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Bi tp Nguyờn hm - Tớch phõn cú li gii
.
TP1: TCH PHN HM S HU T
Dng 1: Tỏch phõn thc
2
Cõu 1.
x2
I =
2
1 x 7 x + 12
2
1
I = 1 +
2
16
9
x
+
+
x x3 x2 + 1
2
1
3
1
3
+ ln( x 2 + 1) = ln 2 + ln 5 +
2
2
2
8
2x
1
1
2
3x 2 + 1
3
2
x 2 x 5x + 6
4
= ( x + 1)2 ( x + 1)3 I = ( x + 1)2 ( x + 1)3 dx =
3
3
0
8
( x + 1)
( x + 1)
Dng 2: i bin s
Cõu 5.
I =
1
Cõu 6.
I =
( x 1)2
(2 x + 1)4
dx
( 7 x 1)99
101
0 ( 2 x + 1)
1
x7
0 (1 +
x 2 )5
99
7x 1
1 1 7x 1
=
d
( 2 x + 1)2 9 0 2 x + 1 2 x + 1
dx
100
Cõu 7.
3
x 1
1 x 1
.
I =
+C
9 2x + 1
2x + 1
1 1
dt = .
2 1 t5
4 25
Bi tp Nguyờn hm - Tớch phõn cú li gii
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1
Cõu 9.
I = x 5 (1 x 3 )6 dx
0
t t = 1 x 3 dt = 3x 2dx dx =
4
Cõu 10. I =
3
1
2
3 7 8 168
1
2
3
1
t
1
3
t t 2 + 1 dt = 4 ln 2
1
1 32
dt
I = 5 10
. t t = x I =
2
2
5
1 128 1 t
I = 7
dx . t t = x I =
dt
7
7 1 t (1 + t )
1 x .(1 + x )
dx
(1 x 7 ).x 6
7
dx
6
x (1 + x 2 )
1
t : x =
1
I =
t
3
3
2
x 2001
Cõu 14. I =
1 (1 +
2
x 2 )1002
2
x 2004
I =
.dx
3
2 1002
1 x (1 + x )
Cỏch 2: Ta cú: I =
.dx =
1
x2
+ 1 dt =
2
x3
dx .
11
x 2000 .2 xdx
. t t = 1 + x 2 dt = 2 xdx
2
2000
2
2
2 0 (1 + x )
(1 + x )
1 2 (t 1)1000
1 2 1
I = 1000 2 dt = 1
21 t
2 1 t
t
Cõu 15. I =
.dx . t t =
1
t 2
1
I= 2
=
dt
=
.ln
=
ln
2
2 2
2 2 1t 2 t + 2
t + 2 1 2 2 2 + 1
1 t 2
dt
1
1
1
Bieõn soaùn: Thay Tran Sú Tuứng - Trang 2
1
1
dt
= x
. t t = x + dt = 1 dx I =
.
2
2
1
x
2
1+ x
x
t
+
2
2
x +
x2
du
5
5
t t = 2 tan u dt = 2
; tan u = 2 u1 = arctan 2; tan u = u2 = arctan
2
2
2
1
2
dx
dx
x6 + 1
x4 + 1
1
1
2
1
4
x
Ta cú: I =
dx . t t = x + I = ln
1
x
5
1
+x
x
2
x2
x6 + 1
=
1
x2 + 1
+
x2
x6 + 1
1 1 d (x3 )
1
I = 2 dx + 3 2 dx = + . =
3 0 (x ) + 1
4 3 4 3
0 x +1
Cõu 19.
1
3
3
I=
x2
1+ 5
2
Ta cú:
1
0t
dx =
x2 + 1
2
x +1
4
2
x x +1
0
1
1
3
t t = x 2 I =
.
x 4 x2 + 1
1
I =
2
2
Cõu 20. I =
Cõu 21. I =
dx
x2 +
1
x2
1
x2
1
4
Bieõn soaùn: Thay Tran Sú Tuứng - Trang 3
2
=
6 3
Bi tp Nguyờn hm - Tớch phõn cú li gii
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TP2: TCH PHN HM S Vễ T
Dng 1: i bin s dng 1
x
Cõu 22. I =
dx
2
3x + 9 x 1
x
I =
x + x
1+ x x
x2
dx =
x2
+ I1 =
1+ x x
2
dx
1+ x x
1+ x x
x
1+ x x
Vy: I =
x
)
3
4
1 + x x + C1
3
3
t2
1 + t dt =2 + ln 2 .
1
3 1
2 12
t t = 4 x + 1 . I = ln
1
t: t = 1 x 2 I = ( t 2 t 4 ) dt =
0
1+ x
Cõu 25. I =
1+ x x
dx . t t= 1 + x x t 2 1 = x x x 3 = (t 2 1)2 x 2dx =
4 2
4 3 4
4
3 (t 1)dt = 9 t 3 t + C = 9
+ I2 =
x
dx +
0
2
.
15
dx
1 3
1
t +t
2
t(t 1)dt
3
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Bi tp Nguyờn hm - Tớch phõn cú li gii
2
t t = x + 1 2tdu = dx I =
2t 3 8t
1t
0
x.
Cõu 29. I =
3
2
+ 3t + 2
2
5
x2 + 1
1
x 3x + 1
Cõu 30. I =
2
3
dx
2
t2 1
+1
4 3
2tdt
2tdt
t t = 3x + 1 dx =
I = 2
.
3
3
t 1
(
t
1)
dt
+
2
2
92
2 t 1
dx
x + 1 = t x = t 2 1 dx = 2tdt
t
2
2(t 2 1)2 + (t 2 1) 1
I =
2tdt
t
1
1
2
2
Cõu 33. I =
0
.2tdt =2
2
1
x +1
2
1 + 2x )
(1 +
2
2
t3
1
1
16 11 2
t
dt
=
22
22
2 2
t t2
t2
t2
=
Cõu 34. I =
8
3
1 t2
2
1
3t + 4 ln t + = 2 ln 2
2 2
t
4
x 1
2
8
3
= 1 + ln
(
3 + 2 ) ln ( 8 + 3)
1
Cõu 35. I = ( x 1)3 2 x x 2 dx
0
1
3
I = ( x 1)
0
2
Cõu 36. I =
2 x x dx = ( x 2 2 x + 1) 2 x x 2 ( x 1)dx . t t = 2 x x 2 I =
0
2 x3 3x 2 + x
0
x 3dx
3
4 + x2
3
t t = 4 + x 2 x 2 = t 3 4 2 xdx = 3t 2dt I =
3 2 4
38
(t 4t )dt = + 4 3 2
23
25
4
Cõu 38. I =
1
dx
1 11
1
1
+ 1 dx = ln x + x |1 = 1
2 1 x
2
1
1
1 + x2
dx . t t = 1 + x 2 t 2 = 1 + x 2 2tdt = 2 xdx I2=
2x
Vy: I = 1 .
2
t 2dt
2
2 2(t 1)
=0
dx
4 x2
dx
x
Ta cú: I =
I=
1
1
3 1
1
Ta cú: I = 2 1 . 3 dx . t t = 2 1 I = 6 .
x
x
1 x
3
2
0
1
t(tdt )
4 t2
x
3
3
0
2 3
= 3 + ln
2
+
3
3
Bieõn soaùn: Thay Tran Sú Tuứng - Trang 6
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.
Cõu 41. I =
2 5
0
dx
t t = x + x + x + 1 I =
1+ 3
1+
2dt
= ln(2t + 1)
1
2t + 1
1
3
x2
1 + x )2 (2 + 1 + x )2
0 (1 +
4
4
2t (t 2 1)2 dt
t t = x + 1 I =
t(t + 1)2
1
Cõu 46. I =
3
2 2
x x 3 + 2011x
x4
1
Ta cú: I =
3
2 2
1
1
1
x2
x3
dx +
2 2
dx =
1
2
2
2
= 2 (t 1)2 dt = (t 1)3 =
1
3
3
1
3
7
2
t 3dt =
0
2 2
2011
2011x dx =
2 x2 1
3
3
=
213 7
128
14077
16
3
2
2 5
2t
1
dt = 5 1 +
dt = 5 3 1 + ln
3 12
t t2 + 1 t2 + 1
t(t 2 + 1)
1
2
Cõu 44. I =
2
t3 2
1
Cõu 43. I =
3t
dt
dt
1
2
t .(t
3
2
1) 3
Bieõn soaùn: Thay Tran Sú Tuứng - Trang 7
Bi tp Nguyờn hm - Tớch phõn cú li gii
3
=
2
1
3
2 1 3
= t
t4
1
dt
1
t 2 . t 3 1 3
t
t u = 1
2
2
3
1
t 4 1 3
t
1
2
I=
1
2
=
3 1
3 0
1
1 2
= u3
=
0
1
3
2
x4
dx
1 2
x x x +1
1
2
2
dt =
19
2 4+ 2
+
ln
3
4 4 2
Dng 2: i bin s dng 2
1
2
x
ln
1
+
x
( ) dx
1+ x
dv = 2 xdx
0
Cõu 50. I =
2
(x
5
K=
1
2
+ x 2 ) 4 x 2 dx
2
I=
2
(x
5
2
2
4 x 2 dx . t x = 2sin t . Tớnh c: B = 2 .
2
2
+ Tớnh B =
2
2
2
+ Tớnh A =
2
2
Vy: I = 2 .
Bieõn soaùn: Thay Tran Sú Tuứng - Trang 8
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.
1 2x
2
+ Tớnh I 2 =
1
I2 =
4
2x4
1
4 x2
2x4
dx .
3 2 4
7
x dx = .
21
16
=
cot 2 t.d (cot t ) =
2
4
8 sin t
8
8
8
sin t
6
Vy: I =
1(
7 2 3) .
16
1
x 2dx
0
4 x6
Cõu 52. I =
Cõu 54. I =
0
Ta cú: I =
0
I =
2
2
3
1
2
Cõu 55.
0
t
2
0
+
3 3
4
2
2
1 2 x 1 x 2 dx
6
t x = sin t I = (cos t sin t )cos tdt =
0
Bieõn soaùn: Thay Tran Sú Tuứng - Trang 9
12
+
3 1
8 8
3
2
I=
3
2
3
x.
2
x 2 1dx
x
x2 1
3
dx = 5 2
Chỳ ý: Khụng c dựng phộp i bin x =
3
2
1
vỡ 2;3 [ 1;1]
cos t
TP3: TCH PHN HM S LNG GIC
Dng 1: Bin i lng giỏc
Cõu 57. I =
8cos2 x sin 2 x 3
dx
sin x cos x
(sin x cos x )2 + 4 cos 2 x
I =
dx = ( sin x cos x 4(sin x + cos x ) dx
sin x cos x
= 3cos x 5sin x + C .
cot x tan x 2 tan 2 x
dx
Cõu 58. I =
sin 4 x
2 cot 2 x 2 tan 2 x
2 cot 4 x
4
cos 2 x +
1
dx
4
dx +
=
2
2 2 1 + sin 2 x +
=
2 2
3
2 2 1 + sin 2 x +
sin x +
4
8
1
3
=
ln 1 + sin 2 x + cot x +
+ C
4
8
4 2
3
2 x
1 cos x +
2sin +
3
3
3
2 6
Cõu 61. I =
6
1
2 sin x
3
0
dx
1
Ta cú: I =
2
dx =
1
2
6
dx
0 sin x sin
3
3
x x
cos +
2 6 2 6
dx
x
2 6
20
x
20
x
sin
cos +
2 6
2 6
6
0
x
ln cos +
2 6
6
0
Cõu 62. I =
x + cos4 x )dx
0
2
0
1
1 2
0
1
5
cos xdx =
2
(1 sin x )
2
A =
0
2
d (sin x ) =
0
8
15
2
2
cos x.dx =
2
x cos 2 xdx
0
2
I = cos2 x cos2 xdx =
0
2
1
12
(1
+
cos
2
x
)
cos2
xdx
=
1 + cos x
sin2 x
I = 2 (4sin x 2sin 2 x )dx = 2
0
Cõu 67. I =
2
1 + sin xdx
0
I=
2
2
2
x
x
x
x
x
sin + cos dx = sin + cos dx = 2 sin + dx
Cõu 68. I =
4
0
dx
6
cos x
4
Ta cú: I = (1 + 2 tan2 x + tan 4 x )d (tan x ) =
0
Dng 2: i bin s dng 1
Bieõn soaùn: Thay Tran Sú Tuứng - Trang 12
28
.
15
1
3
1
t t = tan x . I = t 3 + 3t + + t 3 dt = tan 4 x + tan2 x + 3ln tan x
+C
t
4
2
2 tan2 x
2t
.
Chỳ ý: sin 2 x =
1 + t2
dx
Cõu 71. I =
sin x.cos3 x
dx
dx
dx
2t
I =
. t t = tan x dt =
=
2
; sin 2 x =
sin x.cos x.cos2 x
sin 2 x.cos2 x
cos2 x
1
2011 1
sin 2 x cot xdx =
sin 4 x
Ta cú: I =
t t = cot x I =
2
2011
t
(1 + t 2 )tdt
4024
2011
cot 2 x
sin 4 x
4024
cot xdx
8046
0
2
sin x.cos2 x
(t 1)2
dx . t t = 1 + cos x I = 2
dt = 2 ln 2 1
1
+
cos
x
t
0
1
2
Ta cú: I = 2
Cõu 74. I =
3
sin
2
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1
2
1 u2
3
du = ln 2
u
8
1
I =
sin
Cõu 75. I =
2
x (2 1 + cos2 x )dx
2
2
2
2
2
+ K = sin2 x 2 cos2 x = 2 sin2 x cos xdx = 2 sin2 xd (sin x ) =
I =
2
3
cos2 x
.
4
I=
3
(1 + t 2 )2 dt
t2
1
3
=
1
3
1
1
t3
( 2 + sin x ) dx
2
0
Ta cú: I =
2
sin 2 x
(2 + sin x )2
0
3
I = 2
2
t2
t2
2
sin x
cos 2 x dx
0
I=
6
0
sin x
dx =
cos 2 x
6
sin x
2 cos2 x 1 dx . t t = cos x dt = sin xdx
0
Bieõn soaùn: Thay Tran Sú Tuứng - Trang 14
6
dt =
1
2 2
ln
3
2
2t 2
2t + 2
1
3
2
=
1
2 2
ln
32 2
52 6
2
t t = cos x . I =
3
( + 2)
16
6
Cõu 81. I =
4
sin 4 x
dx
sin6 x + cos6 x
0
4
I=
2
.
3
Cõu 82. I =
2
sin x
( sin x +
0
3 cos x
)
3
dx
Ta cú: sin x + 3 cos x = 2 cos x
dx
I=
=
+
16 0
6
16 0
cos3 x
cos2 x
6
6
Cõu 83. I =
sin x 1 cos2 x
4
sin x
2
cos x
1 cos2 x .dx =
sin x
2
cos x
3
2
4
4
dx +
2
cos x
sin x dx +
3
7
3 1.
12
3
Bieõn soaùn: Thay Tran Sú Tuứng - Trang 15
4
sin x
2
0 cos
x
sin x dx
dx =
20
20
0 sin x + 3 cos x
1 cos2 x +
sin x +
3
3
6
6
6
1
2
I=
2
sin x 3 cos x dx = I =
0
3
sin x 3 cos x dx +
2
sin x
3 cos x dx = 3 3
0
3
(sin t + cos t )3 (sin x + cos x )3
0
2
0
2
12
dx
1
4
1
2I =
=
= cot( x + ) = 1 I =
2
20 2
2
4 0
2
0
sin xdx
( sin x + cos x )
3
;
I2 =
2
cos xdx
0
( sin x + cos x )
3
.
t . Ta chng minh c I1 = I2
=
1
tan( x ) 2 = 1
2
4 0
1
I = 7I1 5I 2 = 1 .
2
Cõu 88. I =
2
3sin x 2 cos x
(sin x + cos x )3 dx
0
Bieõn soaùn: Thay Tran Sú Tuứng - Trang 16
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2
3cos x 2sin x
(cos x + sin x )3 dx
0
3sin x 2 cos x
(sin x + cos x )3
dx +
2
0
dt =
3cos x 2sin x
(cos x + sin x )3
0 1 + cos
2
t
2
1 + cos t
dt =
sin t
2
0 1 + cos t
dt I
2
= + I =
2
4 4
8
4
sin t cos t
cos3 t + sin3 t
dt =
2
sin 4 x cos x
cos3 x + sin3 x dx
0
2
2
2I =
4
20
2
1
4
I= .
2
0
1
cos2 (sin x ) tan
Cõu 91. I =
t x =
2
2
1
I=
2
2
1
1
Do ú: 2I =
+
tan 2 (cos x ) tan 2 (sin x ) dx = 2 dt =
2
2
cos (sin x ) cos (cos x )
0
0
I=
2
1
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du
4u
2
. t u = 2sin t I =
4
2 cos tdt
2
4 4sin t
6
I=
3
3
0
=
dx
sin x
.dx =
cos x 3 + sin2 x
15
2
1 t+2
ln
4 t2
ln
4
15
4
2
3
x
3
cos2 x 3 + sin2 x
sin3 x + sin2 x
3
I =
0
sin x.cos x
x + ( x + sin x )sin x
Cõu 94. I =
2
dx .
15
2
3
1
1
dt
t+2 t2
3+2
1 (
=
ln 15 + 4 ) ln ( 3 + 2 ) .
2
3 2
(
)
dx
dx
3
2
dx
dx
= 3
=4 2 3
x
2
1 + cos x
3 2 cos
2
4 2
+42 3.
2
Cõu 95.
I=
0
2
udu
22
2
dx . t u = 3sin 2 x + 1 I = 3
= du =
u
31
3
1
3sin2 x + 1
2
sin 2 x
tan x
4 dx
cos2 x
tan x
.
=
=
2
t +1 0
2
(t + 1)
dt
0
Cõu 97. I =
3
cot x
dx
sin x.sin x +
4
6
= 2
ln 3
3
3 +1
3 +1
3
3
Cõu 98. I =
3
dx
sin2 x.cos4 x
4
3
=
+
+
t
dt
=
+
t
+
2
2
t
3
t
1
1 t
1
3
=
8 34
3
2
+ 5t + 2
cos4 x (tan 2 x 2 tan x + 5)
4
t t = tan x dx =
Tớnh I1 =
1 1 2
1
1
2
dt = ln 3 ln 2
3 0 t + 2 2t + 1
2
3
I=
1 t
t 1
2
2
2t + 5
= tan u I1 =
1
2
= 2 + ln
0
du =
2
3
2
Cõu 101. I =
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6
I=
2
2
sin x
3sin x 4sin3 x
dx =
dt
2
4t 1
=
1
4
3
2
0
dt
t2
1
4
=
1
ln(2 3)
4
6
Cõu 103. I = 2 1 cos3 x .sin x.cos5 xdx
1
t t = 6 1 cos3 x t 6 = 1 cos3 x 6t 5dt = 3cos2 x sin xdx dx =
2t 5dt
cos2 x sin x
1
1
t 7 t13
12
I = 2 t 6 (1 t 6 )dt = 2
=
7 13 0 91
0
Cõu 104. I =
4
tan xdx
0
dt =
3 2
2
Cõu 105. I =
2
cos2 x
(cos x sin x + 3)3
0
tan x
dx
cos 2 x
4
dx
t 3
1
dt = .
3
cos x. tan x + 1
Ta cú: I =
4
sin 4 x
4
4
sin x + cos x
0
dx . t t = sin 4 x + cos4 x I = 2
2
2
dt = 2 2 .
1
Cõu 108. I =
6
0
1
2
2(2t 1)
1
dt = 2 6 ln .
t +1
3
1
dx . t t = cos2 x I =
tan( x )
4 dx
cos 2 x
1
3
6
tan 3 x
6
6
tan 3 x
Ta cú: I =
dx =
dx .
2
2
2
2
0 cos x sin x
0 cos x(1 tan x)
3
3 t3
1 1 2
t t = tan x I =
dt = ln .
2
6 2 3
0 1 t
3
Cõu 111.
4
4
dx
sin3 x.cos5 x
Ta cú:
3
1
4
1
2
tan x cos x
3
dx .
= 4 ( 8 3 1)
1
Bieõn soaùn: Thay Tran Sú Tuứng - Trang 21
6 2
Bi tp Nguyờn hm - Tớch phõn cú li gii
Cõu 112. I =
x(
0
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cos3 x + cos x + sin x
)dx
u = x
du = dx
+ Tớnh J = x.cos x.dx . t
J = 2
=
dv
cos
xdx
v = sin x
0
x.sin x
+ Tớnh K =
0 1 + cos
2
2
1 + cos ( t )
=
4
2
4
( x ).sin x
0
sin x.dx
2
dt =
x
1 + cos2 x
K=
dx
dt =
0
( x + x ).sin x
0
t x = t dx = dt
dx .
( t ).sin( t )
K=
K=
x
3 + cos 2 x
dx
6
2
Ta cú: I =
sin x cos x
sin x 3 + cos x
2
2
dx . t t = 3 + cos2 x
6
I=
15
2
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.
Bi tp Nguyờn hm - Tớch phõn cú li gii
3 1
3
34
sin t , 0 t I = cos2 tdt = + .
2
2
2 4 2
20
t cos x =
Cõu 115. I =
3sin x + 4 cos x
dx
2
x + 4 cos 2 x
2
2
2
2
2
3
+
cos
3
+
cos
3
+
cos
3
+
cos
4
sin
x
x
x
x
x
0
3 3(1 + tan 2 u )du 3
=
3(1 + tan 2 u )
6
0
6
t t = 3 tan u dt = 3(1 + tan 2 u )du I1 =
2
+ Tớnh I 2 =
0
Vy: I =
1
4 cos x
4dt1
dx . t t1 = sin x dt1 = cos xdx I 2 =
dt1 = ln 3
2
4 sin x
4 t12
0
3
6
6
tan x
cos2 x
t u = tan x du =
dx =
1
2
cos x
1
4
+1
6
tan x
2
3
3
dt = t
7
3
7
3
Cõu 117. I =
2
7
= 3
3
sin x + cos x
( sin x cos x )
t t = 2 tan u I =
1
2
2
arctan
0
+2
dx . t t = sin x cos x I =
1
2(1 + tan 2 u)
1
1
du = arctan
2
Cõu 118. I =
3
x sin x
cos2 x
dx .
S dng cụng thc tớch phõn tng phn ta cú:
I=
3
1
x
xd
=
cos x cos x
3
3
tớnh J ta t t = sin x. Khi ú J =
Vy I =
3
3
2
dx
=
cos x
3
1 + cos x .e
x
dx
x
x
1 + sin x 1 + 2sin 2 cos 2
1
x
Ta cú:
=
=
+ tan
x
x
1 + cos x
2
2 cos2
2 cos2
2
2
2
x
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.
Cõu 120. I =
4
Bi tp Nguyờn hm - Tớch phõn cú li gii
x cos 2 x
(1 + sin 2 x )
0
2
dx
u = x
du = dx
cos2 x
1
dx = +
4+
16 2
2 1 + sin 2 x 0 2 0 1 + sin 2 x
I = x. .
1
.
1
2 cos2 x
4
0
dx
1 1
dt = t 3 t 2 + 2t 2 ln t + 1 + C = e x e x e x + 2 e x 2 ln e x + 1 + C
1+ t
3
3
Cõu 122. I =
I =
( x 2 + x )e x
x + e x
( x 2 + x )e x
x
x+e
dx
Cõu 123. I =
dx
dx =
xe x .( x + 1)e x
xe + 1
x
dx . t t = x.e x + 1 I = xe x + 1 ln xe x + 1 + C .
e2 x + 9
x ln( x 2 + 1) + 2011
Ta cú: I =
dx . t t = ln( x 2 + 1) + 1
2
2
( x + 1) ln( x + 1) + 1
1 t + 2010
1
1
1
I=
dt = t + 1005ln t + C = ln( x 2 + 1) + + 1005ln(ln( x 2 + 1) + 1) + C
2
t
2
2
2
Cõu 124. I =
e
Cõu 125. J =
1
xe x + 1
x (e x + ln x )
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