NGUY
ˆ
E
˜
N THUY
’
THANH
B
`
AI T
ˆ
A
.
P
TO
´
AN CAO C
ˆ
A
´
P
Tˆa
.
p3
Ph´ep t´ınh t´ıch phˆan. L´y thuyˆe
´
t chuˆo
˜
i.
Phu
.

c
10 T´ıch phˆan bˆa
´
tdi
.
nh 4
10.1 C´ac phu
.
o
.
ng ph´ap t´ınh t´ıch phˆan . . . . . . . . . . . . 4
10.1.1 Nguyˆen h`am v`a t´ıch phˆan bˆa
´
td
i
.
nh ....... 4
10.1.2 Phu
.
o
.
ng ph´ap d
ˆo
’
ibiˆe
´
n .............. 12
10.1.3 Phu
.
o

.
n gia
’
n ..... 37
10.2.3 T´ıch phˆan c´ac h`am lu
.
o
.
.
ng gi´ac . . . . . . . . . . 48
11 T´ıch phˆan x´ac d
i
.
nh Riemann 57
11.1 H`am kha
’
t´ıch Riemann v`a t´ıch phˆan x´ac d
i
.
nh . . . . . 58
11.1.1 D
-
i
.
nhngh˜ıa .................... 58
11.1.2 D
-
iˆe
`
ukiˆe

´
´u
.
ng du
.
ng cu
’
a t´ıch phˆan x´ac d
i
.
nh........ 78
11.3.1 Diˆe
.
n t´ıch h`ınh ph˘a
’
ng v`a thˆe
’
t´ıch vˆa
.
tthˆe
’
.... 78
11.3.2 T´ınh d
ˆo
.
d`ai cung v`a diˆe
.
n t´ıch m˘a
.
t tr`on xoay . . 89

.
p...................... 118
12.1.1 Tru
.
`o
.
ng ho
.
.
pmiˆe
`
nch˜u
.
nhˆa
.
t ...........118
12.1.2 Tru
.
`o
.
ng ho
.
.
pmiˆe
`
ncong.............. 118
12.1.3 Mˆo
.
t v`ai ´u
.

12.2.3 .......................... 136
12.2.4 Nhˆa
.
nx´etchung.................. 136
12.3 T´ıch phˆan d
u
.
`o
.
ng..................... 144
12.3.1 C´ac d
i
.
nh ngh˜ıa co
.
ba
’
n .............. 144
12.3.2 T´ınh t´ıch phˆan d
u
.
`o
.
ng .............. 146
12.4 T´ıch phˆan m˘a
.
t ...................... 158
12.4.1 C´ac d
i
.

.
ng...................... 178
13.1.1 C´ac d
i
.
nh ngh˜ıa co
.
ba
’
n .............. 178
13.1.2 Chuˆo
˜
isˆo
´
du
.
o
.
ng.................. 179
13.2 Chuˆo
˜
ihˆo
.
itu
.
tuyˆe
.
td
ˆo
´

.
a ...................... 199
13.3.1 C´ac d
i
.
nh ngh˜ıa co
.
ba
’
n .............. 199
13.3.2 D
-
iˆe
`
ukiˆe
.
n khai triˆe
’
nv`aphu
.
o
.
ng ph´ap khai triˆe
’
n 201
13.4 Chuˆo
˜
iFourier ....................... 211
13.4.1 C´ac d
i

i Fourier . . . 212
14 Phu
.
o
.
ng tr`ınh vi phˆan 224
14.1 Phu
.
o
.
ng tr`ınh vi phˆan cˆa
´
p1 ............... 225
14.1.1 Phu
.
o
.
ng tr`ınh t´ach biˆe
´
n.............. 226
14.1.2 Phu
.
o
.
ng tr`ınh d
˘a
’
ng cˆa
´
p ............. 231

o
.
ng tr`ınh vi phˆan cˆa
´
pcao.............. 259
14.2.1 C´ac phu
.
o
.
ng tr`ınh cho ph´ep ha
.
thˆa
´
pcˆa
´
p .... 260
14.2.2 Phu
.
o
.
ng tr`ınh vi phˆan tuyˆe
´
n t´ınh cˆa
´
p2v´o
.
ihˆe
.
sˆo
´

ng . . . . . . 273
14.3 Hˆe
.
phu
.
o
.
ng tr`ınh vi phˆan tuyˆe
´
n t´ınh cˆa
´
p1v´o
.
ihˆe
.
sˆo
´
h˘a
`
ng290
15 Kh´ai niˆe
.
mvˆe
`
phu
.
o
.
ng tr`ınh vi phˆan d
a

´
p2do
.
n gia
’
n nhˆa
´
t 310
15.3 C´ac phu
.
o
.
ng tr`ınh vˆa
.
tl´y to´an co
.
ba
’
n .......... 313
15.3.1 Phu
.
o
.
ng tr`ınh truyˆe
`
n s´ong . . . . . . . . . . . . 314
15.3.2 Phu
.
o
.

ng ph´ap t´ınh t´ıch phˆan . . . . . . 4
10.1.1 Nguyˆen h`am v`a t´ıch phˆan bˆa
´
td
i
.
nh..... 4
10.1.2 Phu
.
o
.
ng ph´ap d
ˆo
’
ibiˆe
´
n............ 12
10.1.3 Phu
.
o
.
ng ph´ap t´ıch phˆan t`u
.
ng phˆa
`
n..... 21
10.2 C´ac l´o
.
p h`am kha
’

ng gi´ac . . . . . . . 48
10.1 C´ac phu
.
o
.
ng ph´ap t´ınh t´ıch phˆan
10.1.1 Nguyˆen h`am v`a t´ıch phˆan bˆa
´
tdi
.
nh
D
-
i
.
nh ngh˜ıa 10.1.1. H`am F (x)du
.
o
.
.
cgo
.
i l`a nguyˆen h`am cu
’
a h`am
f(x) trˆen khoa
’
ng n`ao d
´onˆe
´

`
su
.
.
tˆo
`
nta
.
i nguyˆen h`am) Mo
.
i h`am liˆen tu
.
ctrˆen
d
oa
.
n [a, b] dˆe
`
u c´o nguyˆen h`am trˆen khoa
’
ng (a, b).
D
-
i
.
nh l´y 10.1.2. C´ac nguyˆen h`am bˆa
´
tk`ycu
’
a c`ung mˆo

.
c˜ung l`a h`am so
.
cˆa
´
p. Ch˘a
’
ng ha
.
n, nguyˆen h`am cu
’
a c´ac h`am e
−x
2
,
cos(x
2
), sin(x
2
),
1
lnx
,
cos x
x
,
sin x
x
,... l`a nh˜u
.

´
td
i
.
nh cu
’
a h`am f(x) trˆen khoa
’
ng
(a, b)v`ad
u
.
o
.
.
ck´yhiˆe
.
ul`a

f(x)dx.
Nˆe
´
u F (x) l`a mˆo
.
t trong c´ac nguyˆen h`am cu
’
a h`am f(x) trˆen khoa
’
ng
(a, b) th`ı theo d

.
p.
C´ac t´ınh chˆa
´
tco
.
ba
’
ncu
’
a t´ıch phˆan bˆa
´
td
i
.
nh:
1) d


f(x)dx

= f(x)dx.
2)


f(x)dx


= f(x).
3)

.
cgo
.
i l`a t´ıch phˆan ba
’
ng) sau d
ˆay:
6Chu
.
o
.
ng 10. T´ıch phˆan bˆa
´
td
i
.
nh
I.

0.dx = C.
II.

1dx = x + C.
III.

x
α
dx =
x
α+1

dx
cos
2
x
=tgx + C, x =
π
2
+ nπ, n ∈ Z.
IX.

dx
sin
2
x
= −cotgx + C, x = nπ, n ∈ Z.
X.

dx
√
1 − x
2
=



arc sin x + C,
−arc cos x + C
−1 <x<1.
XI.


utr`u
.
th`ı x<−1 ho˘a
.
c x>1).
XIII.

dx
1 − x
2
=
1
2
ln



1+x
1 − x



+ C, |x|=1.
C´ac quy t˘a
´
c t´ınh t´ıch phˆan bˆa
´
td
i
.

´
AC V
´
IDU
.
V´ı d u
.
1. Ch´u
.
ng minh r˘a
`
ng h`am y = signx c´o nguyˆen h`am trˆen
khoa
’
ng bˆa
´
tk`y khˆong ch´u
.
ad
iˆe
’
m x = 0 v`a khˆong c´o nguyˆen h`am trˆen
mo
.
i khoa
’
ng ch´u
.
ad
iˆe

´omo
.
i nguyˆen h`am cu
’
a n´o trˆen (a, b) c´o da
.
ng
F (x)=x + C, C ∈ R.
2) Ta x´et khoa
’
ng (a, b)m`aa<0 <b. Trˆen khoa
’
ng (a, 0) mo
.
i
nguyˆen h`am cu
’
a signx c´o da
.
ng F (x)=−x+C
1
c`on trˆen khoa
’
ng (0,b)
nguyˆen h`am c´o da
.
ng F (x)=x + C
2
.V´o
.

1
= C
2
th`ı thu du
.
o
.
.
c h`am liˆen tu
.
c y = |x| + C
nhu
.
ng khˆong kha
’
vi ta
.
id
iˆe
’
m x =0. T`u
.
d
´o, theo di
.
nh ngh˜ıa 1 h`am
signx khˆong c´o nguyˆen h`am trˆen (a, b), a<0 <b. 
V´ı d u
.
2. T`ım nguyˆen h`am cu

trong miˆe
`
n x<0mˆo
.
t trong c´ac nguyˆen h`am l`a −e
−x
+ C v´o
.
ih˘a
`
ng
sˆo
´
C bˆa
´
tk`y.
Theo d
i
.
nh ngh˜ıa, nguyˆen h`am cu
’
a h`am e
|x|
pha
’
i liˆen tu
.
cnˆenn´o
8Chu
.

.
vˆa
.
y
F (x)=









e
x
nˆe
´
u x>0,
1nˆe
´
u x =0,
−e
−x
+2 nˆe
´
u x<0
l`a h`am liˆen tu
.
c trˆen to`an tru

.
i x<0th`ıF

(x)=e
−x
= e
|x|
. Ta c`on cˆa
`
n pha
’
i
ch´u
.
ng minh r˘a
`
ng F

(0) = e
0
= 1. Ta c´o
F

+
(0) = lim
x→0+0
F (x)− F (0)
x
= lim
x→0+0


(0) = 1 = e
|x|
.T`u
.
d
´o c ´o t h ˆe
’
viˆe
´
t:

e
|x|
dx = F (x)+C =



e
x
+ C, x < 0
−e
−x
+2+C, x < 0. 
V´ı d u
.
3. T`ım nguyˆen h`am c´o d
ˆo
`
thi

’
a f l`a h`am F (x)=ln|x| + C,
C ∈ R.H˘a
`
ng sˆo
´
C d
u
.
o
.
.
cx´acd
i
.
nh t`u
.
d
iˆe
`
ukiˆe
.
n F (−2) = 2, t´u
.
cl`a
ln2 + C =2⇒ C =2− ln2. Nhu
.
vˆa
.
y

dx.
Gia
’
i. 1) Ta c´o
I =


2
2
x
10
x
−
5
x
5 · 10
x

dx =


2

1
5

x
−
1
5

ln

1
5

−
1
5

1
2

x
ln
1
2
+ C
= −
2
5
x
ln5
+
1
5 · 2
x
ln2
+ C.
2)
I =

dx
=
2
3
x +
5
9
ln



x +
2
3



+ C. 
V´ı d u
.
5. T´ınh c´ac t´ıch phˆan sau d
ˆay:
1)

tg
2
xdx, 2)

1 + cos
2


dx
cos
2
x
−

dx =tgx − x + C.
10 Chu
.
o
.
ng 10. T´ıch phˆan bˆa
´
td
i
.
nh
2)

1 + cos
2
x
1 + cos 2x
dx =

1 + cos
2
x
2 cos

=


(sin x − cos x)
2
dx =

| sin x − cos x|dx
= (sin x + cos x)sign(cos x − sin x)+C. 
B
`
AI T
ˆ
A
.
P
B˘a
`
ng c´ac ph´ep biˆe
´
nd
ˆo
’
idˆo
`
ng nhˆa
´
t, h˜ay du
.
a c´ac t´ıch phˆan d

arctgx)
2.

1+2x
2
x
2
(1 + x
2
)
dx.(D
S. arctgx −
1
x
)
3.

√
x
2
+1+
√
1 − x
2
√
1 − x
4
dx.(DS. arc sin x +ln|x +
√
1+x

+ x
−4
+2
x
3
dx.(DS. ln|x|−
1
4x
4
)
6.

2
3x
− 1
e
x
− 1
dx.(D
S.
e
2x
2
+ e
x
+1)
1
Dˆe
’
cho go

− 1
√
2
x
dx.(DS.
2
ln2

2
3x
2
3
+2
−
x
2

)
8.

dx
x(2 + ln
2
x)
.(D
S.
1
√
2
arctg

x
− 1|)
11.

e
x
dx
1+e
x
.(DS. ln(1 + e
x
))
12.

sin
2
x
2
dx.(D
S.
1
2
x −
sin x
2
)
13.

cotg
2

1
1 + cos x
dx.(D
S. tg
x
2
)
18.

dx
sin x + cos x
.(D
S.
1
√
2
ln



tg

x
2
+
π
8





2 − sin
2
x
dx.(D
S. −ln| cos x +
√
1 + cos
2
x|)
12 Chu
.
o
.
ng 10. T´ıch phˆan bˆa
´
td
i
.
nh
22.

sin x cos x

3 − sin
4
x
dx.(D
S.
1

8
ln(1 + 4x
2
)+
1
3
arctg
3/2
2x)
25.

arc sin x − arc cos x
√
1 − x
2
dx.(DS.
1
2
(arc sin
2
x + arc cos
2
x))
26.

x + arc sin
3
2x
√
1 − 4x

x)
28.

x|x|dx.(D
S.
|x|
3
3
)
29.

(2x − 3)|x − 2|dx.
(D
S. F (x)=





−
2
3
x
3
+
7
2
x
2
− 6x + C, x < 2

3
3
+ C nˆe
´
u |x|  1
x −
x|x|
2
+
1
6
signx + C nˆe
´
u|x| > 1
)
10.1.2 Phu
.
o
.
ng ph´ap d
ˆo
’
ibiˆe
´
n
D
-
i
.
nh l´y. Gia

i
.
nh v`a c´o nguyˆen h`am F (x) trˆen khoa
’
ng X.
Khi d
´o h`am F (ϕ(t)) l`a nguyˆen h`am cu
’
a h`am f(ϕ(t))ϕ

(t) trˆen
khoa
’
ng T .
T`u
.
d
i
.
nh l´y 10.1.1 suy r˘a
`
ng

f(ϕ(t))ϕ

(t)dt = F (ϕ(t)) + C. (10.1)
V`ı
F (ϕ(t)) + C =(F (x)+C)




(t)dt. (10.2)
D
˘a
’
ng th´u
.
c (10.2) d
u
.
o
.
.
cgo
.
i l`a cˆong th´u
.
cd
ˆo
’
ibiˆe
´
n trong t´ıch phˆan
bˆa
´
td
i
.
nh.
Nˆe

Ta nˆeu mˆo
.
t v`ai v´ıdu
.
vˆe
`
ph´ep d
ˆo
’
ibiˆe
´
n.
i) Nˆe
´
ubiˆe
’
uth´u
.
cdu
.
´o
.
idˆa
´
u t´ıch phˆan c´o ch´u
.
a c˘an
√
a
2

.
´o
.
idˆa
´
u t´ıch phˆan c´o ch´u
.
a c˘an
√
x
2
− a
2
, a>0
th`ı d`ung ph´ep d
ˆo
’
ibiˆe
´
n x =
a
cos t
,0<t<
π
2
ho˘a
.
c x = acht.
iii) Nˆe
´


ho˘a
.
c x = asht.
iv) Nˆe
´
u h`am du
.
´o
.
idˆa
´
u t´ıch phˆan l`a f(x)=R(e
x
,e
2x
,....e
nx
)th`ı
c´o thˆe
’
d
˘a
.
t t = e
x
(o
.
’
d

i. Ta c´o

dx
cos x
=

cos xdx
1 − sin
2
x
(d
˘a
.
t t = sin x, dt = cos xdx)
=

dt
1 − t
2
=
1
2
ln



1+t
1 − t



i. ta c´o
I =

1
4
d(x
4
)
x
8
− 2
=

√
2
4
d

x
4
√
2

−2

1 −

x
4
√

√
2 − x
4



+ C. 
V´ı d u
.
3. T´ınh I =

x
2
dx

(x
2
+ a
2
)
3
·
Gia
’
i. D
˘a
.
t x(t)=atgt ⇒ dx =
adt
cos

cos tdt
=ln



tg

t
2
+
π
4




− sin t + C.
V`ı t = arctg
x
a
nˆen
I =ln



tg

1
2
arctg

10.1. C´ac phu
.
o
.
ng ph´ap t´ınh t´ıch phˆan 15
Thˆa
.
tvˆa
.
y, v`ı sin α = cos α · tgα nˆen dˆe
˜
d`ang thˆa
´
yr˘a
`
ng
sin

arctg
x
a

=
x
√
x
2
+ a
2
·

a
+
π
2

sin

arctg
x
a
+
π
2

=
1 + sin

arctg
x
a

− cos

arctg
x
a

=
x +
√

t x = asht. Khi d´o
I =


a
2
(1 + sh
2
t)achtdt = a
2

ch
2
tdt
= a
2

ch2t +1
2
dt =
a
2
2

1
2
sh2t + t

+ C
=

x +
√
a
2
+ x
2
a



v`a do d
´o

√
a
2
+ x
2
dx =
x
2
√
a
2
+ x
2
+
a
2
2

−x
2
+6x − 8
dx.
16 Chu
.
o
.
ng 10. T´ıch phˆan bˆa
´
td
i
.
nh
Gia
’
i. 1) Ta c´o
I
1
=

1+
1
x
2

x
2
− 7+
1




x −
1
x
+

x
2
− 7+
1
x
2



+ C.
2) Ta viˆe
´
tbiˆe
’
uth´u
.
cdu
.
´o
.
idˆa
´

2
=

f(x)dx
= −
3
2

(−x
2
+6x − 8)
−
1
2
d(−x
2
+6x − 8) + 13

d(x − 3)

1 − (x − 3)
2
= −3
√
−x
2
+6x − 8 + 13 arc sin(x − 3) + C. 
V´ı d u
.
6. T´ınh

d(cos x)
cos
2
x − 1
=
1
2
ln
1 − cos x
1 + cos x
+ C.
C´ach II.

dx
sin x
=

d

x
2

sin
x
2
cos
x
2
=





+ C.
10.1. C´ac phu
.
o
.
ng ph´ap t´ınh t´ıch phˆan 17
2) Ta c´o
I
2
=

sin x cos x[(cos
2
x +1)− 1]
1 + cos
2
x
dx.
Ta d
˘a
.
t t = 1 + cos
2
x.T`u
.
d
´o dt = −2 cos x sin xdx.Dod´o

, 2) I
2
=

e
x
+1
e
x
− 1
dx.
Gia
’
i
1) D
˘a
.
t e
x
= t.Tac´oe
x
dx = dt v`a
I
1
=

dt
√
t
2

u
.
o
.
.
c
I
2
=

t +1
t− 1
dt
t
=

2dt
t − 1
−

dt
t
= 2ln|t − 1|−ln|t| + C
= 2ln|e
x
− 1|−lne
x
+ c
=ln(e
x

x
+1)
3
)
Chı
’
dˆa
˜
n. D
˘a
.
t e
x
+1=t
4
.
18 Chu
.
o
.
ng 10. T´ıch phˆan bˆa
´
td
i
.
nh
2.

dx
√

+ln|e
x
− 1|)
4.

√
1+lnx
x
dx.(D
S.
2
3

(1 + lnx)
3
)
5.

√
1+lnx
xlnx
dx.
(D
S. 2
√
1+lnx − ln|lnx| + 2ln|
√
1+lnx − 1|)
6.


√
e
3x
+ e
2x
dx.(DS.
2
3
(e
x
+1)
3/2
)
9.

e
2x
2
+2x−1
(2x +1)dx.(DS.
1
2
e
2x
2
+2x−1
)
10.

dx

12.

2
x
dx
√
1 − 4
x
.(DS.
arc sin 2
x
ln2
)
13.

dx
1+
√
x +1
.(D
S. 2[
√
x +1− ln(1 +
√
x + 1)])
Chı
’
dˆa
˜
n. D

√
ax + b − mln|
√
ax + b + m|

)
10.1. C´ac phu
.
o
.
ng ph´ap t´ınh t´ıch phˆan 19
16.

dx
3
√
x(
3
√
x − 1)
.(D
S. 3
3
√
x + 3ln|
3
√
x − 1|)
17.


)
3/2
.(DS.
1
a
2
sin

arctg
x
a

)
Chı
’
dˆa
˜
n. D
˘a
.
t x = atgt, t ∈

−
π
2
,
π
2

.

.
20.

√
a
2
− x
2
dx.(DS.
a
2
2
arc sin
x
a
+
x
√
a
2
− x
2
2
)
Chı
’
dˆa
˜
n. D
˘a

dˆa
˜
n. D
˘a
.
t x = asht.
22.

x
2
√
a
2
+ x
2
dx.(DS.
1
2

x
√
a
2
+ x
2
− a
2
ln(x +
√
a

n. D
˘a
.
t x =
1
t
ho˘a
.
c x = atgt, ho˘a
.
c x = asht.
24.

x
2
dx
√
a
2
− x
2
.(DS.
a
2
2
arc sin
x
a
−
x

)
20 Chu
.
o
.
ng 10. T´ıch phˆan bˆa
´
td
i
.
nh
Chı
’
dˆa
˜
n. D˘a
.
t x =
1
t
, ho˘a
.
c x =
a
cos t
ho˘a
.
c x = acht.
26.


)
28.

dx
x
2
√
x
2
− 9
.(D
S.
√
x
2
− 9
9x
)
29.

dx

(x
2
− a
2
)
3
.(DS. −
x

2
8
x
√
x
2
− a
2
+
a
4
8
arc sin
x
a
)
31.


a + x
a − x
dx.(D
S. −
√
a
2
− x
2
+ arc sin
x

−
√
x
2
− a
2
+2aln(
√
−x + a +
√
−x − a)nˆe
´
u x<−a)
Chı
’
dˆa
˜
n. D
˘a
.
t x =
a
cos 2t
.
33.


x − 1
x +1
dx

√
x)
10.1. C´ac phu
.
o
.
ng ph´ap t´ınh t´ıch phˆan 21
Chı
’
dˆa
˜
n. D˘a
.
t x = sin
2
t.
35.

√
x
2
+1
x
dx.(D
S.
√
x
2
+1− ln


√
2 − x
2
)
37.


(9 − x
2
)
2
x
6
dx.(DS. −

(9 − x
2
)
5
45x
5
)
38.

x
2
dx
√
x
2



xe
x
1+xe
x



)
Chı
’
dˆa
˜
n. Nhˆan tu
.
’
sˆo
´
v`a mˆa
˜
usˆo
´
v´o
.
i e
x
rˆo
`
id˘a

’
dˆa
˜
n. D
˘a
.
t x = atgt.
10.1.3 Phu
.
o
.
ng ph´ap t´ıch phˆan t`u
.
ng phˆa
`
n
Phu
.
o
.
ng ph´ap t´ıch phˆan t`u
.
ng phˆa
`
ndu
.
.
a trˆen d
i
.

.
c (10.4) d
u
.
o
.
.
cgo
.
i l`a cˆong th´u
.
c t´ınh t´ıch phˆan t`u
.
ng phˆa
`
n.
V`ı u

(x)dx = du v`a v

(x)dx = dv nˆen (10.4) c´o thˆe
’
viˆe
´
tdu
.
´o
.
ida
.

`
n c´o thˆe
’
phˆan th`anh ba nh´om sau d
ˆa y .
22 Chu
.
o
.
ng 10. T´ıch phˆan bˆa
´
td
i
.
nh
Nh´om I gˆo
`
mnh˜u
.
ng t´ıch phˆan m`a h`am du
.
´o
.
idˆa
´
u t´ıch phˆan c´o ch´u
.
a
th`u
.

ra c`on dv l`a phˆa
`
n c`on la
.
icu
’
a
biˆe
’
uth´u
.
cdu
.
´o
.
idˆa
´
u t´ıch phˆan.
Nh´om II gˆo
`
mnh˜u
.
ng t´ıch phˆan m`a biˆe
’
uth´u
.
cdu
.
´o
.

’
abiˆe
’
uth´u
.
cdu
.
´o
.
idˆa
´
u t´ıch phˆan. Sau mˆo
˜
i
lˆa
`
n t´ıch phˆan t`u
.
ng phˆa
`
nbˆa
.
ccu
’
ad
ath´u
.
c s˜e gia
’
mmˆo

`
n ta la
.
ithud
u
.
o
.
.
c t´ıch phˆan ban d
ˆa
`
uv´o
.
ihˆe
.
sˆo
´
n`ao d
´o. D´ol`a
phu
.
o
.
ng tr`ınh tuyˆe
´
n t´ınh v´o
.
iˆa
’

Nhˆa
.
nx´et. Nh`o
.
c´ac phu
.
o
.
ng ph´ap d
ˆo
’
ibiˆe
´
n v`a t´ıch phˆan t`u
.
ng phˆa
`
n
ta ch´u
.
ng minh d
u
.
o
.
.
c c´ac cˆong th´u
.
cthu
.

=
1
2a
ln



a + x
a − x



+ C, a =0.
3)

dx
√
a
2
− x
2
= arc sin
x
a
+ C, a =0.
4)

dx
√
x

’
i. T´ıch phˆan d
˜a cho thuˆo
.
c nh´om I. Ta d˘a
.
t
u(x) = arctg
√
x,
dv =
√
xdx.
Khi d
´o du =
1
1+x
·
dx
2
√
x
, v =
2
3
x
3
2
.Dod´o
I =


dx
=
2
3
x
3
2
arctg
√
x −
1
3
(x − ln|1+x|)+C. 
V´ı d u
.
2. T´ınh I =

arc cos
2
xdx.
Gia
’
i. Gia
’
su
.
’
u = arc cos
2

ng th´u
.
cthud
u
.
o
.
.
ctad
˘a
.
t u =
arc cos x, dv =
xdx
√
1 − x
2
. Khi d´o
du = −
dx
√
1 − x
2
,v= −

d(
√
1 − x
2
)=−

1 − x
2
arc cos x − x + C
2
.
24 Chu
.
o
.
ng 10. T´ıch phˆan bˆa
´
td
i
.
nh
Cuˆo
´
ic`ung ta thu du
.
o
.
.
c
I = xarc cos
2
x − 2
√
1 − x
2
arc cos x − 2x + C. 

2
3

x cos 3xdx = −
1
3
x
2
cos 3x +
2
3
I
1
.
Ta cˆa
`
n t´ınh I
1
.D˘a
.
t u = x, dv = cos 3xdx. Khi d´o du =1dx,
v =
1
3
sin 3x.T`u
.
d
´o
I = −
1

´
ud
˘a
.
t u = sin 3x, dv = x
2
dx th`ı lˆa
`
n t´ıch phˆan t`u
.
ng
phˆa
`
nth´u
.
nhˆa
´
t khˆong d
u
.
ad
ˆe
´
n t´ıch phˆan do
.
n gia
’
nho
.
n.

a
b

e
ax
sin bxdx =
1
b
e
ax
sin bx −
a
b
I
1
.
D
ˆe
’
t´ınh I
1
ta d˘a
.
t u = e
ax
, dv = sin bxdx. Khi d´o du = ae
ax
dx,
v = −
1