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LUYENTHI999.COM ON THI DAI HOC
LUYENTHI999.COM ON THI DAI HOC
DẠNG 1
22
( 1)os
au atgt du dt a tg t dt
c t
2
2 2
( 1) 1
( 1)
a tg t dtI t c
a tg t a
2( 0)
ax
dxI
bx c
2 2
duI
u a
2 2
1ar
du uI ctg c
u a a a
LUYENTHI999.COM ON THI DAI HOC
Ví dụ:22 3 2
dxI
x x
0 Có 2 2
duI
u a
22 2 2
1 13 3 3 72 3 2 2 22( ) 2 1 ( )2 2 4 16
dx dx dx dxI
x x x x x x x
2
3 3( )1 24 4ar ( )3 72 7 7( )4 16 4
d x xctg c
x
LUYENTHI999.COM ON THI DAI HOC
Dạng 2(x )( )
dxI
a x b
1 ( ) ( ) 1
ln(x )( )
x a x b x bI dx ca b a x b a b x a
2( 0)
ax
dxI
bx c
CHÚ Ý Dạng 2
21 2ax ( )( )bx c a x x x x
LUYENTHI999.COM ON THI DAI HOC
Ví dụ22 3 1
dxI
x x
0 1
( 1) ( )1 21 1 122( 1)( ) ( 1)( ) ( 1)( )2 2 2
x xdx dxI dx
x x x x x x
DO Đưa về dạng 2
1 1( ) ( 1)2 2ln1 1 12
d x xd xc
x xx
LUYENTHI999.COM ON THI DAI HOC
2 2
1( 0)
ax
dx duI c
bx c u u
2 2 2
1 (2 1) 1 1
4 4 1 (2 1) 2 (2 1) 2 1
dx dx d xI c
x x x x x
Dạng 3
LUYENTHI999.COM ON THI DAI HOC
DẠNG 42ax
dx eI dx
bx c
2 2 2
(2ax ) 2ax2 2 ( )ax 2 ax 2 ax
d bdb e d b bd dxa a dx dx ebx c a bx c a bx c
2ln ax2
dbx c
a
Áp dụng dạng 1,2,3
LUYENTHI999.COM ON THI DAI HOC
Ví dụ2 2
1 9(4 3)2 3 2 2
2 3 5 2 3 5
xxI dx dx
x x x x
2 2
1 4 3 9
2 2 3 5 2 2 3 5
x dxdx
x x x x
2
2
1 (2 3 5) 952 2 3 5 2 2( 1)( )2
d x x dx
x x x x
2
51 9 2ln 2 3 5 ln2 14 1
xx x c
x
LUYENTHI999.COM ON THI DAI HOC
Dạng 5 2( )(ax )
dxI
x bx c
2 4b ac
21 2
2 20 0
2 '
2 2 2
10
( )(ax )
10
( )(ax ) ( )
1 (ax )0
( )(ax ) ax ax
A B C
x bx c x x x x x
A B C
x bx c x x x x x
A B bx c C
x bx c x bx c bx c
Xét
LUYENTHI999.COM ON THI DAI HOC
Ví dụ 3 1
dxI
x
2( 1)( 1)
dxI
x x x
2 2 2
1 (2 1)0
( 1)( 1) 1 1 1
A B x C
x x x x x x x x
2
2
( 1) (2 1)( 1) ( 1)
( 1)( 1)
A x x B x x C x
x x x
2 01 1 1
0 , ,2 4 4
1
A B
A B C A B C
A B C
LUYENTHI999.COM ON THI DAI HOC
2 2
1 1 (2 1) 1
2 1 4 1 4 1
dx x dxI dx
x x x x x
1ln 12
x 21ln 14
x x
2
1 1( )1 12 2ar ( )1 34 2 3 3( )2 4 2
d x xctg c
x
LUYENTHI999.COM ON THI DAI HOC
2
2( )(ax )
mx nx pI dx
x bx c
2
21 2
2 2 '
2 2 2
2
2 20 0
0( )(ax )
((ax )0
( )(ax ) (ax ) (ax )
0( )(ax ) ( )
mx nx p A B C
x bx c x x x x x
mx nx p A B bx c C
x bx c x bx c bx c
mx nx p A B C
x bx c x x x x x
Dạng 6
LUYENTHI999.COM ON THI DAI HOC
2
3( )
mx nx pI dx
x
2
3 2 3( ) ( ) ( )
mx nx p A B C
x x x x
Dạng 7
Viết
Bài tập2
2
3 4
( 2)( 4 4)
x xI dx
x x x
LUYENTHI999.COM ON THI DAI HOC
Dạng 8 , ( )(ax )
m
n
xI m n
b
ax
t b dtb t x dx
a a
3
2010(2 1)
xI dx
x
Đặt
VÍ DỤ1
2 12 2
t dtt x x dx
3 3 2
2010 2010
1 ( 1) 1 3 3 1
16 16
t t t tI dt dt
t t
2007 2008 20091( 3 )
16t t t dt x dx
LUYENTHI999.COM ON THI DAI HOC
Dạng 92 2( ) ( )
dxI
x a x b
2 2
2 2 2 2 2
2
2 2 2 2
( ) ( )[ ]1 1 ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )( )
1 1 1 1 1 1 1
( ) ( ) ( ) ( )( ) ( )
x a x bx a x ba b
x a x b x a x b a b x a x b
a b x b x a a b x b x a x b x a
2
duI
u 1
lnx b
Ib a x a
LUYENTHI999.COM ON THI DAI HOC
Ví dụ2 2( 4 3)
dxI
x x
2
2 2 2 2
1 [( 3) ( 1)]
( 1) ( 3) 4 ( 1) ( 3)
dx x x
x x x x
2 2
1 ( 3) ( 1) 1 1 1
4 ( 1)( 3) 4 1 3
x x
x x x x
2 2
1 1 1
4( 1) 2( 1)( 3) 4( 3)x x x x
2 24( 1) 2( 1)( 3) 4( 3)
dx dx dxI
x x x x
LUYENTHI999.COM ON THI DAI HOC
1 1 3 1ln
4( 1) 4 1 4( 3)
xI c
x x x
LUYENTHI999.COM ON THI DAI HOC
Dạng 102
, 2(ax )n
dxI n
bx c
2
bt x dt dx
a 2
1
( )n n
dtIa t k
2
1
( )nu
t k
dv dt
LUYENTHI999.COM ON THI DAI HOC
2 3( 4 3)
dxI
x x
2( 4 3)n n
dxI
x x
1 2 1
1 1 1 1 11 ln
( 4 3) ( 1)( 3) 2 1 3 2 3
dx dx xn I dx c
x x x x x x x
21 2
( 1)n n
dtn t x dt dx I
t
22 2 1
2 2 1
1 2
( 1) ( 1) 2( 1) ( 1)
n nn n n
ntdtu du t t dt
t t I nt t
dv dt v t
2
2 2 1 2 2 2 1
( 1) 12 2 2
( 1) ( 1) ( 1) ( 1) ( 1)n n n n n
t t dt t dt dtn n n
t t t t t
LUYENTHI999.COM ON THI DAI HOC
2
1 12
(1 2 )( 1)
2 2( 1) 2
n n
n n n nn
tn I
t tI nI nI I
t n
1 22 2 2
2 3
3( 1) ( 1)2 4
t tI I
t tI I
1 2 2 2
3 3
8 8( 1) 4( 1)
t tI
t t
2 2 2
3 1 3ln
16 3 8( 1) 4( 1)
x t tc
x t t
LUYENTHI999.COM ON THI DAI HOC
Dạng 112
ex
(ax )nd
I dxbx c
2 2 2
ex (2ax )
(ax ) (ax ) (ax )n n n
d A b B
bx c bx c bx c
2 2
(2ax )
(ax ) (ax )n n
b dx dxI A B
bx c bx c
Viết
Suy ra
Dạng 10du
u
LUYENTHI999.COM ON THI DAI HOC
1 ( )
( )
k kk
k
x P xI dx x t
Q x
3
8 2( 4)
x dxI
x
4 3 34
4
dtt x dt x dx x dx
3
4 2 2[( ) 4]
x dx
x
DẠNG 12
Ví dụ
2 2 2 2
1 1
4 ( 4) 4 ( 2) ( 2)
dt dtI
t t t
DẠNG 9
LUYENTHI999.COM ON THI DAI HOC
'( ) [ ( )]
[ (x)]
x P x dxI
Q
( )t x
4 3 2
(2 1)
2 3 2 3
x dxI
x x x x
2 2
(2 1)
( 1) 4
x dx
x x
Đặt
Ví dụ
Đặt 2 1 (2 1)t x x dt x dx
2
1 2ln
4 ( 2)( 2) 4 2
dt dt tI c
t t t t
Ví dụ2
4
1
1
xI dx
x
2 2
2 22
1 11 1
1 1( ) 2
x xdx dx x
x x
LUYENTHI999.COM ON THI DAI HOC
2
1 1(1 )t x dt dx
x x
2
1 2ln
2 ( 2)( 2) 2 2 2
dt dt tI c
t t t t
2
4 2
1
1
xI dx
x x
2 2
2 2 22
1 1 11 1 ( )
1 1 11 ( ) 1 ( ) 1
d xx x xI dx
x x xx x x
Ví dụ
2
1 1( )
1 1ln
1 2 1
x t d x dtx xdt t
I ct t
LUYENTHI999.COM ON THI DAI HOC
Ví dụ4
6
1
1
xI dx
x
2 2 2 2 2 2
2 3 2 4 2
( 1) 2 ( 1) 2
( ) 1 ( 1)( 1)
x x x xdx
x x x x
2 2
4 2 2 4 2
12
1 ( 1)( 1)
x x dxdx
x x x x x
3 32
6 3 22 2
2
1 11 ( )2 2
1 13 1 3 ( ) 11 ( ) 1
d xdx dxx xdx dxx xx x
x x
31 2ar ( ) ar ( )
3ctg x ctg x c
x
LUYENTHI999.COM ON THI DAI HOC