math grade 12
DESCRIPTION
N/ATRANSCRIPT
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2014-2015
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Tel: 015 581 133
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015 58 11 33 1
1.
2. ?
3.
4.
5.
6.
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015 58 11 33 2
7.
8.
9.
10.
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015 58 11 33 3
.
I. 2 3Z i
II. Z a bi
2 3Z i 2 3Z i
III. Z a bi
2 3Z i 2 3Z i
IV. Z a bi
3 4Z i 2 2 2 23 4 25 5r Z a b
V. Z a bi W c di
Z W a ca bi c dib d
.
Z a ib
Z a ib
Z a ib
2 2r Z a b
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015 58 11 33 4
sin( 2 ) sin ; cos( 2 ) cosk k
tan( 2 ) tan ; cot( 2 ) cot ;k k k
tan( ) tan ; cot( ) cotk k
I. cos
cos sin
sin
a
rZ r i
b
r
Edl
4 4Z i
22 2 24 4 4 2r Z a b
4 2cos24 2
a
r
4 2sin
24 2
b
r
2cos 02
2sin 02
II3
4
cos sinZ r i
3 34 2 cos sin4 4
Z i
Z a ib
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015 58 11 33 5
II.
1 1 1 1cos sinZ r i 2 2 2 2cos sinZ r i
1 2 1 2 1 2 1 2cos sinZ Z r r i
1 2Z Z 1Z 2Z 1 1Z i 2 1 3Z i
1 1Z i 2 2 2 21 1 1 1 1 1 2r Z a b
1
1
1
1
11
1
1cos
2
1 4sin
2
a
r
b
r
2 1 3Z i 222 2
2 2 2 2 1 3 2r Z a b
22
2
1cos
2
a
r
22
2
3sin
2
b
r
2 1cos 02
2
3sin 0
2 2 II 2
2
3
1 2 1 2 1 2 1 2cos sin
2 22 2 cos sin
4 3 4 3
11 112 2 cos sin
12 12
Z Z r r i
i
i
1 2 11 112 2 cos sin12 12
Z Z i
III.
1 1 1 1cos sinZ r i 2 2 2 2cos sinZ r i
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015 58 11 33 6
1 1 1 2 1 22 2
cos sinZ r
iZ r
2 0Z
12
Z
Z 1 1 3Z i 2 1Z i
1 1 3Z i
222 2
1 1 1 1 1 3 2r Z a b
11
1
1cos
2
a
r
11
1
3sin
2
b
r
1 1cos 02
1
3sin 0
2 1 II 1
2
3
2 1Z i 2 2 2 22 2 2 2 1 1 2r Z a b
2
2
2
2
22
2
1cos
2
1 4sin
2
a
r
b
r
1 1 1 2 1 2
2 2
cos sin
2 2 2 5 5cos sin 2 cos sin
3 4 3 4 12 122
Z ri
Z r
i i
12
5 52 cos sin
12 12
Zi
Z
IV. n
cos sinZ r i
cos sin cos sinnn nZ r i r n i n
n
cos sin cos sinni n i n (Demoivre)
-
015 58 11 33 7
13Z 3Z i
2
2 2 23 1 2r Z a b
3 3cos 02 2
a
r
1sin 0
2
b
r II
5
6
5 53 2 cos sin6 6
Z i i
5 5 5 58192 cos 10 sin 10 8192 cos sin6 6 6 6
i i
13 4096 3 4096Z i
V.n
cos sinZ r i n Znn
2 2cos sin , 0 , 1nk k kW r i k nn n
Edl
5 1Z i
22 2 21 1 2r Z a b
1 2cos 022
a
r 1 2sin 0
22
b
r
II 34
3 32 cos sin4 4
Z i
13
13 135 5 13 5 13 52 cos sin 2 cos sin6 6 6 6
Z i i
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015 58 11 33 8
53 3
2 24 42 cos sin 0 , 4
5 5k
k k
W i k
Edl
100
3 30 2 cos sin
20 20k W i
10 101
3 32 2
11 114 41 2 cos sin 2 cos sin5 5 20 20
k W i i
10 102
3 34 4
19 194 42 2 cos sin 2 cos sin20 20 20 20
k W i i
10 103
3 36 6
27 274 43 2 cos sin 2 cos sin5 5 20 20
k W i i
10 104
3 38 8
35 354 44 2 cos sin 2 cos sin5 5 20 20
k W i i
1. 1 2 1 2 1 2, ,z z z z z z 12
z
z
) 1 22 5 , 1 7z i z i ) 1 22 3 , 2 3z i z i
) 1 23 2 , 1 7z i z i ) 1 23 5 , 3 5z i z i
2.
) 2
1 213 12
8 6 2
iiz
i i
) 3 2 2 31
i iz
i
-
015 58 11 33 9
) 1 12 1 2 1
i iz
) 41 63 1 6
50 1 7
i iz
i
)
2 3
3 2
1 2 1
3 2 2
i iz
i i
)
2
1 3
2
iz
i
3. ) 1 21 , 1 3z i z i ) 1 2z z ) 1
2
z
z ) 21z
) 2 2z i ) 3 3z i ) 2 2 3z i ) 1z i
) 3z i ) 4z i )
13
7
1
1
iz
i
4. ) 15z ) 20z i ) 4 3 4z i ) 5 5 3z i ) 2 3 2z i ) 4 4z i ) 1z i ) 3z i ) 3z i ) 7z 5. 1 2z z 1
2
z
z
) 1 21 , 1z i z i ) 1 23 , 3z i z i ) 1 22 2 3 , 5z i z i ) 1 25 5 , 3z i z i 6. ) 121 i ) 101 i ) 53 3i )
3
1 3i ) 5
4 3 4i ) 102 2i )
7
3 i ) 50
3 i ) 25
2 2i ) 15
2 2i
7.
) 3 2 2i ) 4 4i ) 4 1 3i ) 4 2 2 3i ) 5 1 i
) 6 1 ) 8 i ) 4 8 8 3i ) 45 2 2i ) 4 4 3 4i
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015 58 11 33 10
. lim ( ) 0; 0 : ( )
x af x L x a f x L
2
lim 2 1 3x
x
0
2
2
1lim
2x x
0M
2
2
1lim
2x x
2
3 1lim
2x
x
x
2 0x
( )f x L
(2 1) 3 2 4 2 2 22
x x x x
02
2x
2
lim 2 1 3x
x
lim ( ) 0 ; 0 : ( )x a
f x M x a f x M
( )f x M
2
2 21 1 12 2
2M x x
x M M
12x
M
10
M 0 2x
lim ( ) 0 ; 0 : ( )x a
f x M x a f x M
2x 2 0x
0M 3 1 3 6 5
( )2 2
x xf x M M M
x x
-
015 58 11 33 11
2
3 1lim
2x
x
x
3 5lim 32x
x
x
3 5lim 32x
x
x
4 1lim 22 1x
x
x
1 1 12 12 2
x x
4 1lim 22 1x
x
x
5 5 53 3 3
2 2 2M M M
x x x
2 1 52
5 3 3
xx
M M
50
3M
2 x
lim ( ) 0; 0 : ( )x
f x L N x N f x L
0 3 5 3 5 3 6
( ) 32 2
x x xf x L
x x
1 1 12
2 2x
x x
x
1 12 2x x
12 0N
x N
lim ( ) 0; 0 : ( )x
f x L N x N f x L
0 4 1
( ) 22 1
xf x L
x
4 1 4 2 1 1 12 1
2 1 2 1 2 1
x xx
x x x
x1 1
02 2
N
x N
-
015 58 11 33 12
25 4 1
lim2x
x x
x
0M
11 02x
25 4 1
lim2x
x x
x
22
lim3 5x
x x
x
0M
22
lim3 5x
x x
x
3limx
x
3( )f x M x M 3x M
lim ( ) 0; 0 : ( )x
f x M N x N f x M
25 4 1( )
2
x xf x M M
x
2 5 2 6 2 115 10 6 12 112 2
x x xx x xM M
x x
115 6
2x M
x
x
65 6 5 6
5
Mx M x M x
60
5
MN
x N
lim ( ) 0; 0 : ( )x
f x M N x N f x M
22( )
3 5
x xf x M M
x
11 55
3 9 9 3 5
xM
x
x
55
09 3 5x
11 11 113
3 9 3 9 3
x xM M x M
113 0
3N M x N
lim ( ) 0; 0 : ( )x
f x M N x N f x M
0M
-
015 58 11 33 13
3x M 3 0N M x N
3limx
x
lim ( ) 0; 0 : ( )x
f x M N x N f x M
lim 1x
x
0M 2( ) 1 1f x M x M x M
2 21 1x M x M 2 1 0N M x N
lim 1x
x
lim ( ) 0 ; 0 : ( )
lim ( ) 0 ; 0 : ( )
lim ( ) ; 0 : ( )
lim ( ) 0 ; 0 : ( )
x a
x a
x
x
f x L x a f x L
f x M x a f x M
f x L N x N f x L
f x M N x N f x M
L f , ( )x a x a
0 ; 0 0 ( )a x f x L lim ( )x a
f x L
2
lim 2x
x
0 ( ) 2f x L x
2 2
2 2 2 2 2x x x
2 2 2 22 2 2 2 2 2 2 2 2 2 2x x
22 2 2x 22 2 0 0 2 x
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015 58 11 33 14
2
lim 2x
x
L f , ( )x a x a
0 ; 0 0 ( )x a f x L lim ( )x a
f x L
4
lim 2x
x
0 ( ) 2 2f x L x x
2 22 2 2 2x x
2 2 2 24 4 4 4 4 4 4x x 24 4x
2 4 0 0 4x
4
lim 2x
x
lim ( )x a
f x L
lim ( ) lim ( )x a x a
f x f x L
2 4
( )2
xf x
x
4
2
2 2 2 2
2 24lim ( ) lim lim lim 2 4
2 2x x x x
x xxf x x
x x
2
2 2 2 2
2 24lim ( ) lim lim lim 2 4
2 2x x x x
x xxf x x
x x
22 2
lim ( ) lim ( ) lim ( ) 4xx x
f x f x f x
lim[ ( ) ( )] lim ( ) lim ( )
lim ( ) lim ( ) ( )
lim[ ( ) ( )] [lim ( )] [lim ( )]
x a x a x a
x a x a
x a x a x a
f x g x f x g x
kf x k f x k
f x g x f x g x
CacnM Ynefr
lim ( )( )
lim( ) lim ( )
x a
x a
x a
f xf x
g x g x
-
015 58 11 33 15
lim ( )
( )
lim[ ( )] [lim ( )] ( )
lim ( ) lim ( )
lim lim
lim[ln ( )] ln[lim ( )]
x a
n n
x a x a
n nx a x a
f xf x
x a x a
x a x a
f x f x n
f x f x
k k k k
f x f x
nig
Edl CacMnYnKtr; uWLaTbI viCm an
( ) ( ) ( ) ] , [ lim ( ) lim ( )
] , [ lim ( )
( ) ( ) ] , [ ] , [ lim ( ) lim ( )
( ) , lim ( )
x a x a
x a
x a x a
x a
f x g x h x x f x h x L
a g x L
f x g x x a f x g x
f x D a D f x f
I
I
eb ceM BaH ehyI
Edl
ebI ceM BaH eb
ebI
enaHeK)an
enaHeK)an
manEdnkMNt; enaHeK)an ( )
( ) ( ) lim ( ) lim ( )x x
a
f x g x x A g x f x
Ieb cMeBaH Edl enaHeK)an
( ) ( ) lim ( ) lim ( )
( ) ( ) ( ) lim ( ) lim ( ) lim ( )
x x
x x x
f x g x x A g x f x
f x g x h x x A f x h x L g x L
ebI ceM BaH Edl
ebI ceM BaH Edl
enaHeK)an
enaHeK)an
( ) ( ) lim ( ) lim ( )
( ) ( ) lim ( ) lim ( ) ( )
lim [ ( )] ( )
x x
x a x L
x a
f x g x x A f x L g x L
f x g x g x L f x f L
f g x f L
ebI cMeBaH Edl
ebI ngi
ngi enaHeK)an
nig CaGnKu mnE dlmanlmI tIenaHeK)an
lim ( ) lim ( )
1 1 1lim 0 lim 0
( ) ( )
1 1lim 0 lim 0
( ) ( )
( ) ( )lim 0 0 lim 0 0
( ) ( )
x a x a
x a x a
x a x a
x a x a
f x L g x M
L Lf x L f x
L Lf x f x
f x f xL m L m
g x g x
ebI nig
kalNa kalNa
kalNa kalNa
kalNa ehIy kalNa ehyI
.
. 0 0
sinlim lim 1
sinx x
ax ax
ax ax
0
sin 3limx
x
x
-
015 58 11 33 16
0 0 0
sin3 sin3 sin3lim lim 3 3lim 3 1 3
3 3x x x
x x x
x x x
. 0 0
1 coslim lim 0
1 cosx x
ax ax
ax ax
0
1 cos 2limx
x
x
0 0 0
1 cos 2 1 cos 2 1 cos 2lim lim 2 2lim 2 0 0
2 2x x x
x x x
x x x
. 0 0
tanlim lim 1
tanx x
ax ax
ax ax
0
tan 5limx
x
x
0 0 0
tan5 tan5 tan5lim lim 5 5lim 5 1 5
5 5x x x
x x x
x x x
. 20
1 cos 1lim
2x
x
x
20
1 cos 2limx
x
x
2 2 20 0 0
1 cos 2 1 cos 2 1 cos 2 1lim lim 4 4lim 4 2
4 4 2x x x
x x x
x x x
. lim xx
e
lim 2 xx
x xe
lim 2 lim 2 lim 2x xx x x
x xe x xe
. lim 0xx
e
2lim 2 1x xx
e e
2
2 22lim 2 1 lim 1 lim 1 0 1 1x x x xx x x
e e e e
-
015 58 11 33 17
. lim , 0x
nx
en
x
2
2
2lim
x
x
e x x
x
2
2 22
2 2 2 2
1 21
2 1 2lim lim lim 1
x
x x
x x x
ex
x x xe x x e
x x x x x
. lim 0 , 0n
xx
xn
e
lim2 1
x
xx
e x
e
1 1
1 0 1lim lim lim
112 1 2 0 222
x
x x x
xx x xx
xx
x xee x e e
ee
ee
. lim lnx
x
1lim 2lnx
xx
1 1lim 2ln lim 2 lim ln 0 2x x x
x xx x
. 0
lim lnx
x
20
lim 1 5lnx
x x x
2 20 0
lim 1 5ln lim 1 5limln 1 5xx x
x x x x x x
. 0
lim ln 0 , 0nx
x x n
0
1lim 4 lnx
x x xx
2 20 0 0 0
1lim 4 ln lim 1 4 ln 1 4 lim lim ln 1 4 0 0 1x x x x
x x x x x x x x xx
-
015 58 11 33 18
. 0
lnlim , 0
nx
xn
x
2
20
4 3 lnlimx
x x x
x
2
2 2
2 2 20 0 0
3 ln4
4 3 ln 3 lnlim lim lim 4 4x x x
xx
x x x xx x
x x x x
. lnlim 0 , 0nx
xn
x
2
2
4 lnlim
4lnx
x x x
x x
2
2 2 2
22
22
4 ln 4 ln1 14 ln 1 0 0
lim lim lim 1lnln4ln 1 0
1 41 4x x x
x xxx x x x x x x
xxx xx
xx
. 0
lim 0 , 0ln
n
x
xn
x
2
20
3 4lnlim
2 3lnx
x x x
x x x
22
2
22 20 0 0
3ln 4 3 4
ln ln3 4ln 0 0 4 4ln lnlim lim lim2 3ln 0 0 3 32
2 3ln 3ln lnln ln
x x x
x x x xxx xx x x x x
x xx x x x xx
x xx x
. lim , 0ln
n
x
xn
x
2 ln
limlnx
x x
x
2
2 2ln 1
lnlnlim lim lim 1
ln ln lnx x x
xx
xx x x
x x x
. lim , 1xx
a a
-
015 58 11 33 19
lim 4 3xx
lim 4 3 4 lim 3x xx x
. lim 0 ,1 0xx
a a
1lim 1 2 lim 1 1 0 12
x
xx x
. lim 0 , 1xx
a a
1 2lim 1 2 2x xx
2
2 21 2lim 1 2 2 lim 1 2 lim 1 2 1 0 1x x x xx x x
. lim ,1 0xx
a a
1 2
1 1lim 1
2 2x xx
22
2
1 2
1 1 1 1lim 1 lim 1 lim 1 1
2 2 2 2x x x xx x x
. lim , 1x
nx
aa
x
2
2
3 4lim
2 1
x x
x
x
x x
2
2 22 2 2
22
22
3 4 3 41 13 4 1
lim lim lim2 12 12 1 1 0 0
11
x xx x
x x
x x x
xx xx x x
x xx
x xx x
. lim 0 , 1 0x
nx
aa
x
2
2
3 4lim
3 2
x x
x
x
x x
-
015 58 11 33 20
2
2 22 2 2
22
22
3 4 3 41 13 4 1 0 0
lim lim lim 13 23 23 2 1 0 0
11
x xx x
x x
x x x
xx xx x x
x xx
x xx x
. 1
0
1lim 1 lim 1
x
x
x xx e
x
8lim2
x
x
x
x
8 8 101 1 12 2 2
x x
x x x
102 2
10108 10 10lim lim 1 lim 1
2 2 2
xx xx x
x x x
xe
x x x
2
2
20
1lim
1
x
x
x
x
2 2
2 2 2
1 1 11 1 1
1 1 1
x x
x x x
2
22 2 2 2
20
1 12 lim1 0
2 2 20 0 0
1 1 1lim lim 1 lim 1 1
1 1 1
x
x
xx x x x
x
x x x
xe e
x x x
. 0 0
ln 1 1lim lim 1
ax
x x
x e
x ax
2
30
3 4lim
2 3
x x
x xx
e e
e e
22
2 0
33 30 0
0
1 11 1 lim 2 3321 3 3 1 2 3 1
lim lim 11 11 2 2 1 3 2 11 1
2 lim 3 23
x xx x
x x x
x xx x x xx x
x
e ee ex xe e x x
e ee e e e
x x x x
3
0
ln 1 3 1lim
sin 4
x
x
x e
x
-
015 58 11 33 21
3 3
3
0 0 0
ln 1 3 1 ln 1 3 13 3ln 1 3 1 1 3 1 3 33 3lim lim lim
sin 4 sin 4sin 4 1 4 24
4
x x
x
x x x
x e x ex e x x x
x xx
x x
.0
1lim ln
x
x
aa
x
0
7 8lim
3 4
x x
x xx
0 0 0
515
44 145 4 ln 5 ln 4
lim lim lim3 4 ln 3 ln 433
14 144
x
x
x
x x
xx x xx x xx
x
x
.0
(1 ) 1lim
n
x
xn
x
3 220
3 3lim
2x
x x x
x x
3
33 2 3 2
2 22 20 0 0 0
1 1
1 13 3 3 3 1 1 3lim lim lim lim
2 2 1 1 21 1 1 1x x x x
x
xx x x x x x x
x x x x x x
x
. ( )lim( )x a
f x
g x 0
0 0
0
- ( )x a 2( ) ............x a
- ( )x a ( x a x a 0x a ( )x a )
- x a
3
0
1 1lim
3x
x
x
00
-
015 58 11 33 22
23 3 33
0 0 23 3
1 1 1 1 11 1
lim lim3 3 1 1 1
x x
x x xx
x x x x
0 02 23 33 3
1 1 1 1lim lim
93 1 1 1 3 1 1 1x x
x
x x x x x
3 31 1u x x u 0x 1u
3
3 2 20 1 1 1
11 1 1 1 1lim lim lim lim
3 93 1 3 1 1 3 1x u u u
ux u
x u u u u u u
3
0
1 1 1lim
3 9x
x
x
.
2
2 2
3 3 2 2
5 5 4 3 2 2 3 4
1 2 1
3 3 2 2
5 5 4 3 2 2 3 4
1 2
( )( )
( )( )
( )( )
( )( ............ )
( )( )
( )( )
( )(
nn n n n n
n n n n n
a b a b a b
a b a b a ab b
a b a b a a b a b ab b
a b a b a a b ab b
a b a b a ab b
a b a b a a b a b ab b
a b a b a a b a
3 2 1.......... ( 1) .... )p p n p nb a b b n CacnM nY Kte; ss
n nn nnnn nn n
n nn n
nn
bbabaa
baba
1221 ........................
-
015 58 11 33 23
- x ( )f x ( )g x
- ( 0x x )
- x
3 2
3 2
4 5 1lim
3 5 6 7x
x x x
x x x
23
3 3 33 2 2 3
3 2 23
2 33 3 3
4 5 1 4 5 11 14 5 1 1
lim lim lim4 6 73 4 6 7 34 6 7 33
x x x
x xx
x x xx x x x x x
x x x x xx
x x xx x x
3 2
3 2
4 5 1 1lim
3 5 6 7 3x
x x x
x x x
1 11( ) ...................
n n n n
n nP x a x a x a x a
x nnxaxP )( . x
2 2lim 5 1 3 2x
x x x x
2 2 2 2
2 2
2 2
5 1 3 2 5 1 3 2lim 5 1 3 2 lim
5 1 3 2x x
x x x x x x x xx x x x
x x x x
2 2
2 2
2 2
32
5 1 3 2lim lim
5 1 3 25 1 3 21 1
x x
xx x x x x
x x x xx
x x x x
x x x x x x
2 22 2
3 32 2
lim lim 15 1 3 25 1 3 2
1 11 1x x
xx x
xx x x xx x x x
-
015 58 11 33 24
2 2 2 2
3 32 2
lim lim 15 1 3 2 5 1 3 2
1 1 1 1x x
xx x
xx x x x x x x x
2 21
lim 5 1 3 21x
xx x x x
x
ebI
ebI
.
) 0x ( x )
. 0 0
sinlim lim 1
sinx x
ax ax
ax ax .
0 0
1 coslim lim 0
1 cosx x
ax ax
ax ax
. 0 0
tanlim lim 1
tanx x
ax ax
ax ax .
20
1 cos 1lim
2x
x
x
+ x 0 0
sin tanlim lim
180x x
x x
x x
+ x 0 0
sin tanlim lim
200x x
x x
x x
) 0x x ( x ) 0x x
0u x x ( 0u x x ) 0 0x x u 0x u x 0x x u
.
0 0
sinlim lim 1
sinx x
ax ax
ax ax .
0 0
1 coslim lim 0
1 cosx x
ax ax
ax ax
. 0 0
tanlim lim 1
tanx x
ax ax
ax ax .
20
1 cos 1lim
2x
x
x
2
tanlim
2x
x
x
, 2
1
1lim
sinx
x
x
, 2
2
1 sinlim
2
x
x
x
-
015 58 11 33 25
1. 2 2u x x u 2x 0u
2 0 0 0
tan 2 tan 2 tan 2tanlim lim lim lim
2x u u u
u u ux
x u u u
0 0
tan tanlim lim 1u u
u u
u u
2
tanlim
2x
x
x
2. 1 1u x x u 1x 0u
2
1 1 0
1 1 1 11lim lim lim
sin sin sinx x u
x x u ux
x x u
0 0
2 2 2 0 2lim lim 1
sin sinu u
u u u u
u u
2
1
1 2lim
sinx
x
x
3.2 2
u x x u
2
x
0u
2 2 20 0
2
1 sin1 sin 1 cos 12
lim lim lim2
2
u ux
ux u
u ux
I KNnalImItxageRkam
-
015 58 11 33 26
1, 2
0
1lim
1 1x
x x
x x
21 2 3 2
2, lim1 2 1x
x x x
x x
2 23, lim 8 1 3
xx x x
4, 2
0limx
x x
x
5,
2
3
2
4 9lim
2 3x
x
x
6,
22
2lim
2 2x
x
x
7,
2
24
2 4 24 1lim
4x
x x
x x
8, 31
1lim
1x
x
x
9,
24
2lim
3 2 1x
x
x
10,
31
2 2lim
26 3x
x
x
11,
41
1lim
17 2x
x
x
12,
4lim
4x
x
x
13, 0
3lim 4x x
14,
1lim 1x
x x
15, 1
lim 2 1x
x x
16, 2
0
1 1limx x x
II KNnalImItxageRkam
1, 5 2
3
2 1lim
1x
x x
x
2,
2
2020lim
2010x
x
x 3,
2 3
5
2 3 3 2lim
4444444x
x x
x
4,
2 3
2
2 1 5 1lim
100 100 1979x
x x
x x
5,
2 4
3 6
2 5 3 1lim
1 4 3 2x
x x x
x x x
III KNnalImItxageRkam
1, 2 2lim 1 1x
x x x x
2, 2lim 3x
x x x
3, lim 3 5x
x x
4, lim 1x
x x
5, 2 2lim 2 4x
x x x
6, 2lim 2 1 4 4 3x
x x x
IV KNnalImItxageRkam
1/ 0
sin 7lim
49x
x
x 2/
0
sin 2010lim
sin 2008x
x
x 3/
0
1 coslim
sin 5x
x
x
4/
2
20lim
sin 3x
x
x 5/
0
1 cos 4limx
x
x
6/
0
3 1 coslimx
x
x
7/
2
2lim
cosx
x
x
8/
20
1 2lim cos 3
cosxx
x x
9/
2
1
sinlim
1x
x
x
10/ 2
0
1 coslimx
x
x
11/
2
1 sin coslim
1 sin cosx
x x
x x
12/ 2
1lim 1 cosx
xx
13/
20
2 1 coslim
sinx
x
x
-
015 58 11 33 27
14/ 2
2
1 sinlim
2
x
x
x
15/ 20
1 cos cos 2limx
x x
x
16/
0
1 sin coslim
1 sin cosx
x x
x x
17/
2
2
6
2sin 3sin 1lim
4sin 1x
x x
x
18/ 20
1 sin cos 2lim
sinx
x x x
x
19/
0
sin 3lim
2 2x
x
x 20/
2
0
sinlim
1 sin cosx
x
x x x 21/
0
sinlim
1 cosx
x ax
bx
22/
1 sin2lim
x
x
x
23/
1
cos2lim
1x
x
x
24/
1 coslim
sinx
x
x
25/ 2
2
2
sinlim
cosx
xtg x
x
26/
4
sin 2 cos 2 1lim
cos sinx
x x
x x
27/
0
1sin
6 2limx
x
x
V KNnalImItxageRkam
1/ lim 1xx
e
2/ lim 200xx
e
3/ lim 4xx
e
4/ lim 3 5xx
e x
5/ lim 7777xx
e x
6/ 3lim xx
x x e
7/ 3lim xx
x x e
8/ 1
32limx
xe
9/
3lim
1 xx e
10/ lim 2 xx
x xe
11/ 1
lim1xx e
12/ 0
1lim
1xx e
13/ 2lim 1xx
e x
14/ 2lim x
xe
15/ 2lim x
xx xe
16/ lim2 1
x
xx
e x
e
17/
0
2lim
3
x
x
x
x
18/ lim
1
x
x
x
x
19/
21
lim3
x
x
x
x
20/
20
sin 1lim
x
x
x e
x x
21/
sin sin 2
0lim
x x
x
e e
x
22/ 0
1 sinlim
1 1
x
x
e x
x
23/
2 2
0lim
sin 2
x x
x
e e
x
24/
2
0
sin 1lim
tan
x
x
e
x x
25/
2
20
coslim
x
x
e x
x
-
015 58 11 33 28
VI KNnalImItxageRkam
1/ 20
lim 3 lnx
x x x
2/ 0
lim 2 lnx
x x
3/ 0
1lim 2lnx
xx
4/ 20
lim 1 5lnx
x x x
5/ 0
lim ln ln 3x
x x 6/ 0
1lim
lnx
x
x
7/ 2lim 3 lnx
x x x
8/ lim 2 lnx
x x
9/ 1
lim 2lnx
xx
10/ 2lim 1 5lnx
x x x
11/ lim ln ln 3x
x x
12/ 1
limlnx
x
x
13/ 0
lim1x
x
x
14/ 2lim ln 1
xx x
15/ lim ln 1 ln
xx x x
16/ 2ln 2 2 1
limx
x x x
x
17/
2
ln 1lim , 0
x
x
e xx
x
18/
ln 1lim
2010xx
x x
e
19/
3
2
lnlimx
x
x 20/
3
2
lnlim
1x
x
x 21/
0
ln 2lim
1x
x
x
VII . KNnalImItxageRkam
2
2
2
sin 14lim
2
x
x x
x
3 2
1
2lim
sin 1xx x
x
1
0lim 1 sin 2 xx
x
0
lim sintgx
xx
2 1
2lim
1
x
x
x
x
2
3 5lim
3 1
x
x
x
x
0
1 coslim
1 cosxx
x
2
20
1 coslimx
x x
x
2
0
1 cos 2lim
sinxx
x x
IIX . KNnalImItxageRkam
-
015 58 11 33 29
2
0
1lim
3
x
x
e
x
sin
sin
0
sinlim
x
x x
x
x
x
0
ln 2010 1limx
x
x
1
lim1
x
x
e e
x
2
2
2
1lim
1
x
x
x
x
1
0lim cos sin xx
x x
IX . KNnalImItxageRkam
1 . sin sinlimx a
x a
x a
2 > cos coslim
x a
x a
x a
3>
cot cotlimx a
x a
x a
4 > tan tan
limx a
x a
x a
5 >
3
sin3
lim1 2cosx
x
x
6 >
3
3
tan 3tanlim
cos6
x
x x
x
7> 3
3
4
1 cotlim
2 cot cotx
x
x x
8>
3
20
cos coslim
sinx
x x
x
9 >
0
1 cos cos 2 cos3lim
1 cosx
x x x
x
10> 30
1 tan 1 sinlimx
x x
x
11>
2
0lim
1 sin cosx
x
x x x
12>
20
sin 2 2sin sinlimx
a x a x a
x
13>
20
cos 2 2cos coslimx
a x a x a
x
14>
20
tan 2 2 tan tanlimx
a x a x a
x
15>
20
cot 2 2cot cotlimx
a x a x a
x
16> 2
0
sin sin 2 sinlimx
a x a x a
x
17>
220
tan tan 2 tanlimx
a x a x a
x
18> 20
cos cos3limx
x x
x
19 >
4
lim tan 2 tan4x
x x
20>
1lim 1 tan
2xx x
21>
2 1
1
0lim 2 1
xx xx
xe
22>
sec2
1lim 2
x
xx
23>
cot
1lim 1 sin
x
xx
24> 3
1
sin
0
1 tanlim
1 sin
x
x
x
x
25 >
1
sinlim
sin
x a
x a
x
a
26> 2
1
0
coslim
cos 2
x
x
x
x
-
015 58 11 33 30
27 > tan 2
4
lim tanx
x
x
28> tan
2
lim sinx
x
x
29 > cot
0lim tan
4
x
xx
30> 0
lim cosxx
x
31> 1 1
lim sin cos
x
x x x
32>
2cot2
0lim 1
x
xx
33> 0
lim 1 2xx
x
34> tan 2
4
lim tan8
x
x
x
35> lim , 0
x a
x a
a xa
x a
36 > lim , 0x a
x a
x aa
x a
37>
ln lnlimx a
x a
x a
38>
1
0lim x xx
x e
X . KNnalImItxageRkamedayeRbIniymny
1> 5
lim 5x
x
2> 2 3lim 23x
x
x
3>
22lim
3 6x
x x
x
4> 3
lim 2 3 3x
x
5> 4
lim 3 5x
x
6> 22 3
lim 24x
x
x
7>4
lim 2 1 3x
x
8>4
2 3lim
4x
x
x
9>
2
1
2 3lim
1x
x
x
-
015 58 11 33 31
I. ( )f x x a ( )
lim ( ) ( )x a
f a
f x f a
kNM t); an
f 26 3
( ) 93
3
x
f x xx
x
ebI
ebI
(3) 6 (1)f
2
3 3 3
9lim ( ) lim lim 3 6 ....(2)
3x x x
xf x x
x
1 & 2 3
lim ( ) (3) 6x
f x f
f 3x
II.
( )f x x a ( )
lim ( ) ( )x a
f a
f x f a
kNM t); an
21 0
( )0
x
f x xx x
x
ebI
ebI
f 0x
(0) 1 .....(1)f
2
0 0 0lim ( ) lim lim 1 1 ..... 2x x x
xf x x x
x
1 & 2 0
lim ( ) (0) 1x
f x f
f 0x
-
015 58 11 33 32
( )f x x a ( )
lim ( ) ( )x a
f a
f x f a
kNM t); an
3 1( )2 5 1
xf x
x x
ebI
ebI
f 1x
( 1) 3 ...... 1f
1 1
lim ( ) lim 2 5 3 ...... 2x x
f x x
1 & 2 1
lim ( ) ( 1) 3x
f x f
f 1x
( )f x x a ( )f x ( )f x x a
( )f x x a lim ( ) ( ) lim ( ) ( )x a x a
f x f a f x f a
2
2
11
1
( ) 2 1
3 1
xx
x
f x x
x x
ebI
ebI
ebI
f 1x ?
( 1) 2 ........(1)f
2
1 1 1
1lim ( ) lim lim 1 2 ..... 2
1x x x
xf x x
x
21 1
lim ( ) lim 3 2 ....... 3x x
f x x
1 , 2 & 3 1 1
lim ( ) lim ( ) ( 1) 2x x
f x f x f
f 1x
-
015 58 11 33 33
III.
( )f x ( , ) ( ) ( , )f x x CabR; tg;
3( )4
xf x
x
0 , 1
0 1 0 , 12
x 0x
1
31 52 ...... 1
12 94
2
f
1 1
2 2
3 5lim ( ) lim ....... 2
4 9x x
xf x
x
1 1
2 2
3 5lim ( ) lim ........ 3
4 9x x
xf x
x
1 , 2 & 3 0 , 1
( )f x ( ) ( , )
[ , ] ( )
( )
f x
f x
f x
Cabk; ugcena HCabx; agsa M nCabx; ageqgV n
1 cos 2
sin 1( ) sin
2 0
xx x
f x x
x
ebI
ebI
f
[ , ]2 2
(0) 2f 2 sin1 cos 2
( ) sin sinsin sin
xxf x x x
x x
0x
sin 2 0
( )sin 2 0
x xf x
x x
ebI
ebI
0 0
lim ( ) lim sin 2 2 (0)x x
f x x f
0x
-
015 58 11 33 34
0 0
lim ( ) lim sin 2 2 (0)x x
f x x f
0x
f 0 ,2 2
x
f ,2 2
f
1 ln 0
1 0
( ) sin1
1
1
x x x
x
f x xx
x
x
ebI
ebI
ebI
ebI
f 0 , 1
f 0 , 1
(0) 1f (1)f
0 0
lim ( ) lim 1 ln 1 (0)x x
f x x x f
1 1
sinlim ( ) lim
1x x
xf x
x
1 1u x x u 1x 0u
1 0 0
sin sinlim ( ) lim lim (1)x u x
u uf x f
u u
f 0 , 1
IV.
( )f x x a lim ( )x a
f x L
( )f x
x a ( )
( )f x x a
g xL x a
ebI
ebI
f 2 9
( )3
xf x
x
f
3x ?
f 3 0 3x x f {3}D
2
3 3 3
9lim ( ) lim lim 3 6
3x x x
xf x x
x
-
015 58 11 33 35
g
2 93
( ) 3
6 3
xx
g x x
x
ebI
ebI
g f 3x
V.
f g x a
. ( ) ( )f x g x x a . ( ) ( )f x g x x a
. ( )k f x x a k . ( )( )
f x
g x x a ( ) 0g x
11( ) ......
n n
n nP x a x a x a
1, ,...... ,n na a a 0na x
( )( ) , ( ) 0( )
P xR x q x
q x x
( ) ( )nf x P x x
sin , cosy x y x
tany x Zkkx ,2
coty x Zkkx ,
-
015 58 11 33 36
.
cos
2( ) 2
12
xx
xf x
x
ebI
ebI
f
.2 1
( )1
x xf x
Ax B x
ebI
ebI A Bngi
f
.
2sin2
( ) sin2 2
cos2
x x
f x A x B x
x x
ebI
ebI
ebI
A Bngi f 2
x
. 2sin ] , 2 [
3
( ) 2 [ 2 , 4 ]
cos ] 4 , [2
x x
f x Ax Bx x
x x
ebI
ebI
ebI
A Bngi f
.2 1 2
( )2011 2
x xf x
x A x
ebI
ebI A f 2x
.1 cos 2
sin 0( ) sin
2 0
xx x
f x x
A x
ebI
ebI
A f 0x
-
015 58 11 33 37
.
2 2 1
( ) sin 1 22
2
Ax x
f x x x
Ax B x
ebI
ebI
ebI
A Bngi f
. 34 0
( )0
sin
x x
A x
f x e ex
x
ebI
ebI A f 0x
.2
2 0( )
4 0
x xe ex
f x x
A x
ebI
ebI
A f 0x
-
015 58 11 33 38
I. -
0,x x 0( ) , ( )y f x y f x 0,x x 0x x x
0 0 0 0( ) ( ) ( ) ( )y y y f x f x f x x f x 0] , [x x
II.()
0 0, ( )fx x D yx
0x (
00
( ) ( )f x f x
x x
0x x ) 0x
0 'y 0'( )f x 0
0 0 00
0 00
( ) ( ) ( ) ( )'( ) lim lim lim
x x x x
f x x f x f x f xyf x
x x x x
0
0
0
( ) ( )limx x
f x f x
x x
( )f x 0x
2( ) 2 1f x x x 0 2x
2 2 22 2 2
2 1 2 2 2 1( ) (2) 2 1 9'(2) lim lim lim
2 2 2x x x
x xf x f x xf
x x x
2
2 2 2
2 2 22 2 2 24 2 4lim lim lim
2 2 2x x x
x xx x xx x
x x x
2
lim 4 6x
x
f 0 2x '(2) 6f
III.-
- ( )f x 0x 00
( ) ( )f x f x
x x
-
015 58 11 33 39
0x x ( )f x 0x
0
00
0
( ) ( )'( ) lim
x x
f x f xf x
x x
- ( )f x 0x 00
( ) ( )f x f x
x x
0x x ( )f x 0x
0
00
0
( ) ( )'( ) lim
x x
f x f xf x
x x
( )f x 0x
0 0
0 0'( ) '( )...... 1
lim ( ) lim ( )..... 2x x x x
f x f x
f x f x
f 32
0( )
0 0
xx
xf x
x
ebI
ebI
f 0x ?
32
32
22 0
2( ) 2 0
0 0
xx x
x
xf x x x
x
x
ebI
ebI
ebI
2
0 0 0
( ) (0) 2'(0 ) lim lim lim 2 0
0x x x
f x f xf x
x x
2
0 0 0
( ) (0) 2'(0 ) lim lim lim 2 0
0x x x
f x f xf x
x x
'(0 ) '(0 )f f 0x
A Bngi 2
2
1( )
1
x xf x
x Ax B x
ebI
ebI
1x
f 1x f
-
015 58 11 33 40
2 21 1
1 1
1 1
lim limlim ( ) lim ( )
( ) (1) ( ) (1)'(1 ) '(1 ) lim lim
1 1
x xx x
x x
x x Ax Bf x f x
f x f f x ff f
x x
2 2
1 11 1
2 ..... 1 2 .... 1
1 1 lim 1 lim 1lim lim1 1 x xx x
A B A B
x x Ax B A B x A xx x
2 ..... 1 422 2
A B A
BA
4 , 2A B
IV.
-
015 58 11 33 41
.2
2
2 2 1( )
1
x xf x
x Ax B x
ebI
ebI A Bngi f 1x
. f ]0 , [ ln
1 ; ,( )
3 2 0 1
xAx B x A B
xf x
x x
ebI
ebI
-
015 58 11 33 42
A Bngi f 1x
. f 2 2 0
( )
6 2 0
Ax Bx xf x
x x
ebI
ebI
A Bngi f 1x
. A Bngi 2
2
1( )
1
x xf x
x Ax B x
ebI
ebI 1x
. A Bngi 22
2 2 1( )
1
x xf x
x Ax B x
ebI
ebI 1x
. A Bngi 2 1
( )
1
x xf x
Ax B x
ebI
ebI 1x
. 180cm120cm
?
. A M 3 /km h 5 /km h A B
2km B C 6km M
A M M C
. A B I ( )XY
33 ,2
BK m AH 9KH m I ( )XY AI IB
IH
A
B
I HK YX
-
015 58 11 33 43
I. x ( )f x
( )y f x D ( )y f x ( ) 0f x
( )
( )
f xy
g x ( ) 0g x ( )
( )
f xy
g x ( ) 0g x
log ( )ay f x ( ) 0f x , (0 1 )xy a a x
sin
cos
xy
x
x tany x kkx ,2
coty x kkx ,
( )f x 1D ( )g x 2D
( ) ( )f x g x 1 2D D ( ) ( )f x g x 1 2D D
+
x
+ x
) 2 21 1( ) 1 2 1 1 2 12 2
f x x x x x
) 2 2( ) 2 1 3 2 4f x x x x x
-
015 58 11 33 44
)
2 21 1( ) 1 2 1 1 2 12 2
f x x x x x
2 2
2 2 2 2 2 2 2 21 1 1 11 2 1 1 2 1 1 12 2 2 2
x x x x x x x x x x x x
2 21 11 12 2
x x x x
21 0 1 1x x
1 , 1fD
)
2 2 2 2( ) 2 1 3 2 4 1 2 1 1 4 2 4 1f x x x x x x x x x
2 2
2 21 1 4 1 1 1 4 1x x x x
2
11 0
1 44
4
xx
x xx
x
f [4 , )fD
2 21. ( ) ln 3 2 4 5f x x x x x
2
12. ( ) ln
6
xf x
x x
23. ln 1 lg 5 6x x
-
015 58 11 33 45
2
2
5 4 104. ( )
4 5
x xf x
x x
25. ( ) 2 3f x x x
26. ( ) 3 5 4f x x x x 2
27. ( )
1 3
xf x
x x x
18. ( )2
xf x
x
3 2
29. ( )
4 3f x
x x x
2
110. ( ) ln
6
xf x
x x
2 211. ( ) ln 12 ln 1f x x x x x
12 2 1
12. ( ) ln2 1
x
x
xf x
1 1
2 12 213. ( ) ln 4 3 3 2x x
x xf x
1 1
2 12 214. ( ) 4 3 3 2x x
x xf x
3 115. ( )27 3
x
xf x
2216. ( ) ln 12f x x xx x
217. ( ) ln 1 7 25f x x x x
2 3 2
18. ( ) ln1
x xf x
x
219. ln
2
xy
x
20. lg 1y x 1 221. lg
3
xy
x
4
22. log 2x
y
2223. log 8 15xy x x
32 lg 2 lg24. lg 8 4x xy 225. lg lg 5lg 6y x x
-
015 58 11 33 46
II.
f ,a b
+ f ,a b 1 2, ,x x a b 1 2 1 2( ) ( )x x f x f x
3 2( )f x x mx m m 1 , 2
fD
2'( ) 3 2f x x mx
20
'( ) 0 3 2 0 2
3
x
f x x mx mx
0m 2'( ) 3 0 ,f x x x f 1 , 2 0m
0m
0m
-
015 58 11 33 47
1 , 2 1
2
0 1
322
3
x
mmx
0m
0m 1 , 2
1 , 2 3m
+ f ,a b 1 2, ,x x a b 1 2 1 2( ) ( )x x f x f x
22 1 1
( )x m x m
f xm x
m
2 ,
0m x x m
{ }fD m
2
2
2 2 2
2
2 2
2
4 1 2 1 1'( )
4 4 2 1 1
2 4 2 1
x m m x x m x mf x
m x
mx x m x m mx x m x m
m x
x mx m m
m x
-
015 58 11 33 48
2 2( ) 2 4 1 2h x x mx m m
2 , '( ) 0 , 2 ,f x x
2 22 4 2 1 0x mx m m 2
0 , fm x x D
2 2 2 2
22
' 4 2 1 2 4 2 4 2
2 4 2 2 1 0
m m m m m m
m m m
2
2
2 4 1 2 11 ' 0 '( ) 2 0 , 2 ,
1
x xm f x x
x
2 ,
1 ' 0 '( ) 0m f x 1 2x x
2 ,
2
2
1 2
1 0' 0 1
2 (2) 0 10 7 0 2
2 0 5 3 2 5 3 22 02
m m
x x ah m m m
S m m m
rW
5 3 2m
5 3 2m
+
-
015 58 11 33 49
f ] , [a b
. f ] , [ '( ) 0 , ] , [a b f x x a b
. f ] , [ '( ) 0 , ] , [a b f x x a b
. f ] , [ '( ) 0 , ] , [a b f x x a b
+ ' '( )y f x f
+ '( ) 0f x
+ f
III.m
. m ()
+ f '( ) 0 ,f x x m m '( ) 0f x
2'( )f x ax bx c
2 0'( ) 00
af x ax bx c
3 2( ) 1 2f x m x mx x m
f
2'( ) 3 1 2 2f x m x mx '( ) 0f x
222
13 1 0 1
6 6 0 3 34 24 1 0
ma m m
m m mm m
1 3 3 3 33 3 3 3
mm
m
-
015 58 11 33 50
f 3 3 , 3 3m
+ f '( ) 0 ,f x x 2
0'( ) 0
0
af x ax bx c
3 2( ) 1 3 6 1f x m x mx mx m f
2'( ) 3 1 6 6f x m x mx m '( ) 0,f x x
2
13 1 0 1
0 02 0' 2 1 0
2
ma m m
m mm mm m m
m
f 0m
. m f
f '( ) 0f x
2 0'( ) 00
af x ax bx c
m 3 2 2 2( ) 2 2 1 1f x x mx m x m m
2 2'( ) 3 4 2 1f x x mx m '( ) 0f x '( )f x
2 2 26 6
' 4 3 2 1 0 2 3 02 2
m m m m
6 6,2 2
m
-
015 58 11 33 51
. m f ( )
20
'( ) 00
af x ax bx c
b
m 3 2( ) 4 3 3 1 1f x m x m x m x
2'( ) 3 4 2 3 3 1f x m x m x m '( ) 0f x
4 0 43 0
mm
m
. m f ( )
2 0'( ) 00
af x ax bx c
b
m
2
2
2 2( )
1
mx xf x
x
2 2 2
4 4
4 2 1 2 2 2 2 2 2 1 4 2'( )
1 1
mx x mx x x m x mxf x
x x
41 0 , {1}x x '( )f x
2 1 0 1
2 0 2
mm
m
IV.
a) '( )f x 0x x f 0x x
b) '( )f x 0x x f 0x x
2 5 7
( )2
x xf x
x
-
015 58 11 33 52
2 0 2x x {2}fD
2
2 2
2 11'( ) 1
2 2
xf x
x x
2
'( ) 0 2 1 0 3 1 0 1 , 3f x x x x x x
1x 1(1) 1 3 31 2
f
3x 1(3) 3 3 13 2
f
(1) 3f (3) 1f
f 0x x 0'( ) 0f x 0'( ) 0f x f 0x x ( '( )f x )
V.
- x a lim ( )x a
f x
2 3 3
( )2
x xf x
x
2 0 2x x 2
2 2
3 3lim ( ) lim
2x x
x xf x
x
2x
-
015 58 11 33 53
- y b lim ( )x
f x b
2
2
3 4( )
2 1
x xf x
x
12 1 02
x x
2
2
3 4 1lim ( ) lim
4 4 1 4x x
x xf x
x x
14
y
- ( ) :D y ax b lim[ ( ) ( )] 0x
f x ax b
( )lim , lim[ ( ) ]x x
f xa b f x ax
x
22 3 5
( )1
x xf x
x
2( ) 2 3 5lim lim 2
1x x
f x x xa
x x x
22 3 5 5
lim ( ) lim 2 lim 11 1x x x
x x xb f x ax x
x x
y ax b 2 1y x
+ ( )f x ( ) ( )f x ax b x lim ( ) 0x
x
y ax b ( )f x
+ ( )y f x lim ( )x
f x
-
015 58 11 33 54
- ( )limx
f xa
x lim[ ( ) ]
xf x ax b
(
) y ax
- ( )limx
f x
x ( )
( )oy
- 2( ) , ( 0)f x ax bx c a ( ) ( )2
bf x a x x
a
lim ( ) 0x
x
2b
y a xa
( )C
-
( ) : ( )C y f x ( ) :D y ax b
( )C ( ) ( )x f x y
+ lim ( ) 0x
x
( )C
+ lim ( ) 0x
x
( )C
VI.
. () xoy ( , )M x y 0 0( , )x y XOY ( , )M X Y ( , ) / ( , ) /M x y xoy M X Y X Y
00
x x X
y y Y
. ( )y f x ( )C 0x () ( )y f x ( )C (0 , 0)O () ( ) : ( )C y f x 0x x
-
015 58 11 33 55
- 00
x x X
y y Y
- ( )y f x 0( ) ( )Y f x X g X
- ( )Y g X
1x
2
2
1( )
2 3
xy f x
x x
1x X
y Y
2 2
2 2
1 1( )
41 2 1 3
X XY g X
XX X
2
2( ) ( )
4
Xg X g X
X
1x ( )y f x
. 0 0( , )x y
- 00
x x X
y y Y
- ( )y f x 0 0( ) ( )Y f x X y g X
- ( )Y g X
1 , 4I C1
( ) 31
y f x xx
14
x X
y Y
1 14 1 3 ( )1 1
Y X Y X g XX X
-
015 58 11 33 56
1 1( ) ( )g X X X g X
X X
( )Y g X
1 , 4I C ( )y f x
. 0x x ( )y f x
0 0( ) ( )f fx x D x x D 0 0( ) ( )f x x f x x
0(2 )f fx D x x D 0(2 ) ( )f x x f x
1x C
4 3( ) 4 2 12 1y f x x x x x
D
1 1x D x D
4 3 2
2 3 4 2 3 2
4 3 2 3 2 2
4 2
(1 ) 1 4 1 2 1 12 1 1
1 4 6 4 4 1 3 3 2 1 2 12 12 1
4 6 4 1 4 12 12 4 2 4 2 12 12 1
8 8
f x x x x x
x x x x x x x x x x
x x x x x x x x x x
x x
4 3 2
4 3 2 3 2 2
4 2
(1 ) 1 4 1 2 1 12 1 1
4 6 4 1 4 12 12 4 2 4 2 12 12 1
8 8
f x x x x x
x x x x x x x x x x
x x
1 1f x f x
1x C ( )y f x
-
015 58 11 33 57
. ( , )I a b ( )y f x
( ) ( )f fa x D a x D ( ) ( ) 2f a x f a x b
(2 )f fx D a x D (2 ) ( ) 2f a x f x b
C2 3
( )1
xy f x
x
1 , 2I
{1}D
1 1x D x D
2 1 3 2 5 51 2
1 1
x xf x
x x x
2 1 3 2 5 51 2
1 1
x xf x
x x x
5 51 1 2 2 2 2 4f x f xx x
1 , 2I 2f a x f a x b
1 , 2I C ( )y f x
C2 1
( )2
x xy f x
x
2
2 2
1lim ( ) lim
2x x
x xf x
x
2x
( )f x 2 1 3 3( ) 12 2
x xf x x
x x
3lim ( ) lim 02x x
xx
1y x
-
015 58 11 33 58
2 2 1 3x y 2 , 3I
2 ( ) 2f a x f x b
3 34 4 1 5
4 2 2f x x x
x x
3 3
4 5 1 6 22 2
f x f x x x bx x
2 , 3I C
. m 2 1 1
( )x m x m
f xx m
. 4 3 2( ) 4 6 4 12y f x x x x x 1x
. 2 1
( )1
x xy f x
x
VII.
. ( )
( ) : ( , )mC y f x m
xoy ( , )y f x m
m m
( , )y f x m ( )mC
0 0( , )M x y ( )mC
m ( )mC ( )mC
0 0( , )M x y ( )mC
( ) : ( , )mC y f x m m
-
015 58 11 33 59
+
+
+
4: ( ) , ( 2)mmx
C y f x mx m
0 0,A x y 0 0, mA x y C
00 0 0 0 0 00
44 0 ...(1) ,
mxy y x m x y x m
x m
1 , 2m m
0 00 0 0 020 0 0
02
4 0 4
y xy xy x
x y x
2m mC 2 , 2 & 2 , 2A B
0 0( , )M x y ( )mC
0 0 0 0( , ) ( ) ( , ) (1)mM x y C y f x m
(1) 0 (2)Am B 2 0 (3)Am Bm C
(2) (3)......
0(2) :
0
A
B
0
(3) : 0
0
A
B
C
0 0( , )x y
( )mC
-
015 58 11 33 60
+
+
+
2: ( ) 2 2 1 4mC y f x mx m x m mC m
0 0,A x y mC 0 0, mA x y C
2 2
0 0 0 0 0 0 0
2
0 0 0
2
0
2 2 1 4 4 4 2
2 2 .......(1)
'( ) 2
y mx m x m y x x m x
y x m x F m
F m x
20 0 0( ) '( ) 0 2 0 2F m y F m x x efr
0 02 2 2 4x y
mC 2 , 4A
.
-
-
0 0( , )M x y ( )mC 0 0 0 0( , ) ( ) ( , ) (1)mM x y C y f x m
0( , ) ( )f x m F m (1) 0 ( )y F m '( ) 0F m
'( ) 0F m
( )mC
( )mC 0 0( , )M x y 0 0( , )M x y
( )mC 0 0( , )y f x m
0 0( , )M x y
0 0( , )y f x m 0 (1)Am B 2 0 (2)Am Bm C
(1) (2).....
-
015 58 11 33 61
2: ( ) 1 2 3 1 5 2mC y f x m x m x m
mC m
0 0, mA x y C
20 0 01 2 3 1 5 2y m x m x m
2 2
0 0 0 0 0
2 2
0 0 0 0 0
2 2 3 5
2 3 5 2 0
y x x x x m
x x m y x x
2
0 00 0
220 0 0
0 0 0
51 ,2 3 5 0
22
2
x xx x
y x xy x x
00
1
0
x
y
0
0
5
2
7
4
x
y
mC
1x 1 , 0A
52
x 5 7,2 4
B
23 1
: ( ) , 0mm x m m
C y f x mx m
0 00 , , mm A x y C
200
0
3 1m x m my
x m
20 0 03 1y x m m x m m
0(1) :
0
A
B
0
(2) : 0
0
A
B
C
-
015 58 11 33 62
2 0 0 0 03 1 1 0 ..... 1m y x m x y
1 0
2
0 0 0 0
2 2
0 0 0 0 0 0
2 2
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0
3 1 4 1 0
1 6 1 9 4 1 0
1 9 1 1 9 0
1 9 1 9 1 0
9 1 1 0
9 1 0 9 1
1 0 1
9 1 0 9
1 0
y x x y
y x y x x y
y x y x y x
y y x x y x
y x y x
y x y x
y x y x
y x y x
y x
0 0
1
1y x
,I II ( 9 1y x 1y x )
0
mC ,I II
-
-
.
( )
( )
-
-
( ) :D y ax b
( , )A x y ( )D y ax b
( , )A x y ( )D y ax b
( )mC
m
0 0 0( , ) ( ) : ( , )mM x y C y f x m 0 0( , )y f x m
0 (1)Am B 2 0 (2)Am Bm C
-
015 58 11 33 63
-
2 2 21 2
( )mx m m x m m
y f xx m
0 0,A x y 0 1x
0 0,A x y
m 0 0,A x y
2 2 2
0 0
0 0
0
2 2 2
0 0 0 0 0
2 2
0 0 0 0 0 0 0
22
0 0 0 0 0 0 0
2 2 4 3 2
0 0 0 0 0 0 0 0
1 2,
1 2
1 1 2 0 ...(1)
1 4 1 2
2 1 2 3 6 7
mx m m x m my x m
x m
x y my mx m m x m m
x m x x y m x y x
x x y x x y x
y x x y x x x x
0y
22 4 3 2
0 0 0 0 0 0
4 3 2 4 3 2
0 0 0 0 0 0 0 0
0 0 0
' 1 2 3 6 7
2 3 2 1 2 3 6 7
8 8 8 1 0 , 1
x x x x x x
x x x x x x x x
x x x
' 0 0 0 1x 1 1 2&m m
mC 0 0,A x y
(1) (2)..... m (1) (2).....
0 0( , )M x y
-
015 58 11 33 64
. 4: ( )mmx
C y f xx m
m mC
. 2 21
: ( )1
m
mx m x mC y f x
x
. 0
( )1
mxy f x
x m
. 2 21
: ( )mm x m
C y f xx m
0 0,A x y 0 0x
mC 0 0,A x y
IIX.
- ( )
- y ( ) y y
- ()
a)
( , )M x y
( , )M x y
( )
( )
x f mM
y g m
m
m x m x y x x
( )y f x ( )y f x ( )C M
M
( )
( )
x c cM
y g m
CacMnYnefr
-
015 58 11 33 65
x c
( )
b)
y c
( )
2 2
: ( )1 3
m
mx mC y f x
m x
I
mC m
3 3
1 .....(1)1
2.....(2)
1 2
dx m
c m x
a m yy m
c m y
1 & 2
3
1 2 3 3 2 32
22
3
yy x xy xy xy y x
y x
y x
I 2
: 23
D y x
0 , 2x y
M ( )D mM ( )x f m
( )
( )
x f m
y c c
CacnM nY efr
M ( ')D m
M ( )x f m
-
015 58 11 33 66
IX.
( )
.
-
-
3 2: ( ) 3 2C y f x x x
x
2'( ) 3 6 ''( ) 6 6
''( ) 0 6 1 0 1
f x x x f x x
f x x x
''( ) 0f x 1 , ( 0)x y 1 , 0I
C
1 , 0I '(1) 3 6 3f
1 , 0I 0 0 0'( )y y f x x x
: 3 3D y x
( ) : ( )C y f x
( )y f x 0x x
0 0 0( , )M x y
0'( ) tank f x
( )C 0 0 0( , )M x y 0 ( )M C ( )y f x
0x ( )C 0 0 0( , )M x y 0 0 0'( )( )y y f x x x
'( )f x 0'( )f x
0 0 0'( )( )y y f x x x
-
015 58 11 33 67
8
: ( ) 1C y f x xx
C
0 0,A x y C
C
28
'( ) 1 , 0f x xx
A 1 0 20
8'( ) 1k f x
x
1y x 2 1k
A 1 2 020
81 1 1 2k k x
x
82 2 1 72
x y 82 2 1 52
x y
1 0'( ) 1k f x
1 : 5D y x 2 : 3D y x
.
a)
-
-
-
b)
-
-
k
( )C 0 0 0( , )M x y k
0 0( )y y k x x
0 0 0( , )M x y M 0'( )k f x
0'( )k f x 0x 0x ( ) : ( )C y f x 0y
0 0( )y y k x x
( )C ( ) :D y kx b
( ) : ( )C y f x ( ) :D y kx b ( ) (1)f x kx b
-
015 58 11 33 68
-
2 3 1
: ( )2
x xC y f x
x
2k
2x
2
1'( ) 1
2f x
x
0 0,A x y
0 2
0
1'( ) 1
2k f x
x
2
0 02
0
11 2 2 1 1
2x x
x
0 3x
0 01
1 1 1 1 1 , 11 2
x y A
0 01
3 3 1 1 3 , 13 2
x y B
1 , 1A 1 : 2( 1) 1 2 1L y x x
3 , 1B 2 : 2 3 1 2 5L y x x
2k
1 2: 2 1 ; : 2 5L y x L y x
C :D y ax b 2a k
: 2D y x b
D C
( )C ( )D (1) b
-
015 58 11 33 69
2
2 23 1 2 3 1 2 4 22
x xx b x x x x bx b
x
2 1 2 1 0 .....(1)x b x b
D C 1 0
2 2
22
1 4 2 1 0 2 1 8 4 0
6 5 0 3 4 1 ; 5
b b b b b
b b b b b
2k
1 2: 2 1 ; : 2 5D y x D y x
.
a)
- ,
-
-
b)
-
-
-
( ) : ( )C y f x 0 0( , ) ( )M x y C
( )C 0 0( , ) ( )M x y C
( )D
1 1( , )N x y ( )C N
1 1 1( ) : '( )( ) (1)D y y f x x x 1 1( )f x y
( )D 0 0( , )M x y 0 1 1 0 1'( )( ) (2)y y f x x x
(2) 1 1 1, '( ) , ( )x f x f x (1) ( )D
k 0 0( , )M x y
( )C 0 0( )y y k x x 0 1( ) : ( ) (1)D y k x x y
( )D ( ) : ( )C y f x 0 0( ) ( ) (2)f x k x x y
( )D ( )C (2)
k k (1) ( )D
-
015 58 11 33 70
-
-
C 3( ) 2y f x x x
C 1 , 1A
2'( ) 6 1f x x 1 1,B x y
31 1 12f x x x 2
1 1' 6 1f x x C B
1 1 1 1 1 1' 'y y f x x x y f x x x y
31 1 1 1: ' 2D y f x x x x x D 1 , 1A
1 , 1A D
3
1 1 1 1
2 3
1 1 1 1
2 3 3
1 1 1 1 1
3 2
1 1
23 2
1 1 1 1
1 1
1 , 1 1 ' 1 2
1 6 1 1 2
1 6 6 1 2
4 6 2 0
2 3 1 0 2 1 1 0
1; 1
2
A D f x x x x
x x x x
x x x x x
x x
x x x x
x x
1 11 1x f x A
C 1 , 1A
1( ) : ( )C y f x 2( ) : ( )C y g x 0 0( , )M x y
0 0'( ) '( )f x g x
1( ) : ( )C y f x 2( ) : ( )C y g x 0 0( , )M x y
0 0'( ) '( ) 1f x g x
-
015 58 11 33 71
1 11 1
2 4x f x 1
1'
2f x
1 1 1 1: 1
2 2 4 2D y x x
C 1 , 1A 1
: 12
D y x
3 2( ) 3 4y f x x x C
C 1 , 0A
k 1 , 0A
: 1D y k x
3 2 3 2
2
2
3 4 1 1 3 3 1 0
1 4 4 0
1
2 0 ....(1)
x x k x x x k x
x x x k
x
x k
D C 1 1x
2
1 1 2 0 9x k k
: 9 1D y x
' 4 4 0 0o k k
1 : 0D y ()
.22
( )1
x mx my f x
x
C m
0 , 1A
) C
-
015 58 11 33 72
) C
) C
) C
.2 1
( )1
x xy f x
x
C
) C 1 , 1I
) C C
) C
y x
.
-
-
1( )1
xy f x
x
C
: 1D y mx m
m ( )D ( )C
( )C ( )D ( ) 0g x
( )D ( )C ( ) 0g x
1 2x x ( ) 0a g
-
015 58 11 33 73
2
11 1 1 1
1
( ) 2 0 .....(1)
xmx x x mx
x
g x mx mx
1
1 21 (1) 0 2 0 0x x a g m m m m
.
-
-
1( )1
xy f x
x
C
: 1D y mx m
2
11 1 1 1
1
( ) 2 0 .....(1)
xmx x x mx
x
g x mx mx
1
2
1 2
0 8 01 8
(1) 0 0
m mx x m
a g m
m ( )D ( )C
( )C ( )D ( ) 0g x
( )D ( )C ( ) 0g x
1 2x x 1 20
( ) 0x x
a g
-
015 58 11 33 74
.
.
- -
- -
.
-
1( ) 1
2y f x x
x
{2}D
22
111'( ) 1 0 ;( 2)2
f x x Dxx
( )f x
1lim ( ) lim 12x x
f x xx
1lim ( ) lim 12x x
f x xx
-
015 58 11 33 75
( )
( )
2 2
1lim ( ) lim 12x x
f x xx 2 0
1 1lim lim
2x ux u
2 2
1lim ( ) lim 12x x
f x xx 2 0
1 1lim lim
2x tx t
2x
1lim lim 0( )
2x xf x y
x
1y x
( ' )y oy1
02
x y
( ' )x ox 23 5
0 3 1 02
y x x x
-
015 58 11 33 76
2 5 6
1( ) 1
x x
xy f x x e
{1}fD
2 2 225 6 5 6 5 6
1 1 12
2 1 1 2'( ) 1
11
x x x x x x
x x xx x x x
f x e x e exx
'( ) 0 1 2 0 1 , 2f x x x x x
( 61 ( 1) 2x f e 02 (2) 1x f e )
2 5 6 2
41 1
1 1 1
lim ( ) lim 1 lim 1
x x
xx x
x x x
f x x e x e e
2 2
4 31 1
1 1 1
lim lim 1 lim 1x x xx x x
e x e e x e
-
015 58 11 33 77
2 211
k xx k
1x k
2
1
1
2 1lim 1 lim 2 lim 0kx
kk kx
x e ek k e
1
lim ( ) 0x
f x
2 5 6 2
31 1
1 1 1
lim ( ) lim 1 lim 1
x x
x x
x x x
f x x e e x e
2 211
k xx k
1x k
2
1
1
2lim 1 lim 2 lim
kkx
k kx
ex e e
k k
1
lim ( )x
f x
1x
2 5 6
1lim ( ) lim 1 lim
x x
xx
x x xf x x e xe
k x x k
lim lim lim 0x k kx k kk
xe kee
lim ( ) 0x
f x
2 5 6
1lim ( ) lim 1 lim
x x
xx
x x xf x x e x e
-
015 58 11 33 78
60x y e
ln( ) xy f x xx
0 ,fD
2
2 2
1 ln 1 ln'( ) 1
x x xf x
x x
2 0 , fx x D
'( )f x 2( ) 1 lng x x x
21 2 1
'( ) 2x
g x xx x
0 , fx x D
2 2'( ) 0 2 1 02
g x x x
-
015 58 11 33 79
( ) 0 , fg x x D '( ) 0 , ff x x D
0 0
lnlim ( ) lim 0x x
xf x x
x
2ln ln
lim ( ) lim lim 1 1x x x
x xf x x x
x x
) 2 2 4
( )1
x xy f x
x
)
2
2
1( )
2 5 2
xy f x
x x
) 2 3 3
( )2
x xy f x
x
) 3( ) ln
2
xy f x
x
) 1( )1
x
x
ey f x
e
) 2 1( ) ln
2y f x x x
-
015 58 11 33 80
I. .
.
3( ) 3 7F x x x 2( ) 9 7,f x x x
2( ) '( ) 9 7f x F x x
C
2( ) 3 5f x x x
2 3 21 3( ) ( ) 3 5 53 2
F x f x dx x x dx x x x C
3 21 3( ) 53 2
F x x x x C
( )F x ( )f x '( ) ( )F x f x x f
( )F x ( )f x ( )f x
( )F x C C
( )F x ( )f x
( )F x ( )f x
( )F x ( )f x
( ) ( )f x dx F x C
( )F x
( )F x ( )f x , '( ) ( ) , ( , )a b F x f x x a b
( )F x ( )f x'( ) ( ) , ( , )
[ , )'( ) ( )
F x f x x a ba b
F a f a
( )F x ( )f x'( ) ( ) , ( , )
( , ]'( ) ( )
F x f x x a ba b
F b f b
-
015 58 11 33 81
+
+
( )F x ( ) xf x x e (1) 1F e
21( ) ( )2
x xF x f x dx x e dx x e C
1 1(1) 1 12 2
F e e C e C
21 1( )2 2
xF x x e
II.
.
.
). ). ).
).
( )F x ( )f x
'( ) ( ) , ( , )
[ , ] '( ) ( )
'( ) ( )
F x f x x a b
a b F a f a
F b f b
( ) ( ) ( ) ( )'( ) lim , '( ) lim ,
x a x b
F x F a F x F bF a x a F b x b
x a x b
ngi
f C
C
( )F x ( )f x ( )f x
( ) ( )f x dx F x C C
( ) ( )d f x dx f xdx
( ) ( )f x dx F x C ( ) ( )d f x dx f x dx
( ) ( ) ( ) ( )f x g x dx f x dx g x dx
-
015 58 11 33 82
).
( )f x 3 21 1( ) 33 4
f x dx x x x C
2 21 1( ) 3 2 3 33 4 2
xf x x x x
2( ) 32
xf x x
( )f x
3 2) ( ) 2a f x dx x x x c
2( ) 3 4 1f x x x
2( ) 3 4 1f x x x
2) ( ) 2 3 5b f x dx x x c
2 2
2 3 2 3( ) 2
2 3 5 3 5
x xf x
x x x x
2
2 3( )
3 5
xf x
x x
1 1
( ) ( ) ,n n
i i i i
i i
k f x dx k f x dx k
( )f x ( )f x dx ( ) ( )d
f x dx f xdx
( ) ( ) '( ) ( )f x dx F x C F x f x
( ) ( ) '( ) ( )f x dx F x C F x f x
( ) ( ) '( ) ( )f x dx F x C F x f x
( ) ( ) '( ) ( )f x dx F x C F x f x
-
015 58 11 33 83
.
-
015 58 11 33 84
.
.
+
+
+
21x x dx
2 2 211 1 12
x x dx x d x
2 21 2 2t x tdt xdx xdx tdt
( )t f x '( )dt f x dx ( )f x t
'( )f x dx dt t ( )f x
x
( )I f x dx
( ) [ ( )] '( )f x dx g h x h x dx
( ) '( )u h x du h x dx
( ) ( ) ( )I g u du G u C F x C
-
015 58 11 33 85
32 2 3 21 11 13 3
x x dx t tdt t dt t C x C
32 211 13
x x dx x C
+
+
.
xxe dx
u x du dx x x xdv e dx v e dx e
x x x x xxe dx udv uv vdu xe e dx xe e C
1x xxe dx e x C
( )I f x dx
( ) '( )x h t dx h t dt
[ ( )] '( ) ( ) ( ) ( )I f h t h t dt g t dt G t C F x C
( ) '( ) ( ) ( ) ( ) '( )f x g x dx f x g x g x f x dx
udv uv vdu ( ) '( )u f x du f x dx '( ) ( )dv g x dx v g x
( )I f x dx
( )f x dx udv u v
( ) ,' x dx v dvdu u
udv uv vdu
-
015 58 11 33 86
.
a).
+
( ) ( ) ( )sin cos axx x xI P axdx P axdx P e dx ( )xPa
( ) ( ) ( )
sin cos '( ) sin cosax ax
u P x f x P x
dv axdx axdx e dx g x ax ax e
rW
rW
( ) ( )log ln( )x xI P xdx P ax b dx ( )xP
( ) ( )'( )x xdv P dx g x P rW
log ln( ) ( ) log ln( )u x ax b f x x ax b rW
sin cosax axI e axdx e axdx
( )ax axu e f x e rW sin cos '( ) sin cosdv axdx axdx g x ax ax rW
1
1
1
sin
cos( )
tan
.............
x
xI P x dx
x
1 1
1 1
1 1
sin sin
cos cos( )
tan tan
............. .............
x x
x xu f x
x x
rW ( ) ( )'( )x xdv P dx g x P rW
sin cos ; sin sin ; cos cosax bxdx ax bxdx ax bxdx
1
sin sin cos( ) cos( )2
a b a b a b
-
015 58 11 33 87
+
+
sin3 sin5x xdx
1 1 sin 2 sin8sin5 sin3 cos 2 cos82 2 2 8
x xx xdx x x dx C
cos5 cos3x xdx
1 1 sin8 sin 2cos5 cos3 cos8 cos 22 2 8 2
x xx xdx x x dx C
sin 2 cos3x xdx
1 1 cos5sin 2 cos3 sin5 sin cos2 2 10
xx xdx x x dx x C
b).
-
( )
3 2sin cosx x dx
cos sint x dt xdx
3 5
3 2 2 2 2 2sin cos 1 cos cos sin 13 5
t tx xdx x x xdx t t dt C
5 3
3 2 cos cossin cos5 3
x xx xdx C
-
1
cos cos cos( ) cos( )2
a b a b a b
1
sin cos sin( ) sin( )2
a b a b a b
sin cosm nx xdx
m n m cost x n
sint x cos , sinu x u x
m n 21 cos2
sin2
xx
2
1 cos2cos
2
xx
1sin cos sin 2
2x x x
-
015 58 11 33 88
-
c).
- ( )
- ( ) ( )
d).
e).
.
(
)
m n cos , sint x t x
m n 2 2sin cos 1x x
m n 21 cos2
cos2
xx
2
1 cos2sin
2
xx
tan , cotn nxdx xdx tant x cott x
sin , cosR x x dx
2
2tan
2 1
xt dx dt
t
22
sin1
ta
t
2
2
1cos
1
ta
t
2
2 2 2
2 1 2sin , cos ,
1 1 1
t tR x x dx R dt
t t t
( )
( )
x
x
Rdx
S
( )
( )( )
x
x
Rf x
S ( ) ( ),x xR S ( )xR
( )xS ( )xR ( )xS( )
( )( )
( )x
xx
Pf x W
S ( )xS
( )xP ( )xP ( )xW
( ) ( )( )
( ) ( )
x P xx
x x
Rdx W dx dx
S S
22 2
1 1 1
4
2 4
I dx dxax bx c a b ac b
xa a
-
015 58 11 33 89
+
+
+
;
2
1
3 4dx
x x
2
1 1 1 1 1ln ln
3 4 4 5 49 16
x xdx C C
x x x x
2 4 0b ac 2
bu x
a
22 4 0
4
ac bk
a
1
2 2 2
1 1 1 1tan
uI dx dx C
ax bx c a u k ak k
21
2 2
4 2 2tan
2 4 4
ac b ax bk I C
a ac b ac b
2 4 0b ac 2
bu x
a
22 2
1 1 1 1 1 1 2
2
2
I dx dx du C Cax bx c a a u au ax bb
xa
2 4 0b ac 2ax bx c
12
bx
a
2
2
bx
a
1 2x x
a
2
1 2( )( )ax bx c a x x x x
2
1 2 1 2 1 2
1 1 1 1 1 1
( )( ) ( )I dx dx dx
ax bx c a x x x x a x x x x x x
1
1 2 2
1ln
( )
x xI C
a x x x x
1 2x x
a
1
22
1ln
4
x xI C
x xb ac
-
015 58 11 33 90
2
Ax B M N
ax bx c x p x q
2
5 13
5 6
xdx
x x
2
5 13
5 6 3 2
x a b
x x x x
5 13 2 3 5 13 2 3x a x b x x a b x a b
5 22 3 13 3
a b a
a b b
2
5 132 3 2ln 3 3ln 2
5 6 3 2
x dx dxdx x x C
x x x x
2
5 132ln 3 3ln 2
5 6
xdx x x C
x x
( )
24 2
1 2 3
x xdx
x x x
24 2
1 2 3 1 2 3
x x L M N
x x x x x x
24 2 2 3 1 3 1 2x x L x x M x x N x x
1x 16 32
L L
2
Ax BI dx
ax bx c
2
( )( )( )
Ax Bx CI dx
x x x
2
( )( )( )
Ax Bx C L M N
x x x x x x
, ,L M N
-
015 58 11 33 91
2x 415 125
M M
3x 3710 3710
N N
24 2 1 4 37ln 1 ln 2 ln 3
1 2 3 2 5 10
x xdx x x x C
x x x
( )
2
3
1
1 3
xdx
x x
2
3 2 3
1
1 31 3 1 1
x L M N P
x xx x x x
2 32 1 1 3 1 3 3 1x L x x M x x N x P x
2 3 21 3 2 5 3 3 3 3x L P x L M P x M L N P x L M N p
1x 14 22
N N 3x 1064 1064
P P
3 2,x x
10
0 64
3 1 63 1
16
L PL P
L M PM P L
2
3 2 3
1 10 1 6 1 1 1 10 1
64 1 16 2 64 31 3 1 1
xdx dx dx dx dx
x xx x x x
2
3( ) ( )
Ax BI dx
x x
2
3 2 3( ) ( ) ( ) ( )
Ax B L M N P
x x x x x x
, , ,L M N P
-
015 58 11 33 92
2
10 6 1 10ln 1 ln 3
64 16 1 644 1x x C
x x
2
5 1 1 3ln
32 3 8 1
x xC
x x
2
3 2
1 5 1 1 3ln
32 31 3 8 1
x x xdx C
xx x x
( )
)
2
22
2 2 13
2 1
x xA dx
x x
)
2
221 4
xB dx
x x
)
2
22
2 2 13
2 1
x xA dx
x x
2
2 2 22 2
2 2 13
2 12 1 1
x x L mx n px q
x xx x x
22 2 22 2 13 1 2 1 2x x L x mx n x x px q x
2 4 3 22 2 13 2 2 2 2 2 2 2x x L m x n m x L m n p x m n p q x L n q
2x 25 25 1L L
4 3 2, , ,x x x x
2
2 2( )( )
Ax Bx CI dx
x x
2
2 2 2 2 2( )( ) ( )
Ax Bx C L Mx N px q
x x x x x
, ,L M N ,p q
-
015 58 11 33 93
01
2 02 2
2 2 23
2 2 24
2 2 13
L mm L
n mn m
L m n pp
m n p qq
L n q
2 22
2 3 4
2 1 1
dx x xA dx dx
x x x
2 1
22 2
3ln 2 ln 1 2 tan 4
2 1 1
dxx x x
x x
22 1
dx
x
2 22
1 2
1 1
xu du dx
x x
dv dx v x
udv uv vdu
2
2 2 2 2 2 2 22 2
22 2
1 1 1 1 11 1
dx x x dx x dx dx
x x x x xx x
2 2 22
21 11
dx x dx
x xx
1
2 22
24 2 tan
11
dx xx C
xx
1
22
2 3 4ln 4 tan
2 11
x xA x C
xx
)
2
221 4
xB dx
x x
2
2 2 22 21 41 4 4
x A Bx C Dx E
x xx x x
22 2 24 1 4 1x A x Bx C x x Dx E x
-
015 58 11 33 94
2 4 3 28 4 4 4 16 4x A B x C B x A B C D x B C D E x A C E
1x 125 125
A A
4 3 2, , ,x x x x
1025
0
8 4 1
4 4 04
16 4 05
A B B A
C BC B
A B C DD E
B C D E
EA C E
2
2 2 22 2
1 1 1 4 1
25 1 25 4 51 4 4
x dx x xB dx dx dx
x xx x x
1
222 2
1 1 1 2 4ln tan
25 50 2 55 44 4
x x dx
xx x
1
22
1 1 3 4ln tan
25 100 2 10 44
x x xC
xx
( )
3 24
xdx
x
3 2 3 12 14 3 14 3 14ln 44 4 4
x x dxdx dx dx x x C
x x x
( )
Ax BI dx
ax b
Ax B M
Lax b ax b
,L M
( )( )
Ax BI dx
ax b cx d
( )( )
Ax B L M
ax b cx d ax b cx d
,L M
-
015 58 11 33 95
2 9
2 1
xdx
x x
2 9
1 2 1 2
x A B
x x x x
2 9 2 1x A x B x
2 9 2x A B x A B 2 7
2 9 5
A B A
A B B
2 97 5 7ln 1 5ln 2
2 1 1 2
x dx dxdx x x C
x x x x
( )
2 3
dx
x x
1
1 3 22 3 2 3
A BA B x A B
x x x x
0 13 2 1 1
A B A
A B B
3ln
2 3 3 2 2
dx dx dx xC
x x x x x
( )
2
2 3
3 1
xdx
x
( )( )
kI dx
ax b cx d
( )( )
k L M
ax b cx d ax b cx d
,L M
2( )
Ax BI dx
ax b
2 2( ) ( )
Ax B L M
ax b ax b ax b
,L M
-
015 58 11 33 96
2 2
2 3
3 13 1 3 1
x A B
xx x
2 3 3 1 2 3 3x A x B x Ax B A 2
3 2 3
3 11
3
AA
B AB
2 2
2 3 2 11 2 11ln 3 1
3 3 1 3 3 9 3 13 1 3 1
x dx dxdx x C
x xx x
( )
2 4 52
x xdx
x
2
24 5 4 5 2 22 2
x x CAx B x x Ax x B x C
x x
2 21 1
4 5 2 2 2 4 2
2 5 1
A A
x x Ax A B x B C A B B
B C C
2
24 5 12 2 ln 22 2 2
x x dxdx x dx x x x C
x x
( )
2
2
3 4
2 3
x xdx
x
2
22
2 2
3 43 4 2 3 2 3
2 32 3 2 3
x x B CA x x A x B x C
xx x
2Ax Bx CI dx
ax b
2Ax Bx C NLx M
ax b ax b
, ,L M N
2
2( )
Ax Bx CI dx
ax b
2
2 2( ) ( )
Ax Bx C M NL
ax b ax b ax b
, ,L M N
-
015 58 11 33 97
2 23 4 4 12 2 9 3x x Ax A B x A B C
1
4 1 4
12 2 3 3
9 3 4 43
4
AA
A B B
A B CC
2
2 2
3 4 1 433
4 2 3 42 3 2 3
x x dx dxdx dx
xx x
3 43ln 2 3
4 2 8 2 3
xx C
x
( )
2 1
2 3
x xdx
x x
2 1
2 3 2 3
x x B CA
x x x x
2 1 2 3 3 2x x A x x B x C x
2 21 5 6 3 2x x Ax A B C x A B C
1 1 1
5 1 4 3
6 3 2 1 3 2 5 7
A A A
A B C B C B
A B C B C C
2 13 7
2 3 2 3
x x dx dxdx dx
x x x x
3ln 2 7ln 3x x x C
2
( )( )
Ax Bx CI dx
ax b cx d
2
( )( )
Ax Bx C M NL
ax b cx d ax b cx d
, ,L M N
2
2( )( )
Ax Bx CI dx
ax b cx d
-
015 58 11 33 98
( )
2
2
2 7 55
2 3 2
x xdx
x x
2
2 2
2 7 55
2 2 32 3 2 2 3
x x A B C
x xx x x
222 7 55 2 3 2 2 3 2x x A x B x x C x
2 22 7 55 4 2 12 9 6 2x x A B x A B C x A B C
4 2 2 1 2 1 2 1
12 7 14 6 6 14 1
9 6 2 55 21 2 61 21 28 12 61 20
A B B A B A A
A B C A C C A B
A B C A C A A C
2
2 2
2 7 5520
2 2 32 2 3 2 3
x x dx dx dxdx
x xx x x
1 10 2 10ln 2 ln 2 3 ln2 2 3 2 32 3
xx x C C
x xx
.
4sin 3cos 5
dx
x x
2
2 2( )( ) ( )
Ax Bx C L M N
ax b cx d ax b cx d cx d
, ,L M N
1
sin cosI dx
a x b x c
2
2tan
2 1
xt dx dt
t
2
2 2
2 1sin , cos
1 1
t tx x
t t
2
2tan
2 1
xt dx dt
t
2
2 2
2 1sin , cos
1 1
t tx x
t t
-
015 58 11 33 99
2 2
2 2
1 2
4sin 3cos 5 2 1 14 3 5
1 1
dxdt
x x t t t
t t
2
22 2 2
2 1 1 1
4 4 28 3 3 5 5 1
tdt dt C
t t tt t t t
14sin 3cos 5
tan 22
dxC
xx x
( )
sin cossin 2cos
x xdx
x x
sin 2cos cos 2sinsin cossin 2cos sin 2cos
A x x B x xx x
x x x x
1
2 1 5sin cos 2 sin 2 cos
2 1 3
5
AA B
x x A B x A B xA B
B
sin 2cos 'sin cos 1 3 3 ln sin 2cossin 2cos 5 5 sin 2cos 5 5
x xx x xdx dx dx x x C
x x x x
1 1sin cos
sin cos
a x b xI dx
a x b x
1 1s