TÍCH PHÂN HÀM SỐ VÔ TỈ
56
LUYENTHI999.COM 1
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Transcript of TÍCH PHÂN HÀM SỐ VÔ TỈ
LUYENTHI999.COM 1
LUYENTHI999.COM 2
• Đặt , ( ) \ 0sin 2 2
ax t
t
'
2
cos
sin sin sin
a a a tdtdx d dt
t t t
2,
4
dxI
x
TÌM NGUYÊN HÀM
2, ( ) \ 0
sin 2 2x t
t
Dạng 1:
2 2x a t x
Các Em Có Thể Đặt
2 2( , )I f x x a dx
LUYENTHI999.COM 3
2
2cos
sin
tdtdx
t 2
2
2cossin4
4sin
tdttI
t
22
cos2
1sin 4( 1)
sin
tdt
tt
2
cos
sin cot
tdt
t t
2 2
cos2
2sin cot
tdt
t t
LUYENTHI999.COM 4
1 1ln 1 cos ln 1 cos
2 2t t c
TH1: 02
t cot 0t
sin
dtI
t 2
sin cos
sin (1 os )(1 os )
tdt d t
t c t c t
1 cos 1 cos
2 1 cos 2 1 cos
d t d t
t t
1 (1 os ) (1 os )cos
2 (1 cos )(1 os )
c t c td t
t c t
1 (1 cos ) 1 (1 cos )
2 1 cos 2 1 cos
d t d t
t t
LUYENTHI999.COM 5
• Đặt 2( )sin , (0 )2
x a b a t t
2( )sin cosdx b a t tdt 2( )sinx a b a t
2 2( )(1 sin ) ( ) osb x b a t b a c t
2 2 2
2( )sin cos
( ) sin cos
b a t tdtI
b a t t
2 2dt t c
Dạng 2 , ( )( )( )
dxI a b
x a b x
LUYENTHI999.COM 6
2
(4 )( 1)3 4
dx dxI
x xx x
21 5sin , (0 )2
x t t
10sin cos 5sin 2dx t tdt tdt 21 5sinx t 2 24 5 5sin 5(1 sin )x t t
2 2 2 2
5sin 2 sin 2
5sin .5(1 sin ) sin cos
tdt tdtI
t t t t
sin 22
sin cos
tdtdt t c
t t
VÍ DỤ 2:
LUYENTHI999.COM 7
( 1)( 3)I x x dx 1 1, 3 1x t x t
dt dx
2( 1)( 1) 1I t t dt t dt
2 2
2
1 1( )
2 2
y yI y dy
y y
2 2 4 2
3 3 3
( 1) 1 2 1 1 2 1( )
4 4 4
y y ydy dy y dy
y y y y
2
2
1 1ln
8 2 8
yy c
y
2 22
2 2
( 1 ) 1 1ln 1
8 2 8( 1 )
t tt t c
t t
2t x
2 2 2 2 21 1 1 2t t y t y t t y ty t
LUYENTHI999.COM 8
2 2( ( 2) 1 2)
8
x x
( )( )2
a bx a x b dx t x
2
2 2
1 1ln ( 2) 1 2
2 8( ( 2) 1 2)x x c
x x
CẦN NHỚ
LUYENTHI999.COM 9
Dạng 3: 2 2( , )I f x a x dx sin , ( )
2 2x a t t
cosdx a tdt
VÍ DỤ 3:
cosdx tdt
2 2
dxI
x x
21 ( 1)
dx
x
1 sin .( )2 2
x t t
2
cos
1 sin
tdtI
t
cos
cos
tdtdt t c
t
LUYENTHI999.COM 10
Dạng 4: 2 2( , )I f x x a dx
2 4 5
dxI
x x
2( 2) 1
dx
x
2
2 2 2 22 2
2
, ( )2 2 os
2 2
dtx a tgt t dx a
c t
t a t ax a x t x dx dt
t t
VÍ DỤ 4:
2 , ( )2 2
x tgt t
2os
dtdx
c t
2
2
os1
dtc tItg t
cos
dt
t
LUYENTHI999.COM 11
2
cos (sin )
os (1 sin )(1 sin )
tdt d t
c t t t
1 (1 sin ) (1 sin )(sin )
2 (1 sin )(1 sin )
t td t
t t
1 (sin ) (sin )[
2 1 sin 1 sin
d t d t
t t
1[ ln 1 sin ln 1 sin
2t t c
CÁCH 2: 2( 2) 1 ( 2)x t x 2 1
12
tx
t
LUYENTHI999.COM 12
Dạng 5: 2( , ax )I f x bx c dx 21: 0 axTH a bx c t x a
22 : 0 axTH c bx c tx c
21
03:
ax ( )TH
bx c t x x
Chú ý :2 4 6I x x dx
LUYENTHI999.COM 13
2 2x tgt
2 24 6 ( 2) 2I x x dx x dx
2 4 6x x t x
2 4 3I x x dx 2 21 ( 2)x dx
2 sinx t 0& 0a
0& 0a
LUYENTHI999.COM 14
22 3 5I x x dx 2 3 9 492( 2 )
4 16 8x x dx
2 3 9 492( 2 )
4 16 8x x dx 23 49
2 ( )4 16
x dx
73 44 sin
xt
0& 0a
LUYENTHI999.COM 15
DẠNG 6( )( )
dxI
x a x b
0& 0x a x b t x a x b TH1
1[
2 ( )( )
x a x bdt dx
x a x b
, Đặt
2
( )( )
dx dt
tx a x b
2 2lndt
I t ct
TH2: 0& 0x a x b ( ) ( )t x a x b , Đặt
( ) ( )1[
2 ( )( )
x a x bdt dx
x a x b
2
( )( )
dx dt
tx a x b
2 2lndt
I t ct
LUYENTHI999.COM 16
2 2 2 2
dx xdx
a x a x
du
u
DẠNG 7: ( , )a x
I f x dxa x
cos 2x a t 2 sin 2dx a tdt Cách 1: Đặt
2(1 os2 ) 2 osx a a c t ac t 2(1 cos 2 ) 2 sina x a t a t
Đặc biệt2 2
a x a xI dx dx
a x a x
LUYENTHI999.COM 17
DẠNG 8
1 2
1 2ax ax( , , )
a a
b bb bI f x dx
cx d cx d
ax mb
tcx d
1 2( , ....)m bcnn b b x dx Đặt Với
VÍ DỤ 1 223 33 31( ( 1) 1) ( 1) [( 1) 1]
dx dxI
x x x x
Đặt 3 2( 1) 3x t dx t dt 2
2 2
33
( 1) 1
t dt tdtI
t t t
2
22
( 1) 33 ln 1
2( 1) 2
d tt c
t
Suy ra
LUYENTHI999.COM 18
Ví dụ1 1
dxI
x x
1t x x 2 2 1 2 ( 1)t x x x 22 ( 1) (2 1)x x t x
2 24 ( 1) [t (2 1)]x x x 22 1
2
tx
t
2 2
3
( 1)( 1)
2
t tdx dt
t
1 1
dx
x x
2 2 3 2
3 3
( 1)( 1) 1
2 2
t t t t tdt
t t
2 3
1 1 1 1(1 )
2d
t t t
LUYENTHI999.COM 19
2 3 2
1 1 1 1 1 1 1(1 ) ( ln )
2 2 2I dt t t c
t t t t t
LUYENTHI999.COM 20
Dạng9: 2( ) ax
dxI
x bx c
1 1 tt x
x t
2
1( )dx dt
t
2
dtI
mt nt p
LUYENTHI999.COM 21
DẠNG 9 , ( 0& )ax ax
dxI a b c
b c
1
( ax ax )I b c dxb c
1 1
2 21
[(ax ) (ax ) ]dxb cb c
u du
VÍ DỤ
1 1
dxI
x x
1
21 1 1
( 1 1) ( 1) ( 1) ( 1) ( 1)2 2 2
x x dx x d x x d x 3 3
2 21 1
( 1) ( 1)2 2x x c
LUYENTHI999.COM 22
DẠNG 102
( )
( )
f x dxI dx
g x a
2
2 2 2 2
( ) ( ), ,
( )
f x a g a bg ca b c
g x a g a g a g a
2 2
2 2, ln
xdx dxI x a c I x x a c
x a x a
2 2 2ln2 2
x aI x adx x a x x a c
LUYENTHI999.COM 23
Ví dụ2
2
(2 1)
2
x dxI
x x
2 2 2
2 2 2 2 2
2 1 2 1 [( 1) 1] ( 1)
2 ( 1) 1 ( 1) 1 ( 1) 1 ( 1) 1
x x a x b x c
x x x x x x
2
2
ax (2 )
2
a b x b c
x x
Viết
2 2
2 2 2 2
2 1 2[( 1) 1] 4( 1) 5
2 ( 1) 1 ( 1) 1 ( 1) 1
x x x
x x x x x
2& 4& 5a b c
J K L
2
2 2 2
2[( 1) 1] 4( 1) 5d
( 1) 1 ( 1) 1 ( 1) 1
x xI dx dx x
x x x
LUYENTHI999.COM 24
22 2
2
2[( 1) 1]2 ( 1) 1 2 ( 1) 1 ( 1)
( 1) 1
xJ dx x dx x d x
x
2 2 2ln2 2
x aI x adx x a x x a c
2 2( 1) ( 1) 1 ln ( 1) 1J x x x x c 1
2 2 22
2
4( 1)2 [( 1) 1] [( 1) 1] 4 ( 1) 1
( 1) 1
xK d x d x x c
x
Suy ra
Công
thức
2
2
xdxI x a c
x a
Công
thức
Tính
2
5d
( 1) 1L x
x
2
2ln
dxI x x a c
x a
Theo công thức
Tính
Tính
LUYENTHI999.COM 25
25ln ( 1) 1L x x c
LUYENTHI999.COM 26
DẠNG 11 2
( )
( ) ax
P xI
Q x bx c
3
2 2
(6 8 1)
(3 4) 1
x x dxI
x x
3
2 2
6 8 1 12
3 4 3 4
x xx
x x
B1:Nếu bậc của P(x) cao hơn bậc của Q(x) thì chia P(x) cho Q(x)
B2: Lựa chọn phương pháp phù hợp cho mỗi tích phân mới
Ví dụ:
Ta có
Do đó2 2 2 2 2
1 1(2 ) 2
3 4 1 1 (3 4) 1
xdx dxI x dx
x x x x x
Tính
LUYENTHI999.COM 27
2 2(3 4) 1
dxL
x x
22 2 2 2
22( 1)
11
x tt x t x x
tx
2 2( 1) 1
dxdt
x x
2 2
2 2 2 2
( 1) 1
( 1) 1 (3 4) 1
dx x xdt
x x x x
Tính
Đặt
Suy ra
24
dt
t
2
1 ( 2) ( 2) 1 1
4 ( 2)( 2) 4 ( 2)( 2) 4 2 4 2
dt dt t t dt dtL dt
t t t t t t t
1 2ln
4 2
tc
t
LUYENTHI999.COM 28
2
22 2 1
1
xdxK x c
x
LUYENTHI999.COM 29
2
2
2
1
xI dx
x
2 2
2 2 22 2 2 2 2
2 1 2 1 3 1 3(1 )
1 1 12 2 2 ( 1) 2
x x
x x xx x x x x
2 2 23
2 ( 1) 2
dx dxI
x x x
2ln 2x x c
VÍ DỤ
TÍNH2 2
3( 1) 2
dxK
x x
LUYENTHI999.COM 30
'
2 22 2
x xt dx dt
x x
2
2 2
22 2
2
223 ( 2) 3 1 3 3
22 1 2 3 1 ( 3 1)( 3 1)11
tx dt dt dttK dt
tx t t tt
3 (( 3 1) ( 3 1) 3 3 1ln
2 ( 3 1)( 3 1) 2 3 3 1
t t tdt c
t t t
22
2
2
1
tx
t
22
2
2 2 2
222
( 2) ( 2) 2
xx
x dx dt dx dtx x x
LUYENTHI999.COM 31
LUYENTHI999.COM 32
LUYENTHI999.COM 33
Ví dụ 2 1
dxI
x x x
2 21 1x x x t x x t x
2
22
2
2 2 22 2 2(1 2 )
(1 2 )
t tdt
t ttI dt
t t t
22 2 2 1
1 21 2
tx x t tx x x
t
2 2 2
2 2
2 4 2 2 2 2 2
(1 2 ) (1 2 )
t t t t tdx dt dt
t t
LUYENTHI999.COM 34
2 4 2
2 4 2, 3, 3
2
A B
A B C A B C
A
2 2
2 2 2
2 2 2 (1 2 ) (1 2 )
(1 2 ) 1 2 (1 2 ) (1 2 )
t t A B C A t Bt t Ct
t t t t t t t
LUYENTHI999.COM 35
Ví dụ31
dxI
x x
2
3 31
x dx
x x
2 23
3 3
3 21
32 1 1
x xx t dx dt dt
x x
3 33 2
2 21
3 1 3 ( 1)( 1)
dt dtx t I
t t t t
2 2 2
1 (2 1)
( 1)( 1) 1 1 1
a b t c
t t t t t t t t
2
2
( 1) ( 1)(2 1) ( 1)
( 1)( 1)
a t t b t t c t
t t t
LUYENTHI999.COM 36
2 2
2 1 1 2 1 1( )
3 1 3 1 1
tI dt
t t t t t
2 01 2
1 1, ,3 3
0
a b
a b c b a
a b c
2 2
2 2 2 1 2
3 1 9 1 3 1
dt t dtI dt
t t t t t
2ln 1
3t 22
ln 19
t t
14 2ar
3 3 32
ttg c
LUYENTHI999.COM 37
Ví dụ
11
2x
1
31/2 1
xI dx
x
2 2x t dx tdt DO Đặt
1 1 12 2 3
6 3 2 3 22 /2 2 /2 2 /2
2 ( )2 2
1 3( ) 1 ( ) 1
t t dt d tI tdt
t t t
Tích phân I có dạng2
2ln
duu u a c
u a
Suy ra 3 3 22ln ( ) 1
3I t t
2 /2
1
LUYENTHI999.COM 38
Ví dụ3 33 3
4 2 2
x x x x dxI dx
x x x
2
1 dxt dt
x x
33
32 5 3 3 2333
2
1 11 1
( 1)1t tI dt t dt t t dt t t dt
t tt
1 43 2 2 2 23 31 3
1 ( 1) ( 1) ( 1)2 8
t tdt t d t t c
LUYENTHI999.COM 39
Ví dụ2
( 1) 1
xI dx
x x
2 2
3 3
2 2
[( 1) 1] ( 1) 2( 1) 1( 1)
( 1) ( 1)
x x xdx d x
x x
1 1 3
2 2 2( 1) ( 1) 2 ( 1) ( 1) ( 1) ( 1)x d x x d x x d x
3 1 1
2 2 22
( 1) 4( 1) 2( 1)3x x x c
LUYENTHI999.COM 40
Ví dụ2 2
22
1
1
xI dx
x x
2 22 22
2 2 2 22 2
1 1 1( )
21 1
x xxdx d x
x x x x
2 22 2
2 2 22 2
1 1
2 21 1
dx dx
x x x
2 2
22
1
2 1
dxK
x
2 1 22 2 22
22
1( 1) 1 3 5
2x dx x
LUYENTHI999.COM 41
2 2 2 21 1 2t x x t dx tdt
2 2
25 5
1 2
2 ( 1) ( 1)( 1)
tdt dtL
t t t t
2
5
1 1 1 5 1ln (ln 3 ln )
2 1 2 5 1
t
t
2 2
2 22
1
2 1
dxL
x x
LUYENTHI999.COM 42
C2:2 2
22
1
1
xI dx
x x
2 2
2 22 21 1
xdx dx
x x x
2
2
21x K
2
1 dtx dxt t
1
2
21 22 2
1
tdtK
tt
t
LUYENTHI999.COM 43
1/ 22
1/2ln 1t t
2 2
21
1
2
21
n
2
l
1
dxy t t
t tgy
x x ax adt
t
LUYENTHI999.COM 44
VÍ DỤ 2( 1) 1
dxI
x x
sin , ( ) cos2 2
x t t dx tdt
2
cos
(sin 1) 1 sin
tdtI
t t
cos
(sin 1) cos sin 1
tdt dt
t t t
os( ) 12
dt
c t
21 2sin ( ) 1
4 2
dtt
2
2 ( )4 2
2sin ( )4 2
td
t
LUYENTHI999.COM 45
cot( )4 2
tC
• Cách 2:2( 1) 1
dxI
x x
2
1 11
1
dtt x dx
x t t
22 2
1 2 1 21
tx
t t t
2 1 2 1 2 11 , ( )
2
t tx t
t t
LUYENTHI999.COM 46
2 1 2
tdtI
tt
t
1 2
dt
t
1
21
( 1 2 ) ( 1 2 )2
t d t
21 2 1
1t C
x
LUYENTHI999.COM 47
Ví dụ:6
22 3 9
dxI
x x
2
3 3sin
cos os
tx dx dt
t c t
3 2
2
6
3sinos
39
cos
tdt
c tItg t
t
3
3
66
1
3 18dt t
LUYENTHI999.COM 48
Ví dụ1
30
1
(1 )
xI dx
x
21 1 2x t x t dx tdt 0
2 31
2(2 )
ttdtI
t
1 2
2 30
2(2 )
t dt
t
2 sin 2 cost y dt ydy
24
60
2sin 2 cos2
8 os
y ydyI
c y
LUYENTHI999.COM 49
24
30
sin cos2
os
y ydy
c y
24
20
sin2
os
ydy
c y
4
20
12 ( 1)
osdy
c y
402[ ] 2(1 ) 2
4 2tgy y
LUYENTHI999.COM 50
Cách 21
30
1
(1 )
xI dx
x
os2 2sin 2x c t dx tdt 2
2
1 1 os2 2sin
1 1 os2 2cos
x c t t
x c t t
0
6
4
2 sin sin 22
8cos
t tdtI
t
24 4
3 30 0
sin sin 2 2sin cos
os os
t tdt t tdt
c t c t
40
2( ) 22
tgt t
LUYENTHI999.COM 51
C3:1
30
1
(1 )
xI dx
x
1 1
3 20 0
1 1
1 (1 ) (1 )(1 )
x xdx dx
x x x x
2
1 1 21 2
1 1 1 ( 1)
x t dtt x dx
x t t t
0 1
221 0
2 2( 1)1 4 ( 1)
1 ( )1
t dt tI dt
tt t tt
LUYENTHI999.COM 52
1 1 2
20 0
2
( 1)( 1)t yt y ydy
dty yt t
1 1 12
2 20 0 0
1 12 2 2
1 1
y dydy dy
y y
1 1
0 02 2 2
2y arctgy
LUYENTHI999.COM 53
LUYENTHI999.COM 54
LUYENTHI999.COM 55
LUYENTHI999.COM 56
