các tích phân sau đây (sử dụng tích phân từng phần): a. ;b. ;c. ;d. ;e. ;f. ;g. ;h. ;i. ;j. ;k. ;l. ;m. ;n. ;o. ;p. ;q. ;r. ;s. t. (). Hướng dẫn:Câu a: Tính . Đặt Tính .Vậy .Câu b: . Đặt .Câu c: Tính: . Suy ra: .Vậy suy ra: .Câu d: . Đặt . Câu e: …Đặt Câu f: …Đặt .Đặt
Trang 1Bài 1 Tính các tích phân sau đây (sử dụng tích phân từng phần):
e
xdx x
x I
1
2ln
1
;
2 1
)1ln( x dx
12
e e
dx x x
2 1 2
ln
dx x
x
2 1 2)1ln(
1
2)
5 2
)1ln(
3 6
2cos
)ln(sin
dx x
x
2 1
)cos(lnx dx
2 0
(*))
ln(sin
dx x
0
2tan
xdx x
1 0
3 4
2sin
dx x
x
3 0 2cossin
dx x
x x
2 0cos
xdx e
1 0
3 2
dx e x
;
0
2 2 2)1(
2
dx x
e
dx x
x xdx
x xdx x
x I
1 1
1
lnln
1
e
xdx x
I
1
ln 2 2 ln 2 1
ln 2 2 2
1 2 2
ln)(lnlnln
1 1
2 1
e e
e
x x
xd dx
14
2 2
3 3
ln
1ln
1ln
1ln
1ln
1
I dx x
dx x
dx x
dx x x I
e e
e e
e e
dx x x du dx
dv
dx x u dx x
1 ln
1
3
3 3 3
2 1
ln
1ln
ln
e
e e
e e
dx x x
x dx x
3 3 3
3
2 2
1
1 ln
ln
1 ln
e e
e
e e e
e
dx x x
x dx x I
dx x
3
3
.
Trang 2x dx du
2
1ln
1 2 2 1
2 1
x
x
) 1 ln(
1 ) 1
1 2 2 2
ln 2 ln 2 ln 2 ln
Đặt e x xdx x e x dx x e x e
1 2 1 2
1
2
ln 2 2
ln2
lnln
2
1 2 1 2 1
2 2 1
1
2 2
e e
.
Câu g:
e
dx x I
1
2)
1 1 2 2
ln 2 ln 2 ln ln 2 ln
1 1
Câu h:
5
2
)1ln(
) 1 ( ) 1 ln(
) 1 ( 1 2 ) 1 ln(
x dx x x x I x dx du x dv x u
5 2
2 1)ln( 1) ( 1)
2
27 4 ln 24 2
) 1 ln(
)ln(sin
dx x
cos ) ln(sin tan tan
3ln3632
1ln3
32
3ln3)
ln(sin
3 6
3 6
1
) sin(ln ) (ln cos ) cos(ln )
sin(ln
)
cos(ln
dx x x x dx x I x
) cos(ln sin ) sin(ln ) cos(ln )
sin(ln
dx x x x dx x x x du dx
1
2 1
2
)cos(ln
2
1)2cos(ln)
2sin(lnsin
)cos(ln
ln(sin
dx x I
0 sin ) ln(sin sin
)
ln(sin
dx x x x x
)ln(sin
xdx x
x x
Trang 3) ln(sin ) ln(sin cot ) ln(sin cot
dx x x xdx x v dx xdx dv x u
2 0
2
)ln(sin
xdx x
x x
2 0 2
2 0
x x x
xdx x
2
3
cos 3 cos cos
33
3sin12
13
coscos
33cos12
13
cos
0 0
3 0
4 0 2 4
0 2 4
x x dx x
x xdx x
4
)(costan
x
x d x
x
4 0
2ln42cos
lntan
)1(tantan
2 4
0
2 4 0
4 0 4
0
4 0 2 4
t t x x
1 0
sin
dx x
x I
cot cot
3
4 sin
coscot
x
2
3ln2
136
)349(sin
ln36
)349(sin
)(sin
4 3
dx x
x x I
Trang 43 cos 3
23ln21
3cos31sin
1sinln21
3cos3sin1
)(sin
3cos3sin1
3 0
2 3
x d x
0
2 2
2
0
sin sin sin
1cos
2cos
1sin
2 0 2
2 0
2 0 2
2 0 2 2
e xdx e
xdx e
e xdx e
e xdx
Câu r:
1 0 2 1
0
dx xe x dx e x
Đặt tx dt xdx dt xdx
22
0
t t x
x
1 0 1
0 1
0
2
12
2
dt e t dt
e t dx xe
sin )1 2 sin )1 2 sin )3 2 (
cos cos )1 sin )1 cos
sin
1
xdx x x xdx x x
2)1(
2
dx x
1 2
1 ) 1 ( 2
x dx x I x v dx x xdx dv x
t t x
2tan
1
)tan1(1
4 4
4 4
2
2 1
1 2
dt t x
4 1)
(x x
dx
1 0 3
x
I 1
Trang 5k.
4 3
35
dx x x
x
1 0
114
dx x x
2 0 2
2
0
14
1)
4(16
dx x
x
dx I
31
2ln313
32
12
1 0 1
0
2 2
x
dx I
2 2
1 0
2 1
0
2
2
3 2
1 4
3 2 1 1
x
dx x
dx x
x
dx I
.
2
3 tan
2
3 2
4.2
3)1(tan43
)1(tan2
3)
1(tan4
3
)1(tan2
3
3 6
3 6
2
2 3
dt t
x
x B x
A x
B x
A x
2 1 1
2 )
(
B A B
A B A B
A x B
3
11
2ln3
11ln3
12ln3
113
123
1)1)(
2(
5 3
5 3 5
3 5
3 5
3 5
1(
13
4
1
2 2
2 2
2
2 2
2
2 2
2 2
2 4
x
x x
x
x x
x x
x
)3(2
1)
1(2
1
2 2
2 2
1(2
13
0
12
11
12
1
t
t x
4tan
1
)tan1(1
4 0
4
0
2
2 1
34
2
134
Trang 6Câu f: dx
x x
x x x dx x
x
x x
x dx
x x
x x
x x
2
2 4
1
2
2 4
1 2
2 2
4
1
)1)(
1()
1(
)1)(
1()
1(
1)
1(
1 2 4
1
4 1 2 4
1
4 1 2 4
1)
1(
)1)(
1()
1
dx x
xdx x
x d dx x
x x
dx dx
x x
x x dx
31
ln1ln1
1 4
3 2
1
4 1) ( 1)
dx x x
1
t
t x
11
ln4
11
114
1)1(4
1)1( 4
16 1
16 1
16 1
)1)(
1(
31
2 2
x x
C Bx x
A x
x x
Bx Cx Bx A Ax
0 0
C B C
A C B A B A
0
1 0 2
1 0
1 0
2 1
0
21
ln1
21
1
21
11
x x
x x
dx x x
x x
dx dx
x x
x x
2 2
1 0 2
1 0 1
0 2 1
0 2
2 1
0
2
3 2
1 2
3 1 ln
2
1 1 ln 1 2
3 1
) 1 (
x x
x x
dx dx
x x
x x x
2
3 2
2
3 2
x
Suy ra
33
23
2)
1(tan43
)1(tan23
2
32
1
6 6 6
6
2
2 1
0
2 2
dt x x
dx
Vậy
32ln33
2.2
31ln
2
11ln1
0
2 1
0 1
2 2
2
1
0
2 4
3
3 2
1 1
x
xdx dx
x x
x I
2
3 2
tan 2
3 2
x
.
Trang 7Suy ra 33 183
)1(tan43
)1(tan43
3
32
11
3 6 3
6
2
2 1
0
2 2
2
1
0
2 4
xdx dx
x x
0 2 2
0 2 2
0
224
2
1224
2
32
x x
dx dx
x x
x dx
x x
x dx
x x
x I
2 2
2 0
2 2
2 0 2
2 0
2 2
0
2
2
3)1(3ln3)1(42ln3)1(42
)42
(
x
dx x
dx x
x x
dx x
33
3)
1(tan3
)1(tan33
)1(
3 6
3 6
3 6
2
2 2
0
2 2
3ln21
ln22ln71
22
7)
2)(
1(
352
3
4 3
4 3
2 2
1 0 2 1
0 2 1
0
)65(2656
5
522652
152265
114
x x
x x d x
x
dx dx
x x
x dx
x x
x dx
x x
x
2
9ln3ln2ln2ln232
65ln2)3
)(
2
(
1 0
1 0 1
0
1 0
1 0 2
dx x
x
1 0
3
)1
3 2
3
)1
5 1)
(x x
3
)12
2 1 2 212
7x dx x
x
1 0
x
j.
2 1 4
21
1
dx x
x I
2 4
21
1
dx x
1 0
1 0 2 1
0 2
2 1
0
2
2
)2)(
2(
44
44
44
44
x x
dx dx
x
dx dx
x
x dx x
x I
(
x x
x x
Trang 8ln2ln12
2
12
2
1 0
1 0
1 0
5 3
5 3
5 3
5 3
5
102
)2(32
10)2(32
43
x
dx dx
x
dx x
dx x
dx x
x dx
x
x I
3ln1061ln103ln103.35.32ln103
210
x
2 3
2
)1()1()1()1()1()1
B x
A x
C x
B x
A x
1 2
0 )
2
(
2
C B C
B A B A C
B A x B A Ax
1)
1(
1)
1 0 2 3
1 0
2 1
0
3 1
0
2 1
0
1 )
1 ( 2
1 1
1 )
1 (
) 1 ( )
1 (
) 1 ( )
1 ( ) 1 ( )
1
x d x
x d x
dx x
dx dx
x
x I
Câu d:
1
0
3 2
3
)1
1
t
t x
0
2 2 3
3 4
0
2 2
3 4
0
3 2
2 3
cossincos
1.cos
sin)
1(tan
tan)
1(tan
)1.(tantan
t t
t dt
t
t t
dt t t I
3 1
1 1 1
1 0
2 3 1
0 2 1
dx dx
x x
x dx
x x
x dx
x x
x x
x x
4 2
1
2 1 5
5 2
1 5
5 2
1
5
5 5
2
1
)1()
1(
)1()1(
5 5
5 2
33ln5
12ln1
ln5
1ln1
)1(5
1
x x
x
x d
3 1
0
2 1
0
3 1
0
3 1
0
3 1
0
1)12(2
1)12(2
1)
12(
122
1)
12(
1122
1)
1
2
dx x
dx x
dx dx
x
x dx
x
x dx
1 2 ( 2
1 1
2
1 4
1 ) 1 2 (
) 1 2 ( 4
1 )
1
2
(
) 1
0
3 1
2 1
2 1
2 1
2 2
1612
7
127112
x dx
x x
x I
x I
2 2
2 2
x C x
B Ax x
C x
B Ax x
x
x x
C A x
x C x
B Ax
Trang 94 2
0
C B C
x x
x x x
1 0 2 1
0 2 1
0
2
9)1(5
29)
2(5
9)
1(5
292
2
14
x
dx dx
x
x dx
x x
x dx
x x x
x I
0
1 0 2 1
0
1 0 2 1
0
2
2
15
22ln5
91ln10
925
915
21
)1(
10
9
x
dx x
x x
dx x
dx x
1
t
t x
Suy ra
41
tan
)1(tan1
4 0
4 0 2
2 1
272
2
14
1 0
2 3
2 2 2
2
2 2
1
4 2
21
111
11
1
1
dx x
x
x dx
x x
x dx x
x
x dt dx x dt
2 2
1
t
t x
12
2ln22
12
2
2 5 2
2 5 2
2 2
2 2
5 1
1
2 2
2 2
5 1
1
2 2
2
2
2 2
5 1
1
2 4 2
11
11
1
111
1
111
1
x x
x x d dx
x x
x dx
x x
x
x
x dx
x x
x I
x x d t x
5 1
1
t
t x
x
11
1
4 4
1
4 1 2
2 2
dt t
x x
x x d
Bài 4 Tính các tích phân sau: (sử dụng tích phân hàm vô tỉ)
a. I x x dx
2 1
2 1 x dx x
2
1 x
dx x
2
3 1 x dx x
Trang 10j.
1 0
2
2 4 x dx x
2 2
3
2)1
x I
1
ln2
x
dx x
Hướng dẫn:
Câu a:
)47(3
1)3(3
1)3()3(2
1)3(32
1
2 1
3 2 2
2 1
2
1 2 2
2 1 2 2
16 0
16 0
16 0
16
0
99
19
)9()9)(
9(
)9(
dx x x
x x
x x
dx x x
x x
9(27
2)
9()
3 3 16
0 2 1 16
1(4
1)1()1(3
1)1(1
3
11
42 0
42 0
3 3
1 3 42
2
t
t x
x
Vậy
3 2
3 2
3 2
3 2
3
2
t t t
dt dt
t
dt dt
t
t t
dt t
0 2 3 2 3 1
0 2 1 1
0
2 1 1
0 1
0
x x
dx x dx x
dx x x
x x
x
Trang 11
4 0 2 4
0
2 4
0
2 4
2 2
2
2
)2cos1(2
1sin
cos
cossincos
cossinsin
1
cossin1
t
tdt t t
tdt t t
tdt t x
sin4
12
12cos2
2
1)1(21
.2
)1(1
2
3 0
3 3
0 2 3
0
2 1
dx x
4 2 1
0
2 2 0
1 2 1
0
2 2
1
0
2 3
15
2)
()
1()
()1(1
1
t
t x
2 0 2 2
0
2 0 2 2
0
2 1
t tdt
t tdt
t dx
sin4
12
12cos2
12
0
2 0
2 0
2 0
2
t
t x
Vậy
dt t tdt
t tdt
t tdt
t t dx
x x
I 4 4sin 2cos 2cos 4 4sin cos 4 sin 2 2 (1 cos4)
2 0
2 0
2 2
2 0 2 2
0 2 2
0
2 2
x
.
Trang 122 4
2
2
2 4
2
2cot
sin2sin22cos2
sincos1
cos1
t dt
t x
t dt
t t
t
4 2
.sin2cot
dt t t
4 2
2 4
2
)
cos1(
2cos2
2
cos2sin
t dt
t t t
t
2
214
sin 4 2
0 2
1
0 2
1 0 2 1
0
2
1
21
.21
.1
1
x
dx x x
xdx x x
xdx x x
x dx x
2
1
211
21
112
x
dx I
x
dx dx
x dx
x x
21
22
x
dx I
x
dx I
4 0
2 4
0 2 4
0
4 0
2 4
0
2
2 4
2 1
)(sinsin
1
)(sincos
coscos
cos1coscos
1
cos1
tan
)tan1(1
t d t
t d t
tdt t
dt
t t dt
t
t dt
t
dt t x
dx
)(sin)sin1)(
sin1(
sin12
1)(sin)sin1)(
sin1(
sin12
1)(sin)
sin1)(
sin
1
(
)sin1()sin
4
0
t d t t
t t
d t t
t t
d t t
t t
4 0
4 0
4 0
4
sin1ln2
1sin1
)(sin2
11sin
)(sin2
1sin
1
)(sin2
1sin
11ln2
1223ln2
110sin
0sin1ln2
114
sin
4sin
212
12
1
t
t x
x
2 0
2 2
0 4 2
0 6 2
0
3 2 1
sinsinsin
)cos1()
tdt t tdt
x dx
x I
2 0
34
14
cos12
12cos214
12
cos2
12cos
dt t t
dt t t
3 1
Trang 13t
t x
3 2 0 2 2
15124
)32(
)32(5
124
)32
dt t
t
tdt x
x x
dx x x
x x
14
1)1(tan4
)tan1(22
142
0
3 0
3 0
2
2 3
tdt dx dx e tdt e
t e
ln
0
t
t x
0 2 1
0
1 0 2
2
ln
2.1
t
dt t
dt dt
t
dt t dx e
Lại đặt t tanu dt ( 1 tan 2u)du
0
u
u t
t
.
Suy ra:
4)
tan1(
)tan1(1
4 0
4 0
2
2 1
2313
0
t
t x
2 1
0
7 8 8
1
0
8 15
12
1.3
13
13
5 36
1 ) (
36
1
3 5 2
1 ) ln 2 ( 3 2 2
3 ) ln 2 ( ) (ln ) ln 2 ( ) (ln ln
2
1 2 3 1
2 3 1
2 1 1
e e
x x
x d x x
d x
21
2
11
2 1
2 2
Trang 14Đặt 3 9 2 1 6 9 9 2 1 1 (6 2) 2(3 11)
2 2
2 2 2
x t
x x x tx t x
x x
1 1
0
t
t x
x
Vậy
2
126ln3
113
2 2 1
1sin
28
.cossin
217
.cos2
cos7
.cos
dx x x
dx x x
dx x
0
t
t x
2
1
t
dt t
0
u
u t
t
Suy ra
262
12
1cos
2
cos2
2cos
4
cos22
12
2
0
6 0
6 0
2sin
22cos
2tan
dx x
4 3cos1
dx x
3 4 6
2
cossin
dx x
x
4 0 4cos
x
x x
6 sin sin 6
x x
dx
3 4sin
3cos
dx x
x
3 4
2
2 cossin
4cos
4sin
xdx
Trang 15q.
2 0
2cos.4cos
xdx x
r. I 2cos2x(sin x cos x)dx
0
4 4
3sin
5
4 cossin
xdx x
2 0
3
2 cossin
xdx x
xdx I
44
sin4
12
1)4cos1(2
12
0
2 0
sin
3 4
3
4 3
d x
dx x
x xdx
I
3
4 3 4
3
4 3
2 4
3
2 4
dx x dx
x xdx
I
cos1
)(coscos
1
)(cossin
sinsin
2
2 2
3
2 2
3 2 2
x d dx
x
x dx
x I
1sin
)(sin2
11sin
)(sin2
11sin
)(sinsin
1
)(sincos
coscos
3
4 3
4 3 2 4
3
2 4
3 2 4
x d x
x d x
x d dx x
x dx
x I
dx x
3
t
t x
5)
1(cos
.cos
1tancos
.cos
1.cos
1
3 1
3 5 2
2 3
4
2 2
2 3
4
2 2
)tan1(cos
.coscos
4 0
2 4
0
2 2
4 0
2 2 4
dx x x
x
dx x
dx I
Câu i:
3
0
3cos
x dx
Trang 16x xdx du x
2 3
0
3 0
sintancos
tancos
x
x x
dx I
3 0
3 0
3 0 3 3
2 3
0
3
2
cos3
2coscos
32cos
cos132cos
x
dx x
dx dx
x
x x
xdx
23
23ln2
13
21sin
1sinln2
13
2sin1
cos3
2cos
cos3
0
3 0
2 3
3 2 3 ln 2
1 3 2 2 2 3 2 3 ln 2
1 3
0 2 4
1cos
22
cos1
dx x
x dx
x
x dx
dx du
4 0 4
0
4 0
4 0
sintan
tantan
x x
x xdx x
x dx x x
4
18cos
lntancos
)(cos
0 4
x x
x d x
2 6
3 2 6
2)
sin(cos
cos
2
4coscos
dx x
x x
dx dx
x x I
.2
3ln23
3ln2tan
1ln2tan
1
)tan1
(
3 6
x d
3 6
3
6
cossin
3sin26sincos6cossinsin6
sinsin
x x
dx x
x
dx I
3
2lncot
3ln2cot
3
)cot3(2cot
3
sin2cossinsin
3
6 3
6
3 6
2 3
x d
x x dx
x x x
2 3
4
2 3
4
3 3
4
sin
3)sin1(4cossin
)3cos4(cossin
cos3cos4sin
x x
dx x
x x
dx x
x x
3 4
2 3
)sin1(4)(sinsin
3)sin
d x
x x
d x
x
3
32cot
tancos
sincos
sin
)cos(sin
cos
3 4 2 3
4 2 3
4
2 2
2 2
x
dx x
dx x
x
dx x x
x x dx
Trang 17“Note”:
cossin
coscos
sin
sincos
sin
)cos(sin
cossin
2 2
2 2
x
xdx x
x
dx x x
x x
dx
2 0
2 2
0
2 2
0
4
1)
2cos1(4
1cos
dx x x
dx x xdx
2
0 16
34
sin32
12sin4
18
34
cos2
12cos22
34
1
x x
x dx x
2 0
2 2
0
4
12cos22
34
1)
2cos1(4
1sin
dx x x
dx x xdx
I
16
3 4
sin 32
1 2 sin 4
1 8
2 0
2 0
cos2
1)2cos1(4cos2
1cos
.4cos
xdx dx
x x
xdx x
2 0
2 0
2 0
2
0
2cos4
16cos4
14
cos2
12
cos6
cos2
12
14
xdx dx
x x
xdx
0 2
sin 4
1 6 sin 24
4 4
Câu r: I 2cos2x(sin x cos x)dx
0
4 4
34
4cos112sin2
11cossin2)sin(cos
cos
x x
x x
x x
2 0
2
0
4 4
4
2cos4cos4
2cos34
4cos4
32cos)
cos(sin
2cos
dx x dx
x dx
x x
dx x dx
x x
2 0
2 0
2 0
2 0
2
0
6cos8
12
cos8
7)2cos6
(cos8
12
cos4
34
2cos4cos4
sin 48
1 2
2 2
0
2 2
x x
.
2 0
2 2 4
2 0
4 4 2
x xdx
x x xdx
x I
8 6
4 2
0
4 2
4 2
0
2 2
4 (1 sin ) (sin ) sin (1 2sin sin ) (sin ) (sin 2sin sin ) (sin )sin
x d x x
x x
d x x
x x
d x x
315
8 9
sin 7
x
.
Trang 182 0
4 2
2 2
2 0
2 2 2
x d x x
xdx x
x xdx
x
I
15
2 5
sin 3
0
5 3
3cos1
sin4
dx x
x
3 6
4 cossin
x x
dx
2 3
2
)cos1(cos
cos
2 2
x x
dx x x
x x
4 0
2)cos2(sin
x x
dx
3 4
3tan
sin4cos52 0
x x
N
Hướng dẫn giải Câu a:
2
0
3cos1
sin4
dx x
0
t
t x
x
22
4)
1(41
)1)(
1(41
)1(4cos
1
sin.sin
1
2 0
1
0 1
0 1
2 2
6
3
t
t x
t t t
t
dt x
x
xdx x
x
xdx x
2
4 4 2
3
2
3 6
2 4
3 6
2 4 3
6
)1()1()
sin1(sin
coscos
sin
coscos
2
2 3
2
2 2
3
2
2 3
2
2 2
2 3
2
4 2
)1()1()
1(
)1)(
1()
1()
t
t t
dt dt
t t
t t
dt t t
t dt
t
t
t
Trang 19)23(3ln2
1327
261
1ln2
113
1)
1()
1(
2 3
2
2 3
2
1 2
2 3
2
1 4
2 3
x x
0
t
t x
2 2
1 0
2 1
0 2 1
0 2 4
0
2 2
)2()1(2)1(2121
21
tan2
dt t
dt t
t
dt t
t
dt x
dx C
2 2 2 2 ln 2 2
1 2
1 2 1 ln 2
Câu d:
2 3
2
)cos1(cos
12
1
22
3
t
t x
x
Vậy
3
2 2 4
2 2
2 1
3 3
2 2 2
2 2
2 1
3 3
2 2
2 2
2 2
2 1
3 3
2 2 2
2 2
2 2
3
2
)1(4
)1(
121
2
)1(
121
)1()1(
)1(
12
1
111
2.11
)cos
1
(
cos
t t
dt t t
t t
dt t t
t
t t
dt t t
t t t
dt t t
22
11
3
12
11
12
11
2
14
)1(.)
1 3 3
2 4 1
3 3 4
2 1
3
3
4
2 2 2
12
1
22
0
t
t x
0
2 2
2 2 2
2 3
1
2 1
) 1 (
2 3
sin cos
dt t
t t
dt x
x
dx I
2 2 1
0
24
t
dt
Đặt t 1 2 tanu dt 2 (tan 2u 1 )du
t
4 0
1
với tan 21.
4)
1(tan4
)1(tan42)1(
2
4 4
4
2
2 1
Trang 202 2
0
2 2
0
3 2
sin)sin441(1
cos
sin)sin43(1
cos
sin4sin31
x
xdx x
dx x
x x
2 2
0
2
1cos
sin)cos41(1
cos
sin)sin1
0
t
t x
)41(1
cos
sin)cos
4
1
1 2
0 1
0 1
2 2
cos2
x x
2)(
2 2
2
2 2
2
cossin
4
cos)
(sin4
x x
dt t
t t dt
t
t t
2 2 2
2
2 2
2
2 2
2
2 2
2
)(sinsin
4
)(sinsin
4
cossin
4
cossin
4
cos2
x d dx
x
x dx
x
x x
dx x
x x
3 ln 2
1 2
sin
2 sin
dx x x
x x
Ta có
5cos3sin45cos3sin4
sin3cos45
cos3sin
4
6cos7sin
C x
x
x x
B A x
x
x x
C x x
B x
x A x
C A x B A x B A x
5
7 4
3
1 3
4
C B C
A
B A
B A
2
15
cos3sin4
sin3cos415
cos3sin
4
6cos7sin
dx x
x x
x
x x
dx x x
x x
cos3sin4
15
cos3sin
4
sin3cos
x x
d x
dx x x
dx x x
x x
2 0 2
15
cos3sin4ln25cos
x x
dx x
Trang 211
22
0
t
t x
2 1
0 2 1
0 2 1
0
2
2 2
2 2
25
1
1.31
2.4
1
25
cos3sin
4
1
t
dt t
t
dt t
t dt t
t t
dt dx
x x
6
12
1)
2(
)2
0
1 0
9ln25cos3sin4
15
cos3sin4ln2
2 0
x x
x x
dx
4
;0(
1)
2(tan
)2(tan)
2(tancos)
cos2(sin
4 0
4 0
2 4
0
2 2
x d x
x
dx x
2 2
3 4
2 2 3
xdx x
xdx I
2 3
4
2 3
4
2 3
x d x xdx
xdx x
3 4
3 4 3
4
3 3
4
3
4
2 3
4
3
tan3
tan)
1(tan3
x dx
dx x
4 0
4 0
4
.coscos
cossin
1cos
sin1
dx x
x
x x
dx
x x
dx x
x x
dx x
Ta có:
4cos
sin
.cossin
cossin
.sincos
4 0
4 0
)cos(sin
cossin
sin
0 4
0
4 0
x x
x x
d dx x x
x x J
2
182
ln422ln4)(
)
Trang 22Câu l:
4 0
4 0
2 4
0
2 4
x x dx
x x
xdx I
.2
2ln2
1cos
ln2
tancos
)(cos)
(tan.tancos
sin)
0
4 0
4 0
x d x d x dx
x
x x
d
x
3 6
3 6
2 3
6
2 3
x x
dx x
x xdx
3 6
3 6
3 6
3 6
(cotcot
sin
cos)
(cotcot
cot)
1(cot
x
x x
xd xdx
dx x x
2 ln 1 sin
ln 2
)cos(sin
sin4cos52 0
x x
x x
x
2 0
3 1
)cos(sin
sin
x x
x
2 0
3 2
)cos(sin
0
t
t x
t t
t t
d t
t
t x
d x x
3 0
2
3 2
0
3 2
2
cos2
sin
2cos
2
cos2
sin
2cos)
cos(sin
3 2
0
3
sincos
sinsin
cos
sin
N x d x x
x t
d t t
2 2
0
3 2
0
3 1
2
1
4cos
2)cos(sin
)cos(sin
sin)
cos(sin
cos2
x
dx x
d x x
x x
d x x
x N
N
N
14
tan21
4cos
5)
cos(sin
sin4)
cos(sin
cos5)
cos(sin
sin4cos
5
1 2 2
0
3 2
0
3 2