Bài 5. Các phép i bin s c bn và nâng cao tích phân hàm lng giác
169
BÀI 5. CÁC PHÉP I BIN S C BN VÀ NÂNG CAO
TÍCH PHÂN HÀM LNG GIÁC
I. CÁC DNG TÍCH PHÂN VÀ PHÉP BIN I C BN
• t vn :
Xét tích phân dng
( )
I R sin x,cos x dx
=

t
2
2 2 2
2 2 1
2
21 1 1
x dt t t
t tg x arctg t ;dx ; sin x ; cos x
t t t
−
== = = =
+ + +
Khi ó:
( )
2
2
2 1
1 1 1
dt
t t
I R sin x,cos x dx R ,
t t t
−
= =
+ + +
Ta xét 3 trng hp c bit thng gp sau ây mà có th i bin s bng
cách khác hàm s di du tích phân nhn c ơn gin hơn.

(
)
 o
eo
:
(
)
(
)
− −
 o  o
thì cn bin i hàm s và vi phân  thc hin phép i bin t
=
cosx.

(
)
 o
eoo
(
)
(
)
− −
 o  o
thì cn bin i hàm s và vi phân  thc hin phép i bin t
=
sinx.
!
(
)
 o
o"#$%&'
(
)
(
)
− −
 o  o
thì cn bin i hàm s và vi phân  thc hin phép i bin t
=
tgx.
II. CÁC BÀI TP MU MINH HA
()
− −
3sin2x 2cos2x 1
I = dx
3cos2x + 4sin2x + 5
t
2
2 2 2
dt 2t 1 t
t tg x x arctg t ; dx ; sin 2x ; cos 2x
1 t 1 t 1 t
−
== = = =
+ + +
(
)
(
)
( ) ( )
(
)
( )
( )
2 2 2 2
2 2 2 2
2 2 2
3.2t 2 1 t 1 t dt 1 t 6t 3 dt 1 t 6t 3 dt
I2 2
1 t t 4t 4 1 t
3 1 t 4.2t 5 1 t t 2 1 t
− − − + + − + −
= ⋅ = ⋅ =
+ + + +
− + + + + +
Chng II: Nguyên hàm và tích phân
−
−−
−
Trn Phng
170
Gi s
( )
( )
( )
2
2 2 2
2
6 3
21
2 1 2
t t A B Ct D
, t
tt
t t t
+ − +
= + + ∀
++
+ + +
⇔
( )
(
)
(
)
( )( )
2
2 2 2
t 6t 3 A t 2 1 t B 1 t Ct D t 2 , t
+ − = + + + + + + + ∀
(*)
( ) ( ) ( )
( )
2 3 2
t 6t 3 A C t 2A B 4C D t A 4C 4D t 2A B 4D
⇔ + − = + + + + + + + + + + +
Thay t
=
−
2 vào (*) thì
−
11
=
5B
B
=
−
11/5
(*)
A C 0 A C 0 A 34 25
2A B 4C D 1 2A 4C D 16 5 B 11 5
A 4C 4D 6 A 4C 4D 6 C 34 25
2A B 4D 3 2A 4D 4 5 D 12 25
+ = + = = −
+ + + = + + = = −
⇔ ⇔ ⇔
+ + = + + = =
+ + = − + = − =
( )
( )
( )
2
2 2 2
2
1 t 6t 3 34 dt 11 dt 1 24t 12
I dt dt
2 25 t 2 5 25 1 t
t 2 1 t t 2
+ − +
= = − − +
++
+ + +
( )
(
)
( )
( )
( )
( )
2
222
2
2
34 dt 11 dt 12 d t 12 dt
25 t 2 5 25 25
1 t 1 t
t 2
34 11 12 12
ln t 2 ln 1 t arctg t c
25 5 t 2 25 25
34 11 12 12
ln tg x 2 ln 1 tg x x c
25 5 tg x 2 25 25
= − − + +
++ +
+
= − + + + + + +
+
= − + + + + + +
+
()
(
)
(
)
− −
 o  o
•
3 2
2
2
=
− − + −
13 2
sin2xdx
J = cos x sin x 1
sin x cos xdx
cos x cos x
( ) ( ) ( )
3 2
2sin x cos x
R sin x, cos x R sin x, cos x R sin x, cos x
cos x cos x 2
=− = −
+ −
t
t
=
cos x
( )
( )
13 2 2
2
2t dt 2t dt A Bt C
J 2 dt
t 1
t t 2 t 2t 2
t 1 t 2t 2
− − +
= = = − +
−
+ − + +
− + +
Ta có:
( )
( ) ( )
( ) ( )
2
2
2
t A Bt C
t A t 2t 2 Bt C t 1
t 1 t 2t 2
t 1 t 2t 2
+
= + ⇔ = + + + + −
−+ +
− + +
( )
( ) ( )
2
A B 0 A 1 5
t A B t 2A B C t 2A C 2A B C 1 B 1 5
2A C 0 C 2 5
+ = =
⇔ = + + − + + − ⇔ − + = ⇔ = −
− = =
Bài 5. Các phép i bin s c bn và nâng cao tích phân hàm lng giác
171
( )
( )
( )
( ) ( )
12 2
2
2 2
2
2
2 1 t 2 2 dt 1 2t 2 6
J dt dt
5 t 1 5 t 1 5
t 2t 2 t 2t 2
2 dt 1 d t 2t 2 6 dt
5 t 1 5 5
t 2t 2 t 1 1
2 1 6
ln t 1 ln t 2t 2 arctg t 1 c
5 5 5
2 1 6
ln 1 cos x ln cos x 2 cos x 2 arctg 1 cos x c
5 5 5
− + −
= − − = − +
− −
+ + + +
+ +
= − + −
−+ + + +
= − − + + + − + +
= − − + + + − + +
•
(
)
( ) ( )
2 6 2 6 6 2
1 1
sin x dx d cos x dt
sin x cos x cos x cos x t t
−
= = =
− −
26
dx
J = sinxcos x
(
)
( )
6 6 4 2
2 6 3 5
6 2
3 5
t t 1 1 t t 1 t 1 1 1 1
dt dt ln c
t 1 t
t 1 t 3t 5t
t t 1
1 cos x 1 1 1
ln c
1 cos x cos x 3 cos x 5cos x
− − + + −
= = − = + + + +
+
−
−
−
= + + + +
+
•
2
2
2 2 4
22 1
sin x cos x sin x cos x
dx dx
cos x cos x
= = −
3
sinx + sin3x
J = dx
cos2x
( )
2 2
2 2 2 2
4 cos xd cos x 4t dt 2 dt
2 dt 2 dt
1
1 2 cos x 1 2t 1 2t t
2
1 1 2t 1 1 2 cos x
ln 2t c ln 2 cos x c
2 1 2t 2 1 2 cos x
= = = − = −
− − −
−
+ +
= − + = − +
− −
•
(
)
( )
2 2
2 2
0 0
4 4 1
1 1
sin x cos x
sin x dx d cos x
cos x cos x
π π
−
= −
+ +
23
4
0
4sin x
J = dx =
1 + cosx
(
)
( )
( )
0 1
21
2
0
1 0
4 1 t
dt 4 1 t dt 4t 2t 4 2 2
1 t
−
= − = − = − = − =
+
•
2 2 2
2
3 2 2
6 6 6
3 4 3 4 4 1
sin x dx sin x dx sin x dx
sin x sin x sin x cos x
π π π
π π π
= = =
− − −
22
5
6
sin x
J = dx
sin3x
( ) ( )
( )
( )
3 2
6 3 2 3 2
2 2 2
0
2 0 0
d cosx dt 1 d 2t 1 2t 1 1
ln ln 2 3
2 4 2t 1 4
4cos x 1 4t 1 2t 1
π
π
−
= = = = = −
+
− − −
Chng II: Nguyên hàm và tích phân
−
−−
−
Trn Phng
172
()
(
)
(
)
− −
 o  o
•
( )
( )
( )
4 4
8 2 2
20 20 20
1 1cos x sin x t
cos x dx d sin x dt
sin x sin x t
− −
= = =
9
120
cos x
K = dx
sin x
( ) ( ) ( ) ( ) ( )
2 4 6 8
20 19 17 15 13 11
19 17 15 13 11
1 4t 6t 4t t 1 4 6 4 1
dt c
t 19t 17t 15t 13t 11t
1 4 6 4 1
c
19 sin x 17 sin x 15 sin x 13 sin x 11 sin x
− + − + −
= = + − + − +
−
= + − + − +
•
(
)
(
)
( )
2 4 2 4
2 4 2 4
cos x cos x cos x cos x
cos x dx d sin x
sin x sin x sin x sin x
+ +
= =
+ +
3 5
22 4
cos x + cos x
K = dx
sin x + sin x
( )
( )
( )
2
2 2 4 2
2 4 2 2
2 2
1 t 1 t t 3t 2 2 6
dt dt 1 dt
t t t 1 t
t 1 t
2 2
t 6 arctg t c sin x 6 arctg sin x c
t sin x
− + − − +
= = = + −
+ +
+
= − − + = − − +
!()!
(
)
(
)
− −
 o  o
•
( )
( )
( )
6 6
6
0
2
0 0
3 3
1
1 3
1
−
= = = − =
−− −
6
1
0
Ldx
=cosx sinx cosx
d tg x
dx ln tg x ln
tg x
cos x tg x
π π
π
•
( )
( )
3 3 3
3
3 8 2 3
4 4
4
4 4 4
d tg x
dx dx
tg x cos x cos x . tg x
tg x
π π π
π π π
= = =
3
24 3 5
4
dx
L =
sin xcos x
( ) ( ) ( )
( ) ( )
33
311 4 8
4
44
4
tg x d tg x 4 tg x 4 3 1 4 3 1
ππ
−
π
π
= = = − = −
•
( ) ( )
442
4
3 3
02 3
cos x
sin x
dx
cos x
cos x sin x cos x
π
=+
42
33 3
0
sin xdx
L =
cosx 2sin x + 3cos x
( )
(
)
( )
( )
4 4 4 3
2 2
3 2 3 3
0 0 0
4
3
0
d 3 2 tg x
tg x tg x 1
dx d tg x 6
3 2 tg x cos x 3 2 tg x 3 2 tg x
1 1 1 5
ln 3 2 tg x ln 5 ln 3 ln
6 6 6 3
π π π
π
+
= ⋅ = =
+ + +
= + = − =
Bài 5. Các phép i bin s c bn và nâng cao tích phân hàm lng giác
173
II. BIN I VÀ I BIN NÂNG CAO TÍCH PHÂN HÀM S LNG GIÁC
1. DNG 1: MU S LÀ BIU
THC THUN NHT CA SIN
( )
n
dx
sinx
•
()()()
(
)
(
)
()
2
2
3 3 6 3
1
1
2 2
4
2 8
2 2 2 2 2
x x
tg d tg
dx dx
x x x x x
sin cos tg cos tg
+
= = =
13
dx
A = sin x
()()()()
2 4
2
3 2
x x
1 2 tg tg
1 1 1 x 1
2 2 x x
d tg 2ln tg tg c
2 2
4 4 2 2
x x
tg 2 tg
2 2
+ +
−
= = + + +
Cách 2:
(
)
( )
(
)
( ) ( )
[ ]
1
3 4 2 2
2
d sin d d cos d cos
sin sin 1 cos 1 cos
1 cos
x x x x x
Ax x
x x
x
= = = − = −
+ −
−
( ) ( )
( )( ) ( ) ( )
22
1 1 cos x 1 cos x 1 1 1
d cos x d cos x
4 1 cos x 1 cos x 4 1 cos x 1 cos x
− + + −
= = +
+ − − +
( ) ( ) ( )
2 2 2 2
1 1 1 2 cos x 1 1 cos x
d cos x ln c
4 2 1 cos x
1 cos x 2sin x
1 cos x 1 cos x
− − +
= + + = − +
−
−
− +
•
()()()
5 5 10
dx dx
=
x x x x
2 sin cos 32 tg cos
2 2 2 2
=
25
dx
A = sin x
(
)
(
)
()()()
()()()()
4
22 4 6 8
5 5
2 4
4 2
x x x x x x
1 tg d tg 1 4 tg 6 tg 4 tg tg
1 1
2 2
2 2 2 2 x
d tg
2
16 16
x x
tg tg
2 2
1 1 2 x 1
x x
6 ln tg 2 tg tg c
2 2
16 2 4
x x
4 tg tg
2 2
+++++
= =
−
= − + + + +
Cách 2:
25 6
dx sin x dx
A
sin x sin x
= =
(
)
( )
(
)
( ) ( )
3 3
2
d cos x d cos x
1 cos x 1 cos x
1 cos x
= − = −
+ −
−
( ) ( )
( )( ) ( ) ( )
33
1 1 cos x 1 cos x 1 1 1
d cos x d cos x
8 1 cos x 1 cos x 8 1 cos x 1 cos x
− + + −
= = +
+ − − +
( ) ( )
( )
( )
1
2 2 2 4
2
1 1 1 3 d cos x cos x 3
A
8 2 4
4sin x
2 1 cos x 2 1 cos x 1 cos x
− −
= − + = −
− + −