2
1 dx
x
2
ln x dx
x
2
1
ln
x
x
dx
2
x dx
e
x
x dx
e
x
x
(
x
1)
e
x dx
e
x
x
(
x
1)
e
2
2
3
e
lim (
x
x
1)
e
x
2
3
e
2
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lim(1
t
0
2
x
t
t
)
sin
x
(
,
)
( )
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lim(1
t
0
2
x
t
t
)
sin
x
sin
x
2
t x
t
lim
0
t
e
x
e
x
0
( )
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x
0
x
cos
1
x
0
0,
x
0x
f x
0
2
,
x
0
(
0,
0)
'f
x
0
1
0
2
0x
f
x
0
f
0
0
f
0
lim
0
x
0
x
1
cos
x
x
lim
0
x
x
1
cos
1
x
0x
x
f
1
x
cos
1
x
cos
1
x
1
x
sin
1
x
1
x
1
x
sin
1
x
1
1
x
f
0
f
0
lim
0
x
1
x
cos
1
x
0
1
x
cos
1
x
1
x
sin
1
x
=0
= lim
0
x
+
f
x
0x
1 0
f
0
f
x
lim
0
x
+
1 0
( )
f x
,
f
( )
x
y
( )
f x
0
1
2
3
,
f u v
f
x
y
, y
x
2
x
2
y
f
1
u
u
1
v
f
1
u
v
1
v
1 ,02
10, 2
1 ,02
1
0, 2
u
x
,
y v
y
x
x
u
1
v
,
y
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1
v
(
f x
y
,
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x
2
x
2
y
( , )
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1
u
v
2
2
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1
v
v
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u
2(1
1
v
f
u
v
)
2 (1
u
1
v
,
f
v
2
2
u
(1
)
v
2
f
u
0,
u
v
1
1
f
v
u
v
1
1
1
2
D
y
3
x
2
xy
1
4
xy
1
y
x
f x y
,
D
f x y dxdy
,
D
3
4
d
1
sin 2
1
2sin 2
f r
cos , sin
r
rdr
3
4
d
1
sin 2
1
2sin 2
f r
cos , sin
r
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3
4
d
1
sin 2
1
2sin 2
f r
cos , sin
r
dr
3
4
d
1
sin2
1
2sin2
f r
cos , sin
dr
r
D
( ,
r
)
4
3
,
1
2sin 2
r
1
sin 2
( ,
f x y dxdy
)
D
3
4
d
1
n 2
si
1
2sin 2
( cos , sin )
f r
r
A
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1 2
1 4
1
a
2
a
b
1
d
2
d
rdr
1, 2
Ax b
a
d
,
a
d
,
a
d
,
a
d
,
( , )
A b
1 1
1 2
1 4
1
a
2
a
1
d
2
d
1 1
0 1
0 0 (
1
1
a
1)(
a
a
1
1
d
1)(
d
2)
2)
(
d
(
)
r A
(
r A b
, ) 3
1a
2a
1d
d
2
,
f x x
1
2
,
x
3
x Py
2
2 y
1
y
2
2
y
2
3
P
,
(
e e e
1
,
2
)
3
(
Q e
1
,
,
e e
3
2
)
f
(
,
x x x
1
3
,
2
)
x Qy
2
2 y
1
y
2
2
y
2
3
2
2 y
1
y
2
2
y
2
3
2
2 y
1
y
2
2
y
2
3
2
2y
1
y
2
2
2
y
3
)
T
y P AP y
(
T
2
2
y
1
y
2
2
2
y
3
x Py
f
T
x Ax
TP AP
2 0
0 1
0 0
0
0
1
Q P
1
0
0
0
0
0
1
1 0
PC
T
Q AQ C P AP C
(
)
T
T
2
0
0
0
0
1 0
0
1
)
T
y Q AQ y
(
T
2
2
y
1
y
2
2
2
y
3
f
T
x Ax
x
y
t
arctan
3
3
t
t
2
d y
2
dx
1
t
dy
dt
dy
dx
dx
dt
d
[3(1
t
2 2
) ]
2
d y
2
dx
d
dx
2
d y
2
dx
t
1
48
2
3(1
t
2 2
)
3 3
t
1
1
t
[3(1
t
dt
2
2 2
) ]
dx
dt
2
t
) 12 (1
t
t
2 2
)
t
12 (1
1
t
1
2
( )
f x
2
x
2 x
0x
n
nf
(0)
n n
1 ln 2 n
2
f
n
0
C
2
n
2 2
x
(
n
2)
x
0
(
n n
2
x f
x
2
1)
2 ln 2
n
2
(
n n
1) ln 2
n
2
x
0
t dt
1
1,
1
5
1f
f x
2
2
x
( )
f t dt
( )
x
2
x
0
( )
f t dt
2
2
(
x f x
2
)
( )
x
x
( )
t dt
0
1
f
(1)
1,
(1) 1 2 (1) 5,
f
0
f
(1) 2
y
y x
''
y
y
' 2
y
0
0x
y x
y x
e
x
2
x
2
e
y
0
3
y
0
0
2
2
0
21,
1
2
2C
1
1C
2
y
2 x
e
x
2
e
y C e C e
x
1
2
2
x
y
0
3
y
0
0
Z
z x y
,
e
x
2
y
z
3
xyz
1
dz
0,0
1 d
x
3
2d
y
x
0,
y
0
0z
x
2
y
3
z
(3
e
xy
)
x
2
y
3
z
(3
e
xy
)
z
x
z
y
yz
e
x
2
y
3
z
xz
2
e
x
2
y
3
z
1
3
1
3
(0,0)
2
3
dy
,
z
y
2
3
.
(0,0)
d
x
2d .
y
2, 2,1
B A
2
A E
E
3
z
x
dx
dz
|
(0,0)
3
1
3
A
B
A
2, 2,1.
B
3,7,1.
|
B
| 3 7 1 21
( )
f x
x a
ln(1
x
)
bx
sin
x
( )g x
3
kx
( )
f x
( )g x
x
0
,
,a b k
a
1,
k
1
3
,
b
1
2
ln(1
x
)
x
2
x
2
3
x
3
(
o x
3
)
sin
x
x
3
x
3!
3
(
o x
)
1 lim
0
x
( )
f x
( )
g x
lim
0
x
x a
ln(1
)
x
3
kx
bx
sin
x
lim
0
x
(1
)
a x
(
b
3
x
(
o x
3
)
2
a
3
a
)
2
kx
x
3
0
0
1
a
ab
2
a
3
k
1
1
a
1
b
2
1
k
3
lim1
0
x
)(
xf
)(
xg
lim
0
x
ax
1ln(
)
x
3
kx
bx
sin
x