7
.||ln
2
1
|2|ln
14
1
7
Cxx +++?=
3
~ ?yvD ?Dy
è p
3
.
1
1
24
dx
xx

+
dx
xx

+1
1
24
7
t
x
1
=,
1
2
dt
t
dx?=?
dt
t
tt
+
=
∫ 2
24
1
1
11
1

s
¥¨?ú
dt
t
t

+
=
2
3
1
2
2
2
1
2
1
dt
t
t

+
=
2
tu=
~ ?yvD ?Dy

+
= du
u
u
12
1

+

= du
u
u
1
11
2
1

+
+?
+
= )1(1
1
1
2
1
udu
u
() Cuu ++++?= 11
3
1 3
.
11
3
1
2
3
2
C
x
x
x
x
+
+
+
+
=
~ ?yvD ?Dy
a
ü 
?$f
cμ
?
?[
¥
?
T
H V?¨
7

? 1ò?·
¥ Kl

lk
xx,,L
n
tx =
n
è p
.
)1(
1
3
dx
xx

+
3
7
6
tx =
,6
5
dttdx =?
dx
xx

+ )1(
1
3 ∫
+
= dt
tt
t
)1(
6
23
5

+
= dt
t
t
2
2
1
6
~ ?yvD ?Dy

+
+
= dt
t
t
2
2
1
11
6

+
= dt
t
2
1
1
16
Ctt +?= ]arctan[6
.]arctan[6
66
Cxx +?=
~ ?yvD ?Dy

'

s
V
?;|cos|lntan)1(

+?= Cxxdx;|sin|lncot)2(

+= Cxxdx;|tansec|lnsec)3(

++= Cxxxdx;|cotcsc|lncsc)4(

+?= Cxxxdx;arctan
11
)5(
22
C
a
x
a
dx
xa
+=
+

~ ?yvD ?Dy;ln
2
11
)7(
22
C
xa
xa
a
dx
xa
+
+
=
∫;arcsin
1
)8(
22
C
a
x
dx
xa
+=

.||ln
1
)9(
22
22
Caxxdx
ax
+±+=
±
∫;ln
2
11
)6(
22
C
ax
ax
a
dx
ax
+
+
=

~ ?yvD ?Dy
è ps
.
)1(
1
2
dx
xx

dx
xx

2
)1(
1
dx
xxx

+=
1
1
)1(
11
2
dx
x
dx
x
dx
x
∫∫∫
+=
1
1
)1(
11
2
.|1|ln
1
1
||ln Cx
x
x +
=
3
a
ü 
?$f
1μ ?f
H ?¨}?
"
E|μ ?f
1?ss
T-,
~ ?yvD ?Dy
è ps
3
.
)1)(21(
1
2

++
dx
xx
dx
x
x
dx
x
∫∫
+
+?
+
+
=
2
1
5
1
5
2
21
5
4

++
dx
xx )1)(21(
1
2
dx
x
dx
x
x
x
∫∫
+
+
+
+=
22
1
1
5
1
1
2
5
1
)21ln(
5
2
.arctan
5
1
)1ln(
5
1
|21|ln
5
2
2
Cxxx +++?+=
~ ?yvD ?Dy
2
cos
2
sin2sin
xx
x =Q
2
sec
2
tan2
2
x
x
=
,
2
tan1
2
tan2
2
x
x
+
=
,
2
sin
2
coscos
22
xx
x?=
a
ü 
?$f
1 ??f
¥μ ?
T
H
?¨£

T}D,
~ ?yvD ?Dy
2
sec
2
tan1
cos
2
2
x
x
x
=,
2
tan1
2
tan1
2
2
x
x
+
=
7
2
tan
x
u=
,
1
2
sin
2
u
u
x
+
=
,
1
1
cos
2
2
u
u
x
+
=
ux arctan2=
du
u
dx
2
1
2
+
=

£
?D
T
~ ?yvD ?Dy
è ps
.
cossin1
sin

++
dx
xx
x
3
,
1
2
sin
2
u
u
x
+
=
2
2
1
1
cos
u
u
x
+
=
,
1
2
2
du
u
dx
+
=
?£
?D
T

++
dx
xx
x
cossin1
sin
du
uu
u

++
=
)1)(1(
2
2
du
uu
uuu

++
++
=
)1)(1(
112
2
22
~ ?yvD ?Dy
du
uu
uu

++
+?+
=
)1)(1(
)1()1(
2
22
du
u
u

+
+
=
2
1
1
du
u

+
1
1
uarctan= )1ln(
2
1
2
u++
Cu ++? |1|ln
2
tan
x
u=Q
2
x
=
|
2
sec|ln
x
+
.|
2
tan1|ln C
x
++?
~ ?yvD ?Dy
è ps,
sin
1
4

dx
x
3
B
,
2
tan
x
u=,
1
2
sin
2
u
u
x
+
=,
1
2
2
du
u
dx
+
=

dx
x
4
sin
1
du
u
uuu

+++
=
4
642
8
331
C
u
u
uu
+++= ]
3
3
3
3
1
[
8
1
3
3
.
2
tan
24
1
2
tan
8
3
2
tan8
3
2
tan24
1
3
3
C
xx
x
x
+
++?
=
~ ?yvD ?Dy
3
=
??£
?D
T,xu tan=
7
,
1
sin
2
u
u
x
+
=
,
1
1
2
du
u
dx
+
=

dx
x
4
sin
1
du
u
u
u

+
+
=
24
2
1
1
1
1
du
u
u

+
=
4
2
1
C
uu
+=
1
3
1
3
.cotcot
3
1
3
Cxx +=
~ ?yvD ?Dy
3
? V[?¨£
?D
T,

dx
x
4
sin
1
dxxx )cot1(csc
22

+=
xdxxxdx
2
2
2
csccotcsc
∫ ∫
+=
)(cot xd=
.cot
3
1
cot
3
Cxx +=
2
1?[
 ??3E,L?£
?D?B?
^KDZE,# ??μ ?
T¥9
?5 I
n 
m
,?¤X?¨£
?D,
~ ?yvD ?Dy
è ps,
sin3sin
sin1

+
+
dx
xx
x
3
2
cos
2
sin2sinsin
BABA
BA
+
=+

+
+
dx
xx
x
sin3sin
sin1

+
= dx
xx
x
cos2sin2
sin1

+
= dx
xx
x
2
cossin4
sin1

= dx
xx
2
cossin
1
4
1

+ dx
x
2
cos
1
4
1
~ ?yvD ?Dy

+
= dx
xx
xx
2
22
cossin
cossin
4
1

+ dx
x
2
cos
1
4
1
∫∫
+= dx
x
dx
x
x
sin
1
4
1
cos
sin
4
1
2 ∫
+ dx
x
2
cos
1
4
1
∫∫
+?= dx
x
xd
x sin
1
4
1
)(cos
cos
1
4
1
2 ∫
+ dx
x
2
cos
1
4
1
xcos4
1
=
2
tanln
4
1 x
+
.tan
4
1
Cx++
~ ?yvD ?Dy
=al2
 ?sDíE

B X±s

= ??}Da?}Da?
T}D
'sV (2)
~ ?yvD ?Dy
± I5
ps
.)1(ln)ln( dxxxx
p
+

~ ?yvD ?Dy
± I53s
dxxxxd )ln1()ln( +=Q
dxxxx
p
)1(ln)ln( +∴

)ln()ln( xxdxx
p

=
=+
≠+
+=
+
1,)lnln(
1,
1
)ln(
1
pCxx
pC
p
xx
p
~ ?yvD ?Dy
°z A b5
a ? CxFdxxf +=

)()( 7 )(xu Φ= 5
 =

duuf )( @@@@@@@@@@@@@@@ 
a p

>? )0(
22
adxax
H VTM
}D@@@@@@@
@@@@@@@@@@@@@@ ?a ps 
a p

+
dx
xx
2
1
1
H V5
7 =x @@@@@@@@@ 
a =dxx @@@@@ )1(
2
xd?  
a =
dxe
x
2
@@@ )1(
2
x
ed
+  
a =
x
dx
@@@@ )ln53( xd?  

5
~ ?yvD ?Dy
a
2
91 x
dx
+
@@@@ )3arctan( xd 
a =
2
1 x
xdx
@@@@ )1(
2
xd? 
a

=dt
t
tsin
@@@@@@@@@@@@@@@@@
a

=
22
2
xa
dxx
@@@@@@@@@@@@@@@
~ ?yvD ?Dy
?z  p/
??s 
?= ?DíE
a

+
2
1 xx
dx
 
a

+
32
)1(x
dx
 
a

+ x
dx
21
 
a

dx
xa
x
x
2
 
a
!

= xdxI
n
n
tan
p£

2
1
tan
1
1
=
n
n
n
Ix
n
I 
i p

xdx
5
tan 
~ ?yvD ?Dy
5s?
Baa CuF +)(  a tax sec= tax csc= 
a
t
1
 a
2
1
 a a
5
1

a
3
1
 a?  a Ct +cos2 
a Cxa
a
x
a
xa
+ )(arcsin
2
22
2
2

=aa Cxxx +?++ )]1ln([arcsin
2
1
2
 
a C
x
x
+
+
2
1
 
a Cxx ++? )21ln(2  
a )2(2
2
arcsin3
2
xaxa
a
x
a Cxax
xa
+?
)2(
2

~ ?yvD ?Dy
~ ?yvD ?Dy