chuỗi và phương trình vi phân - nguyễn thủy thanh

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
.

o
.
ng tr`ınh vi phˆan
NH
`
AXU
ˆ
A
´
TBA
’
NDA
.
IHO
.
CQU
ˆ
O
´
C GIA H
`
AN
ˆ
O
.
I
Mu
.
clu
.

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

.
ng ph´ap t´ıch phˆan t`u
.
ng phˆa
`
n 21
10.2 C´ac l´o
.
p h`am kha
’
t´ıch trong l´o
.
p c´ac h`am so
.
cˆa
´
p 30
10.2.1 T´ıch phˆan c´ac h`am h˜u
.
uty
’
30
10.2.2 T´ıch phˆan mˆo
.
tsˆo
´
h`am vˆo ty
’
d
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

.
nd
ˆe
’
h`am kha
’
t´ıch 59
11.1.3 C´ac t´ınh chˆa
´
tco
.
ba
’
ncu
’
a t´ıch phˆan x´ac d
i
.
nh . . 59
11.2 Phu
.
o
.
ng ph´ap t´ınh t´ıch phˆan x´ac d
i
.
nh 61
11.3 Mˆo
.
tsˆo

´
´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

11.4 T´ıch phˆan suy rˆo
.
ng 98
11.4.1 T´ıch phˆan suy rˆo
.
ng cˆa
.
n vˆo ha
.
n 98
11.4.2 T´ıch phˆan suy rˆo
.
ng cu
’
a h`am khˆong bi
.
ch˘a
.
n . . 107
2MU
.
CLU
.
C
12 T´ıch phˆan h`am nhiˆe
`
ubiˆe
´
n 117
12.1 T´ıch phˆan 2-l´o

.
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
.

ng du
.
ng trong h`ınh ho
.
c 121
12.2 T´ıch phˆan 3-l´o
.
p 133
12.2.1 Tru
.
`o
.
ng ho
.
.
pmiˆe
`
n h`ınh hˆo
.
p 133
12.2.2 Tru
.
`o
.
ng ho
.
.
pmiˆe
`
ncong 134

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
.

nh ngh˜ıa co
.
ba
’
n 158
12.4.2 Phu
.
o
.
ng ph´ap t´ınh t´ıch phˆan m˘a
.
t 160
12.4.3 Cˆong th´u
.
c Gauss-Ostrogradski . . . . . . . . . 162
12.4.4 Cˆong th´u
.
cStokes 162
13 L´y thuyˆe
´
t chuˆo
˜
i 177
13.1 Chuˆo
˜
isˆo
´
du
.
o

.
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
´

iv`ahˆo
.
itu
.
khˆong tuyˆe
.
tdˆo
´
i . . . 191
13.2.1 C´ac d
i
.
nh ngh˜ıa co
.
ba
’
n 191
13.2.2 Chuˆo
˜
id
an dˆa
´
u v`a dˆa
´
uhiˆe
.
u Leibnitz . . . . . . 192
13.3 Chuˆo
˜
il˜uy th`u

.
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

.
nh ngh˜ıa co
.
ba
’
n 211
MU
.
CLU
.
C3
13.4.2 Dˆa
´
uhiˆe
.
udu
’
vˆe
`
su
.
.
hˆo
.
itu
.
cu
’
a chuˆo
˜

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

14.1.3 Phu
.
o
.
ng tr`ınh tuyˆe
´
nt´ınh 237
14.1.4 Phu
.
o
.
ng tr`ınh Bernoulli . . . . . . . . . . . . . . 244
14.1.5 Phu
.
o
.
ng tr`ınh vi phˆan to`an phˆa
`
n 247
14.1.6 Phu
.
o
.
ng tr`ınh Lagrange v`a phu
.
o
.
ng tr`ınh Clairaut255
14.2 Phu
.

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
´

h˘a
`
ng 264
14.2.3 Phu
.
o
.
ng tr`ınh vi phˆan tuyˆe
´
n t´ınh thuˆa
`
n nhˆa
´
t
cˆa
´
p n
n
n (ptvptn cˆa
´
p n
n
n)v´o
.
ihˆe
.
sˆo
´
h˘a
`

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

.
o h`am riˆeng 304
15.1 Phu
.
o
.
ng tr`ınh vi phˆan cˆa
´
p 1 tuyˆe
´
n t´ınh d
ˆo
´
iv´o
.
i c´ac d
a
.
o
h`amriˆeng 306
15.2 Gia
’
iphu
.
o
.
ng tr`ınh d
a
.
o h`am riˆeng cˆa

´
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 tr`ınh truyˆe
`
n nhiˆe
.
t 317
15.3.3 Phu
.
o
.
ng tr`ınh Laplace . . . . . . . . . . . . . . 320
T`ai liˆe
.
u tham kha
’
o 327
Chu
.
o
.
ng 10
T´ıch phˆan bˆa
´
td
i
.
nh
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 dˆo
’
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
’
t´ıch trong l´o

.
p c´ac h`am
so
.
cˆa
´
p 30
10.2.1 T´ıch phˆan c´ac h`am h˜u
.
uty
’
30
10.2.2 T´ıch phˆan mˆo
.
tsˆo
´
h`am vˆo ty
’
d
o
.
n gia
’
n 37
10.2.3 T´ıch phˆan c´ac h`am lu
.
o
.
.
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
´
u F (x)liˆen tu

.
c trˆen khoa
’
ng d´o v`a kha
’
vi
10.1. C´ac phu
.
o
.
ng ph´ap t´ınh t´ıch phˆan 5
ta
.
imˆo
˜
idiˆe
’
m trong cu
’
a khoa
’
ng v`a F

(x)=f(x).
D
-
i
.
nh l ´y 10.1.1. (vˆe
`

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
.

t h`am l`a chı
’
kh´ac nhau bo
.
’
imˆo
.
th˘a
`
ng sˆo
´
cˆo
.
ng.
Kh´ac v´o
.
id
a
.
o h`am, nguyˆen h`am cu
’
a h`am so
.
cˆa
´
p khˆong pha
’
i bao
gi`o
.

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
.
ng h`am khˆong so

.
cˆa
´
p.
D
-
i
.
nh ngh˜ıa 10.1.2. Tˆa
.
pho
.
.
pmo
.
i nguyˆen h`am cu
’
a h`am f(x) trˆen
khoa
’
ng (a, b)d
u
.
o
.
.
cgo
.
i l`a t´ıch phˆan bˆa
´

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
i

.
nh l´y 10.1.2

f(x)dx = F (x)+C, C ∈ R
trong d
´o C l`a h˘a
`
ng sˆo
´
t`uy ´y v`a d˘a
’
ng th´u
.
ccˆa
`
nhiˆe
’
ul`ad
˘a
’
ng th ´u
.
cgi˜u
.
a
hai tˆa
.
pho
.
.

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)


df (x)=

f

(x)dx = f(x)+C.
T`u
.
d
i
.
nh ngh˜ıa t´ıch phˆan bˆa
´
tdi
.
nh r ´ut ra ba
’
ng c´ac t´ıch phˆan co
.
ba
’
n (thu
.
`o
.
ng d
u
.
o
.
.

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
α +1

+ C, α = −1
IV.

dx
x
=ln|x|+ C, x =0.
V.

a
x
dx =
a
x
lna
+ C (0 <a= 1);

e
x
dx = e
x
+ C.
VI.

sin xdx = −cos x + C.
VII.

cos xdx = sinx + C.
VIII.

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.

dx

1+x
2
=



arctgx + C,
−arccotgx + C.
XII.

dx
√
x
2
± 1
=ln|x +
√
x
2
± 1|+ C
(trong tru
.
`o
.
ng ho
.
.
pdˆa
´
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
.
nh:

10.1. C´ac phu
.
o
.
ng ph´ap t´ınh t´ıch phˆan 7
1)

kf(x)dx = k

f(x)dx, k =0.
2)

[f(x) ± g(x)]dx =

f(x)dx ±

g(x)dx.
3) Nˆe
´
u

f(x)dx = F (x)+C v`a u = ϕ(x) kha
’
vi liˆen tu
.
cth`ı

f(u)du = F (u)+C.
C
´

AC V
´
IDU
.
V´ı du
.
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
’

m x =0.
Gia
’
i. 1) Trˆen khoa
’
ng bˆa
´
t k`y khˆong ch´u
.
ad
iˆe
’
m x = 0 h`am y = signx
l`a h˘a
`
ng sˆo
´
. Ch˘a
’
ng ha
.
nv´o
.
imo
.
i khoa
’
ng (a, b), 0 <a<bta c´o signx =1
v`a do d
´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
.
imo

.
i c´ach cho
.
nh˘a
`
ng sˆo
´
C
1
v`a C
2
ta thu du
.
o
.
.
c h`am [trˆen (a, b)] khˆong c´o d
a
.
o h`am ta
.
idiˆe
’
m x =0.
Nˆe
´
u ta cho
.
n C = C
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 du
.
2. T`ım nguyˆen h`am cu
’

a h`am f(x)=e
|x|
trˆen to`an tru
.
csˆo
´
.
Gia
’
i. V´o
.
i x  0 ta c´o e
|x|
= e
x
v`a do d´o trong miˆe
`
n x>0mˆo
.
t
trong c´ac nguyˆen h`am l`a e
x
. Khi x<0 ta c´o e
|x|
= e
−x
v`a do vˆa
.
y
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
.
o

.
ng 10. T´ıch phˆan bˆa
´
td
i
.
nh
pha
’
i tho
’
am˜andiˆe
`
ukiˆe
.
n
lim
x→0+0
e
x
= lim
x→0−0
(−e
−x
+ C)
t´u
.
cl`a1=−1+C ⇒ C =2.
Nhu
.

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
.

csˆo
´
.Tach´u
.
ng minh r˘a
`
ng F(x) l`a nguyˆen
h`am cu
’
a h`am e
|x|
trˆen to`an tru
.
csˆo
´
. Thˆa
.
tvˆa
.
y, v´o
.
i x>0 ta c´o
F

(x)=e
x
= e
|x|
,v´o
.

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
e

x
− 1
x
=1,
F

−
(0) = lim
x→0−0
F (x) −F (0)
x
= lim
x→0−0
−e
−x
+2− 1
x
=1.
Nhu
.
vˆa
.
y F

+
(0) = F

−
(0) = F


(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 du
.
3. T`ım nguyˆen h`am c´o d
ˆo
`
thi
.

qua diˆe
’
m(−2,2) dˆo
´
iv´o
.
i h`am
f(x)=
1
x
, x ∈ (−∞, 0).
Gia
’
i. V`ı (ln|x|)

=
1
x
nˆen ln|x| l`a mˆo
.
t trong c´ac nguyˆen h`am cu
’
a
h`am f(x)=
1
x
. Do vˆa
.
y, nguyˆen h`am cu
’

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
F (x)=ln|x|+2− ln2 = ln




x
2



+2. 
10.1. C´ac phu
.
o
.
ng ph´ap t´ınh t´ıch phˆan 9
V´ı du
.
4. T´ınh c´ac t´ıch phˆan sau dˆay:
1)

2
x+1
−5
x−1
10
x
dx, 2)

2x +3
3x +2
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


1
2

x

dx
=2


1
5

x
dx −
1
5


1
2

x
dx
=2

1
5

x
ln

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

2

x +
3
2

3

x +
2
3

dx =
2
3

x +
2
3

+
5
6


x +
2
3

dx

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

1 + cos 2x
dx, 3)

√
1 −sin 2xdx.
Gia
’
i. 1)

tg
2
xdx =

sin
2
x
cos
2
x
dx =

1 −cos
2
x
cos
2
x
dx
=


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
2

x
dx =
1
2


dx
cos
2
x
+

dx

=
1
2
(tgx + x)+C.
3)

√
1 − sin 2xdx =


sin
2
x − 2 sin x cos x + cos
2
xdx
=



(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
˜acho

vˆe
`
t´ıch phˆan ba
’
ng v`a t´ınh c´ac t´ıch phˆan d
´o
1
1.

dx
x
4
− 1
.(D
S.
1
4
ln



x − 1
x +1



−
1
2
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
2

|)
4.

√
x
2
+1−
√
1 −x
2
√
x
4
− 1
dx.(D
S. ln|x +
√
x
2
− 1|−ln|x +
√
x
2
+1|)
5.

√
x
4
+ 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
.

n, trong c´ac “D´ap sˆo
´
”cu
’
a chu
.
o
.
ng n`ay ch´ung tˆoi bo
’
qua khˆong viˆe
´
t
c´ac h˘a
`
ng sˆo
´
cˆo
.
ng C.
10.1. C´ac phu
.
o
.
ng ph´ap t´ınh t´ıch phˆan 11
7.

2
2x
−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
lnx

√
2
)
9.

3
√
ln
2
x
x
dx.(D
S.
3
5
ln
5/3
x)
10.

e
x
+ e
2x
1 −e
x
dx.(DS. −e
x
− 2ln|e
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
xdx.(DS. −x − cotgx)

14.

√
1 + sin 2xdx, x ∈

0,
π
2

.(D
S. −cos x + sin x)
15.

e
cosx
sin xdx.(DS. −e
cos x
)
16.

e
x
cos e
x
dx.(DS. sin e
x
)
17.

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





)
19.

1 + cos x
(x + sin x)
3
dx.(DS. −
2
2(x + sin x)
2
)
20.

sin 2x

1 − 4 sin
2
x
dx.(D
S. −
1
2

1 −4 sin
2
x)
21.

sin x


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
2

arc sin

sin
2
x
√
3

)
23.

arccotg3x
1+9x
2
dx.(DS. −
1
6
arccotg
2
3x)
24.

x +
√
arctg2x
1+4x
2
dx.(DS.
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
2

dx.(DS. −
1
4
√
1 −4x
2
+
1
8
arc sin
4
2x)
27.

x + arc cos
3/2
x
√
1 −x
2
dx.(DS. −
√
1 −x
2
−
2
5
arc cos
5/2
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
2

3
x
3
−
7
2
x
2
+6x + C, x  2
)
30.

f(x)dx, f(x)=



1 − x
2
, |x|  1,
1 −|x|, |x| > 1.
(D
S. F (x)=





x −
x
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
’

su
.
’
:
10.1. C´ac phu
.
o
.
ng ph´ap t´ınh t´ıch phˆan 13
1) H`am x = ϕ(t) x´ac di
.
nh v`a kha
’
vi trˆen khoa
’
ng T v´o
.
itˆa
.
pho
.
.
p gi´a
tri
.
l`a khoa
’
ng X.
2) H`am y = f(x) x´ac d
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)


x=ϕ(t)

=

f(x)dx


x=ϕ(t)
cho nˆen d˘a
’
ng th ´u
.
c (10.1) c´o thˆe
’
viˆe
´
tdu
.
´o
.
ida
.
ng

f(x)dx


x=ϕ(t)
=

f(ϕ(t))ϕ


(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
´

u h`am x = ϕ(t) c´o h`am ngu
.
o
.
.
c t = ϕ
−1
(x)th`ıt`u
.
(10.2) thu
d
u
.
o
.
.
c

f(x)dx =

f(ϕ(t))ϕ

(t)dt


t=ϕ
−1
(x)
. (10.3)
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
− x

2
, a>0
th`ı su
.
’
du
.
ng ph´ep d
ˆo
’
ibiˆe
´
n x = a sin t, t ∈

−
π
2
,
π
2

.
ii) Nˆe
´
ubiˆe
’
uth´u
.
cdu
.

´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
´
u h`am du

.
´o
.
idˆa
´
u t´ıch phˆan ch´u
.
a c˘an th´u
.
c
√
a
2
+ x
2
, a>0
th`ı c´o thˆe
’
d
˘a
.
t x = atgt, t ∈

−
π
2
,
π
2


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
ˆay R l`a h`am h˜u

.
uty
’
).
14 Chu
.
o
.
ng 10. T´ıch phˆan bˆa
´
td
i
.
nh
C
´
AC V
´
IDU
.
V´ı d u
.
1. T´ınh

dx
cos x
.
Gia
’
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




+ C =ln



tg

x
2
+
π
4




+ C. 
V´ı d u
.
2. T´ınh I =

x
3
dx
x
8
− 2
.
Gia
’
i. ta c´o

I =

1
4
d(x
4
)
x
8
− 2
=

√
2
4
d

x
4
√
2

−2

1 −

x
4
√
2


2

D
˘a
.
t t =
x
4
√
2
ta thu d
u
.
o
.
.
c
I = −
√
2
8
ln



√
2+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
2

t
.Dod
´o
I =

a
3
tg
2
t ·cos
3
tdt
a
3
cos
2
t
=

sin
2
t
cos t
dt =

dt
cos t
−

cos tdt

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

a
+
π
4




− sin

arctg
x
a

+ C
= −
x
√
x
2
+ a
2
+ln|x +
√
x
2
+ a
2
| + C.
10.1. C´ac phu

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

´
p theo ta c´o
sin

1
2
arctg
x
a
+
π
4

cos

1
2
arctg
x
a
+
π
4

=
1 −cos

arctg
x
a

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

2
+ x
2
a
v`a t `u
.
d
´o suy ra diˆe
`
u pha
’
ich´u
.
ng minh. 
V´ı du
.
4. T´ınh I =

√
a
2
+ x
2
dx.
Gia
’
i. D
˘a
.
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
=
a

2
2
(sht ·cht + t)+C.
V`ıcht =

1+sh
2
t =

1+
x
2
a
2
. e
t
=sht +cht =
x +
√
a
2
+ x
2
a
nˆen
t =ln



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
ln|x +

√
a
2
+ x
2
| + C. 
V´ı du
.
5. T´ınh
1) I
1
=

x
2
+1
√
x
6
− 7x
4
+ x
2
dx, 2) I
2
=

3x +4
√
−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

2
dx =

d

x −
1
x



x −
1
x

2
−5
=

dt
√
t
2
− 5
=ln|t +
√
t
2
− 5| + C =ln




x −
1
x
+

x
2
−7+
1
x
2



+ C.
2) Ta viˆe
´
tbiˆe
’
uth´u
.
cdu
.
´o
.
idˆa
´
u t´ıch phˆan du

.
´o
.
ida
.
ng
f(x)=−
3
2
·
−2x +6
√
−x
2
+6x −8
+13·
1
√
−x
2
+6x −8
v`a thu d
u
.
o
.
.
c
I
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
1)


dx
sin x
, 2) I
2
=

sin x cos
3
x
1 + cos
2
x
dx.
Gia
’
i
1) C´ach I.Tac´o

dx
sin x
=

sin x
sin
2
x
dx =

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
=

d


x
2

tg
x
2
· cos
2
x
2
=

d

tg
x
2

tg
x
2
=ln



tg
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
I

2
= −
1
2

t −1
t
dt = −
t
2
+ln|t|+ C,
trong d
´o t = 1 + cos
2
x. 
V´ı du
.
7. T´ınh
1) I
1
=

e
x
dx
√
e
2x
+5
, 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
+5

=ln|t +
√
t
2
+5| + C =ln|e
x
+
√
e
2x
+5|+ C.
2) Tu
.
o
.
ng tu
.
.
,d
˘a
.
t e
x
= t, e
x
dx = dt, dx =
dt
t
v`a thu d
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
− 1)

2
− x + C. 
B
`
AI T
ˆ
A
.
P
T´ınh c´ac t´ıch phˆan:
1.

e
2x
4
√
e
x
+1
dx.(D
S.
4
21
(3e
x
− 4)
4

(e
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
√
e

x
+1
.(D
S. ln



√
1+e
x
−1
√
1+e
x
+1



)
3.

e
2x
e
x
− 1
dx.(D
S. e
x
+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.

dx

e
x/2
+ e
x
.(DS. −x − 2e
−
x
2
+ 2ln(1 + e
x
2
))
7.

arctg
√
x
√
x
dx
1+x
.(D
S. (arctg
√
x)
2
)
8.

√

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
√

e
x
−1
.(D
S. 2arctg
√
e
x
− 1)
11.

e
2x
dx
√
e
4x
+1
.(D
S.
1
2
ln(e
2x
+
√
e
4x
+1))
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
˘a

.
t x +1=t
2
.
14.

x +1
x
√
x − 2
dx.(D
S. 2
√
x −2+
√
2arctg

x − 2
2
)
15.

dx
√
ax + b + m
.(D
S.
2
a
√

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.

dx

(1 −x
2
)
3/2
.(DS. tg(arc sin x))
Chı
’
dˆa
˜
n. D
˘a
.
t x = sin t, t ∈

−
π
2
,
π
2

)
18.

dx
(x
2
+ a
2
)

3/2
.(DS.
1
a
2
sin

arctg
x
a

)
Chı
’
dˆa
˜
n. D
˘a
.
t x = atgt, t ∈

−
π
2
,
π
2

.
19.


dx
(x
2
− 1)
3/2
.(DS. −
1
cos t
, t = arc sin
1
x
)
Chı
’
dˆa
˜
n. D
˘a
.
t x =
1
sin t
, −
π
2
<t<0, 0 <t<
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
.

t x = a sin t.
21.

√
a
2
+ x
2
dx.(DS.
x
2
√
a
2
+ x
2
+
a
2
2
ln|x +
√
a
2
+ x
2
|)
Chı
’
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
2

+ x
2
)

)
23.

dx
x
2
√
x
2
+ a
2
.(DS. −
√
x
2
+ a
2
a
2
x
)
Chı
’
dˆa
˜
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
a

√
a
2
−x
2
)
Chı
’
dˆa
˜
n. D
˘a
.
t x = a sin t.
25.

dx
x
√
x
2
− a
2
.(DS. −
1
a
arc sin
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.

√

1 −x
2
x
2
dx.(DS. −
√
1 −x
2
x
−arc sin x)
27.

dx

(a
2
+ x
2
)
3
.(DS.
x
a
2
√
x
2
+ a
2
)

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

2
√
x
2
− a
2
)
30.

x
2
√
a
2
− x
2
dx.
(D
S. −
x
4
(a
2
− x
2
)
3/2
+
a
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
a

)
Chı
’
dˆa
˜
n. D
˘a
.
t x = a cos 2t.
32.


x −a
x + a
dx.
(D
S.
√
x
2
− a
2
− 2aln(
√
x −a +
√
x + a)nˆe
´
u x>a,
−

√
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

2
.(DS. arc cos
1
x
−
√
x
2
−1
x
)
Chı
’
dˆa
˜
n. D
˘a
.
t x =
1
t
.
34.

dx
√
x −x
2
.(DS. 2arc sin
√

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




1+
√
x
2
+1
x



)
36.

x
3
dx
√
2 −x
2
.(DS. −
x
2
3
√
2 − x
2
−
4
3
√

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

2
.(DS.
x
2
√
x
2
−a
2
+
a
2
2
ln|x +
√
x
2
− a
2
|)
39.

(x +1)dx
x(1 + xe
x
)
.(D
S. ln



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

t xe
x
= t.
40.

dx
(x
2
+ a
2
)
2
.(DS.
1
2a
3

arctg
x
a
+
ax
x
2
+ a
2

)
Chı
’

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
.
nh l´y sau dˆay.

D
-
i
.
nh l´y. Gia
’
su
.
’
trˆen khoa
’
ng D c´ac h`am u(x) v`a v(x) kha
’
vi v`a h`am
v(x)u

(x) c´o nguyˆen h`am. Khi d´o h`am u(x)v

(x) c´o nguyˆen h`am trˆen
D v`a

u(x)v

(x)dx = u(x)v(x) −

v(x)u

(x)dx. (10.4)
Cˆong th´u
.

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
.
ng


udv = uv −

vdu. (10.4*)
Thu
.
.
ctˆe
´
cho thˆa
´
yr˘a
`
ng phˆa
`
nl´o
.
n c´ac t´ıch phˆan t´ınh d
u
.
o
.
.
cb˘a
`
ng
ph´ep t´ıch phˆan t`u
.
ng phˆa
`

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
.
asˆo

´
l`a mˆo
.
t trong c´ac h`am sau d
ˆay: lnx, arc sin x, arc cos x, arctgx,
(arctg x)
2
, (arc cos x)
2
,lnϕ(x), arc sin ϕ(x),
D
ˆe
’
t´ınh c´ac t´ıch phˆan n`ay ta ´ap du
.
ng cˆong th´u
.
c (10.4*) b˘a
`
ng c´ach
d
˘a
.
t u(x)b˘a
`
ng mˆo
.
t trong c´ac h`am d˜achı
’
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
.
idˆa

´
u t´ıch phˆan
c´o da
.
ng P (x)e
ax
, P(x) cos bx, P(x) sin bx trong d´o P (x)l`adath´u
.
c, a,
b l`a h˘a
`
ng sˆo
´
.
D
ˆe
’
t´ınh c´ac t´ıch phˆan n`ay ta ´ap du
.
ng (10.4*) b˘a
`
ng c´ach d˘a
.
t u(x)=
P (x), dv l`a phˆa
`
n c`on la
.
icu
’

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
.

td
o
.
nvi
.
.
Nh´om III gˆo
`
mnh˜u
.
ng t´ıch phˆan m`a h`am du
.
´o
.
idˆa
´
ut´ıch phˆan c´o
da
.
ng: e
ax
sin bx, e
ax
cos bx, sin(lnx), cos(lnx), Sau hai lˆa
`
n t´ıch phˆan
t`u
.
ng phˆa
`

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
’
n l`a t´ıch phˆan cˆa

`
n t´ınh.
D
u
.
o
.
ng nhiˆen l`a ba nh´om v`u
.
a nˆeu khˆong v´et hˆe
´
tmo
.
it´ıch phˆan
t´ınh d
u
.
o
.
.
cb˘a
`
ng t´ıch phˆan t`u
.
ng phˆa
`
n (xem v´ıdu
.
6).
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
.
`o

.
ng hay su
.
’
du
.
ng sau d
ˆay:
1)

dx
x
2
+ a
2
=
1
a
arctg
x
a
+ C, a =0.
2)

dx
a
2
− x
2
=

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
2

± a
2
=ln|x +
√
x
2
± a
2
| + C.
10.1. C´ac phu
.
o
.
ng ph´ap t´ınh t´ıch phˆan 23
C
´
AC V
´
IDU
.
V´ı du
.
1. T´ınh t´ıch phˆan I =

√
xarctg
√
xdx.
Gia
’

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 =
2

3
x
3
2
arctg
√
x −
1
3

x
1+x
dx
=
2
3
x
3
2
arctg
√
x −
1
3


1 −
1
1+x


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

du = −
2arc cos x
√
1 −x
2
dx, v = x.
Theo (10.4*) ta c´o
I = xarc cos
2
x +2

xarc cos x
√
1 −x
2
dx.
D
ˆe
’
t´ınh t´ıch phˆan o
.
’
vˆe
´
pha
’
id
˘a
’
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
+ C
1
v`a ta chı
’
cˆa
`
nlˆa
´
y v = −
√
1 − x
2
:

xarc cos x
√
21 − x
2
dx = −
√
1 − x
2
arc cos x −

dx
= −
√
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. 
V´ı d u

.
3. T´ınh I =

x
2
sin 3xdx.
Gia
’
i. T´ıch phˆan d
˜a cho thuˆo
.
c nh´om II. Ta d˘a
.
t
u(x)=x
2
,
dv = sin 3xdx.
Khi d
´o du =2xdx, v = −
1
3
cos 3x v`a
I = −
1
3
x
2
cos 3x +
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
3

x
2
cos 3x +
2
3

1
3
x sin 3x −
1
3

sin 3xdx

= −
1
3
x
2
cos 3x +
2
9
x sin 3x +
2
27
cos 3x + C. 
Nhˆa
.
n x´et. Nˆe
´

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.
V´ı d u

.
4. T´ınh I =

e
ax
cos bx; a, b =0.
Gia
’
i. D
ˆay l`a t´ıch phˆan thuˆo
.
c nh´om III. Ta d˘a
.
t u = e
ax
, dv =
cos bxdx. Khi d
´o du = ae
ax
dx, v =
1
b
sin bx v`a
I =
1
b
e
ax
sin bx −
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
b

cos bx v`a
I
1
= −
1
b
e
ax
cos bx +
a
b

e
ax
cos bxdx.