~ ?yvD ?Dy
? 7 ?
??sD?s
￥s?sE
~ ?yvD ?Dy
p?
Ds
^o
I￥
,
s￥ZEè VYV p?E5?,?
￥
'
T?s￥'
T,±s?
T?M
??s?
T?M?,??BDíE,
/H
^e￥ p?E5?s?s E,
~ ?yvD ?Dy
ù5
∫
=?dxxe
x
3 %
±
^ ?¨
?f
e￥ p?E5,
!f
)(xuu = )(xvv = μ ???
,
(),vuvuuv
′
+
′
=
′
(),vuuvvu
′
′
=
′
,dxvuuvdxvu
∫∫
′
=
′
.duvuvudv
∫∫
=
s?s
T
Ba??s￥s?sE
~ ?yvD ?Dy
è ps,cos
∫
xdxx
3
B
3
=
7
,xu= dvxdxdx == sincos
∫
xdxxcos
∫
= xxd sin
∫
= xdxxx sinsin
.cossin Cxxx ++=
7
,cos xu=
dvdxxdx ==
2
2
1
∫
xdxxcos
∫
+= xdx
x
x
x
sin
2
cos
2
22
A ? ê4?? s÷
4é?,vu
′
,
~ ?yvD ?Dy
è ps
.
2
∫
dxex
x
3
,
2
xu=
,dvdedxe
xx
==
∫
dxex
x2
∫
= dxxeex
xx
2
2
)(2
2
dxexeex
xxx
∫
=

Q
P¨s?sE
,xu=
dvdxe
x
=
92 ?$f
^
af
? (? )?f

af
·
f
￥e,ü I
n
!
af
1,
P ?
aBQ (L?
a·
^??
)u
.)(2
2
Cexeex
xxx
+=
~ ?yvD ?Dy
è ps,arctan
∫
xdxx
3
7,arctan xu=
dv
x
dxdx ==
2
2
∫
xdxxarctan
)(arctan
2
arctan
2
22
xd
x
x
x
∫
=
dx
x
x
x
x
2
22
1
1
2
arctan
2 +
=
∫
dx
x
x
x
)
1
1
1(
2
1
arctan
2
2
2
+
=
∫
.)arctan(
2
1
arctan
2
2
Cxxx
x
+=
~ ?yvD ?Dy
è ps
.ln
3
∫
xdxx
3
,ln xu=,
4
4
3
dv
x
ddxx ==
∫
xdxx ln
3
∫
= dxxxx
34
4
1
ln
4
1
.
16
1
ln
4
1
44
Cxxx +?=
92
?$f
^
af

f

a
f
Q ??f
￥eü I
n
!
f
Q ??f
1,u
~ ?yvD ?Dy
∫
xdxarccos
3
,,arccos dxdvxu ==
!
xvdx
x
du =
=,
1
1
2
5
∫∫
+= dx
x
x
xxxdx
2
1
arccosarccos
)1(
)1(
1
2
1
arccos
2
2
2
1
∫
= xd
x
xx
C
x
xx +
=
2
1
)1(
2
1
arccos
2
1
2
Cxxx +=
2
1
)1(arccos
2
è ps
~ ?yvD ?Dy
è ps,)sin(ln
∫
dxx
3
∫
dxx)sin(ln
∫
= )][sin(ln)sin(ln xxdxx
∫
= dx
x
xxxx
1
)cos(ln)sin(ln
∫
+?= )][cos(ln)cos(ln)sin(ln xxdxxxx
∫
= dxxxxx )sin(ln)]cos(ln)[sin(ln
∫
∴ dxx)sin(ln
.)]cos(ln)[sin(ln
2
Cxx
x
+?=
~ ?yvD ?Dy
è ps
.sin
∫
xdxe
x
3
∫
xdxe
x
sin
∫
=
x
xdesin
∫
= )(sinsin xdexe
xx
∫
= xdxexe
xx
cossin
∫
=
xx
xdexe cossin
∫
= )coscos(sin xdexexe
xxx
∫
= xdxexxe
xx
sin)cos(sin
∫
∴ xdxe
x
sin,)cos(sin
2
Cxx
e
x
+?=
?i?ì?
T
~ ?yvD ?Dy
è ps
∫
+
.
1
arctan
2
dx
x
xx
3 ()
,
1
1
2
2
x
x
x
+
=
′
+Q
∫
+
∴ dx
x
xx
2
1
arctan
∫
+=
2
1arctan xxd
)(arctan1arctan1
22
xdxxx
∫
+?+=
dx
x
xxx
2
22
1
1
1arctan1
+
+?+=
∫
~ ?yvD ?Dy
dx
x
xx
∫
+
+=
2
2
1
1
arctan1
7 tx tan=
dx
x
∫
+
2
1
1
∫
+
= tdt
t
2
2
sec
tan1
1
∫
= tdtsec
Ctt ++= )tanln(sec
Cxx +++= )1ln(
2
∫
+
∴ dx
x
xx
2
1
arctan
xx arctan1
2
+=,)1ln(
2
Cxx +++?
~ ?yvD ?Dy
¨ ??}Ds
∫
+
1
arctan
2
dx
x
xx
3=
7
tdtdxxttx
2
sec,arctan,tan ===
e
T
dt
t
tt
tdt
t
tt
∫∫
=
=
2
2
cos
sin
sec
sec
tan
t
1
x
)(cos)cos(cos
12
ttdttdt
∫ ∫

=?=
∫∫
=?= tdt
t
t
dt
tt
t
sec
coscos
1
cos
Ctt
t
t
++?= |tansec|ln
cos
Cxxxx +++?+= |1|ln1)(arctan
22
2
1 x+
~ ?yvD ?Dy
è X? )(xf ￥B?ef
^
2
x
e
, p
∫
′
dxxfx )(,
3
∫
′
dxxfx )(
∫
= )(xxdf,)()(
∫
= dxxfxxf
,)(
2
∫
+=∴
Cedxxf
x
( ) ),()( xfdxxf =
′
∫
Q
H]
H p?,¤
x
,2)(
2
x
xexf
=
=
′
∴
∫
dxxfx )(
∫
dxxfxxf )()(
2
2
2
x
ex
=,
2
Ce
x
+?
~ ?yvD ?Dy
[

 è0?
^¨s?s￥?? ?
?,
 ?
af
D ??f
a
af
DQ ?
?f
a
af
D
f
a
af
D·
f
Me
H ??
X±s51s?sb
Bt?
^??￥1¨s?s￥5μ
H
9 Vs?syN1
2?¨??s￥
ò?ZEb
è 10 p dxxa
∫
+
22
N5 V¨ ??}D,??eL, V
k¨s?
s,
~ ?yvD ?Dy
!
dx
xa
x
xaxdxxaI
∫∫
+
+=+=
22
2
2222
dxxaI
∫
+=
22
dx
xa
aax
xax
∫
+
+
+=
22
222
22
∫∫
+
++?+= dx
xa
a
dxxaxax
22
2
2222
,)ln(
22222
cxaxaIxax ++++?+=
.)ln([
2
1
22222
CxaxaxaxI +++++=∴
3
~ ?yvD ?Dy
.
)(
11
22∫
+
=
n
n
ax
dx
I p è
,,,
)(
1
22
xvdvdx
ax
u
n
==
+
=
7
∫ +
+
+
+
=
122
2
22
)(
2
)(
nn
n
ax
dxnx
ax
x
I5
n
ax
x
)(
22
+
=
+
+
=
+
+
+
∫ + nn
ax
x
dx
ax
aax
n
)()(
2
22122
222
+
∫ n
ax
dx
n
)(
2
22
=
+
∫ +122
2
)(
2
n
ax
dx
na,22
)(
1
2
22
+
+
+
nn
n
InanI
ax
x
μ?w
TyN,
3
~ ?yvD ?Dy
].)12(
)(
[
2
1
222
1 n
n
n
In
ax
x
na
I?+
+
=
+
=
+
==
∫ 222
2
)(
,1
ax
dx
In
H?
)(
2
1
22222 ∫
+
+
+ ax
dx
ax
x
a
c
a
x
a
ax
x
a
++
+
= )arctan
1
(
2
1
222
,
n
I} ??w
T V p¤ ?
~ ?yvD ?Dy
=a?s￥s?sE
?s￥s?s
T
ü??s￥s?s
T¨??
s
 ?
L
^?s￥s?s
T
:,],[)(),( 5
μ ??￥?

! baxvxu
.)()()()()()(
∫∫
=
b
a
b
a
b
a
xduxvxvxuxdvxu
~ ?yvD ?Dy
è 9
.arcsin
2
1
0
∫
xdx
3
7
,arcsin xu=,dxdv =
,
1
2
x
dx
du
=
,xv =
∫
2
1
0
arcsin xdx
[ ]
2
1
0
arcsin xx=
∫
2
1
0
2
1 x
xdx
62
1 π
=
)1(
1
1
2
1
2
0
2
2
1
xd
x
+
∫
12
π
= [ ]
2
1
0
2
1 x?+,1
2
3
12
+=
π
5
~ ?yvD ?Dy
è 9
3
.
2cos1
4
0
∫
π
+ x
xdx
,cos22cos1
2
xx =+Q
∫
π
+
∴
4
0
2cos1 x
xdx
∫
π
=
4
0
2
cos2 x
xdx
()xd
x
tan
2
4
0
∫
π
=
[]
4
0
tan
2
1
π
= xx xdxtan
2
1
4
0
∫
π
[]
4
0
secln
2
1
8
π
π
= x,
4
2ln
8
π
=
~ ?yvD ?Dy
è 9
3
.
)2(
)1ln(
1
0
2
∫
+
+
dx
x
x
∫
+
+
1
0
2
)2(
)1ln(
dx
x
x
∫
+
+?=
1
0
2
1
)1ln(
x
dx
1
0
2
)1ln(
+
+
=
x
x
∫
+
+
+
1
0
)1ln(
2
1
xd
x
3
2ln
=
dx
xx
∫
+
+
+
1
0
1
1
2
1
xx +
+ 2
1
1
1
[]
1
0
)2ln()1ln(
3
2ln
xx +?++?=
.3ln2ln
3
5
=
~ ?yvD ?Dy
è
! p
3
∫
=
2
1
,
sin
)(
x
dt
t
t
xf,)(
1
0
∫
dxxxf
y1
t
tsin
àμ??
T￥ef

íE°¤ p )(xf 
[?¨s?sE
∫
1
0
)( dxxxf
∫
=
1
0
2
)()(
2
1
xdxf
[ ]
1
0
2
)(
2
1
xfx=
∫
1
0
2
)(
2
1
xdfx
)1(
2
1
f=
∫
′
1
0
2
)(
2
1
dxxfx
~ ?yvD ?Dy
∫
=
2
1
,
sin
)(
x
dt
t
t
xfQ
,
sin2
2
sin
)(
2
2
2
x
x
x
x
x
xf =?=
′
∫
∴
1
0
)( dxxxf
)1(
2
1
f=
∫
′
1
0
2
)(
2
1
dxxfx
∫
=
1
0
2
sin2
2
1
dxxx
∫
=
1
0
22
sin
2
1
dxx
[]
1
0
2
cos
2
1
x=
).11(cos
2
1
=
,0
sin
)1(
1
1
∫
== dt
t
t
f
~ ?yvD ?Dy
è ￡
ü?s
T
∫∫
ππ
==
22
00
cossin xdxxdxI
nn
n


=
n
n
n
n
n
n
n
n
n
n
,
3
2
5
4
2
31
,
22
1
4
3
2
31
L
L
π
1?
}
1v? 1￥ 
￡
!,sin
1
xu
n?
=,sin xdxdv =
,cossin)1(
2
xdxxndu
n?
=,cos xv?=
~ ?yvD ?Dy
[] dxxxnxxI
nn
n
∫
π
π

+?=
2
2
0
22
0
1
cossin)1(cossin
x
2
sin1?
0
dxxndxxnI
nn
n ∫∫
=
22
00
2
sin)1(sin)1(
ππ
nn
InIn )1()1(
2
=
2
1
=
nn
I
n
n
I s1?/S￥?w
T
n
I
42
2
3

=
nn
I
n
n
I
,LL
°?/Sh? 011?
~ ?yvD ?Dy
,
2
1
4
3
6
5
22
32
2
12
02
I
m
m
m
m
I
m

= L
,
3
2
5
4
7
6
12
22
12
2
112
I
m
m
m
m
I
m

+
=
+
L
),2,1( L=m
,
2
2
0
0
π
==
∫
π
dxI,1sin
2
0
1
==
∫
π
xdxI
,
22
1
4
3
6
5
22
32
2
12
2
π

= L
m
m
m
m
I
m
.
3
2
5
4
7
6
12
22
12
2
12

+
=
+
L
m
m
m
m
I
m
?
^
~ ?yvD ?Dy
?ê4 ? ?
P¨s?
s
T
vu
′
,
dxvuuvdxvu
∫∫
′
=
′
?al2
?s￥s?s
T
[],
∫∫
=
b
a
b
a
b
a
vduuvudv

?iD??ss?sE￥ uY
~ ?yvD ?Dy
?$f
^
??] ??f
￥e
, O?L?Dís
H, V I
n¨s?s,
1???￥f
^ ( p(x)
^[
T ):
xxpxfexpxf
baxxpxfbaxxpxf
ax
arctan)()(,)()(
),cos()()(),sin()()(
==
+=+=
,sin)(),ln()()( axexfbaxxpxf
bx
=+=
7 u,v￥e5
^,? dv^? p v,
:,5¨
T÷ ?^ p1 ?
∫∫
udvvdu
∫ ∫
= vduuvudv
~ ?yvD ?Dy
± I5
1a¤ ?+Q?¨s?s
T
H
??i
I
1$
2a
! )(xf
′′
 [ ]1,0
 ?? O 1)0( =f 
3)2( =f  5)2( =
′
f  p
∫
′′
1
0
)2( dxxfx,
~ ?yvD ?Dy
± I53s
?i -a+Q
ê￥ ?1] ??f
,
u
è
∫
xdxe
x
cos
?BQ
H ?ê
xu cos
1
=
∫
xdxe
x
cos
dxxexe
xx
∫
+= sincos
?=Q
H ˉ?ê xu sin
2
=
Qs?a, V
Ce ?1 p￥s,(
H?
|"
?] ),YV3Z?31 p￥s,
~ ?yvD ?Dy
± I53s
∫
′′
1
0
)2( dxxfx
∫
′
=
1
0
)2(
2
1
xfxd
[]
∫
′
′
=
1
0
1
0
)2(
2
1
)2(
2
1
dxxfxfx
[]
1
0
)2(
4
1
)2(
2
1
xff?
′
=
[])0()2(
4
1
2
5
ff=
.2=
~ ?yvD ?Dy
BaA b5
a
∫
=xdxxsin @@@@@@@@@@@@@@@@
a
∫
=xdxarcsin @@@@@@@@@@@@@@@
a9
∫
xdxx ln
2
 =u V
! @@@@@
=dv @@@@@@@@
a9
∫
xdxe
x
cos  =u V
! @@@@
=dv @@@@@@@@
a9
∫
xdxx arctan
2
 =u V
! @@@@
=dv @@@@@@
a9
∫
dxxe
x
 =u V
! @@@@@@
=dv @@@@@@@@@@
=a p/
??s
a
∫
dx
x
x
2
cos
22
 a
∫
dx
x
x
2
3
)(ln

5B
~ ?yvD ?Dy
3a
∫
nxdxe
ax
cos  4a
∫
dxe
x3

5a
∫
dxx)cos(ln  6a
∫
+
dx
x
xe
x
2
3
2
arctan
)1(
,
?a X?
x
xsin
^ )(xf ￥ef
 p
∫
dxxxf )(
'

1a
!
∫
+= CxFdxxf )()(  )(xf V± O )(xf ￥Q
f
)(
1
xf
i5
[]CxfFxxfdxxf +?=

∫
)()()(
111

~ ?yvD ?Dy
BaA b5
a
!O1? 
5 =
∫
π
2
0
sin xdx
n
@@@@@@@@@@@
a
!O1?
}
5
∫
π
2
0
cos xdx
n
@@@@@@@@@@@
a =
∫
dxxe
x
1
0
@@@@@@@@@@@@@@
a =
∫
e
xdxx
1
ln @@@@@@@@@@@@@
5a =
∫
1
0
arctan xdxx ____________,
=a 9
/
?s
a
∫
e
dxx
1
)sin(ln  a
∫
e
e
dxx
1
ln 
5=
~ ?yvD ?Dy
a
∫
π
=
0
sin)( xdxxmJ
m

m11 ?

a
∫
π
+
0
1
)1cos(sin xdxnx
n

?aX? xxf
2
tan)( =
p
∫
π
′′′
4
0
)()( dxxfxf 
1a ? [ ]π,0)( xf
′′
??,1)(,2)0( =π= ff
￡
ü 3sin])()([
0
=
′′
+
∫
xdxxfxf
π
.
~ ?yvD ?Dy
Baa Cxxx ++? sincos 
a Cxxx +?+
2
1arcsin 
a dxxx
2
,ln  a,
x
e
 xdxcos 
a dxxx
2
,arctan  a dxex
x?
,
=a 1a Cxxxxx
x
+?++ sincossin
2
1
6
2
3

2a Cxxx
x
++++? ]6ln6)(ln3)[(ln
1
23

3a Cnxnnxa
na
e
ax
++
+
)sincos(
22
4a Cxxe
x
++? )22(3
3
3 2
3

5Bs?
~ ?yvD ?Dy
a Cxx
x
++ )]sin(ln)[cos(ln
2

a Ce
x
x
x
+
+
arctan
2
12
1

a Cexe
x
ex
xx
x
+?+
+ 2
2

?a C
x
x
x +?
sin2
cos 
~ ?yvD ?Dy
Baa
!!
!)!1(
n
n?
 a
2!!
!)!1( π
n
n
 a
e
2
1? 
a )1(
4
1
2
+e  a
2
3
ln
2
1
)
9
3
4
1
( +π? 
=aa
2
11cos1sin +?ee
 a )
1
1(2
e

5=s?
 a
>π?


π


=
1 
1
}
1,
531
)1(642
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