˜
’ THANH
ˆ N THUY
NGUYE

` TA
ˆP
BAI
.
´ CAO CA
ˆ´P
TOAN
Tˆa.p 3
Ph´ep t´ınh t´ıch phˆan. L´
y thuyˆe´t chuˆ˜o i.
Phu.o.ng tr`ınh vi phˆan

’ N DAI HOC QUO
` XUA
ˆ´T BA
ˆ´C GIA HA
` NO
ˆ. I
NHA
.
.


Mu.c lu.c
10 T´ıch phˆ
an bˆ

ac di.nh Riemann
11.1 H`am kha’ t´ıch Riemann v`a t´ıch phˆan x´ac d i.nh . . .
- i.nh ngh˜ıa . . . . . . . . . . . . . . . . . .
11.1.1 D
- iˆ
`eu kiˆe.n dˆe’ h`am kha’ t´ıch . . . . . . . . . .
11.1.2 D
11.1.3 C´ac t´ınh chˆa´t co. ba’n cu’a t´ıch phˆan x´ac di.nh
11.2 Phu.o.ng ph´ap t´ınh t´ıch phˆan x´ac d i.nh . . . . . . .
11.3 Mˆo.t sˆo´ u
´.ng du.ng cu’a t´ıch phˆan x´ac d i.nh . . . . . .
11.3.1 Diˆe.n t´ıch h`ınh ph˘a’ng v`a thˆe’ t´ıch vˆa.t thˆe’ . .

30
30
37
48
57

. .

58

. .
. .

58
59

. .

an h`
am nhiˆ
e´n
12.1 T´ıch phˆan 2-l´o.p . . . . . . . . . . . . . .
`en ch˜
u. nhˆa.t . . .
12.1.1 Tru.`o.ng ho..p miˆ
`en cong . . . . . .
12.1.2 Tru.`o.ng ho..p miˆ
12.1.3 Mˆo.t v`ai u
´.ng du.ng trong h`ınh ho.c
12.2 T´ıch phˆan 3-l´o.p . . . . . . . . . . . . . .
`en h`ınh hˆo.p . . .
12.2.1 Tru.`o.ng ho..p miˆ
`en cong . . . . . .
12.2.2 Tru.`o.ng ho..p miˆ
12.2.3
. . . . . . . . . . . . . . . . . .
12.2.4 Nhˆa.n x´et chung . . . . . . . . . .
12.3 T´ıch phˆan d u.`o.ng . . . . . . . . . . . . .
12.3.1 C´ac di.nh ngh˜ıa co. ba’n . . . . . .
12.3.2 T´ınh t´ıch phˆan du.`o.ng . . . . . .
12.4 T´ıch phˆan m˘a.t . . . . . . . . . . . . . .
12.4.1 C´ac di.nh ngh˜ıa co. ba’n . . . . . .
12.4.2 Phu.o.ng ph´ap t´ınh t´ıch phˆan m˘a.t
12.4.3 Cˆong th´
u.c Gauss-Ostrogradski .
12.4.4 Cˆong th´
u.c Stokes . . . . . . . . .


146
158
158
160
162
162

˜i
13 L´
y thuyˆ
e´t chuˆ
o
13.1 Chuˆ˜o i sˆo´ du.o.ng . . . . . . . . . . . . . . . . . . . . . .
13.1.1 C´ac di.nh ngh˜ıa co. ba’n . . . . . . . . . . . . . .
13.1.2 Chuˆo˜ i sˆo´ du.o.ng . . . . . . . . . . . . . . . . . .
13.2 Chuˆ˜o i hˆo.i tu. tuyˆe.t d ˆo´i v`a hˆo.i tu. khˆong tuyˆe.t d ˆo´i . . .
13.2.1 C´ac di.nh ngh˜ıa co. ba’n . . . . . . . . . . . . . .
13.2.2 Chuˆo˜ i dan dˆa´u v`a dˆa´u hiˆe.u Leibnitz . . . . . .
13.3 Chuˆ˜o i l˜
uy th`
u.a . . . . . . . . . . . . . . . . . . . . . .
13.3.1 C´ac di.nh ngh˜ıa co. ba’n . . . . . . . . . . . . . .
- iˆ
`eu kiˆe.n khai triˆe’n v`a phu.o.ng ph´ap khai triˆe’n
13.3.2 D
13.4 Chuˆo˜ i Fourier . . . . . . . . . . . . . . . . . . . . . . .
13.4.1 C´ac di.nh ngh˜ıa co. ba’n . . . . . . . . . . . . . .

177
178

.
.
.
.
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.
.
.
.
.
.
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.
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.

.
.
.
.
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.

.
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.

.
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.

.

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´n t´ınh . . . . . . . . . . . . . 237
14.1.4 Phu.o.ng tr`ınh Bernoulli . . . . . . . . . . . . . . 244
`an . . . . . . . . 247
14.1.5 Phu.o.ng tr`ınh vi phˆan to`an phˆ
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´p cao . . . . . . . . . . . . . . 259
14.2.1 C´ac phu.o.ng tr`ınh cho ph´ep ha. thˆa´p cˆa´p . . . . 260
14.2.2 Phu.o.ng tr`ınh vi phˆan tuyˆe´n t´ınh cˆa´p 2 v´o.i hˆe.
sˆo´ h˘a`ng . . . . . . . . . . . . . . . . . . . . . . 264
`an nhˆa´t
14.2.3 Phu.o.ng tr`ınh vi phˆan tuyˆe´n t´ınh thuˆ
cˆa´p n (ptvptn cˆa´p n ) v´o.i hˆe. sˆo´ h˘a`ng . . . . . . 273
14.3 Hˆe. phu.o.ng tr`ınh vi phˆan tuyˆe´n t´ınh cˆa´p 1 v´o.i hˆe. sˆo´ h˘a`ng290
`e phu.o.ng tr`ınh vi phˆ
15 Kh´
ai niˆ
e.m vˆ
an da.o h`
am riˆ
eng
15.1 Phu.o.ng tr`ınh vi phˆan cˆa´p 1 tuyˆe´n t´ınh dˆo´i v´o.i c´ac da.o
h`am riˆeng . . . . . . . . . . . . . . . . . . . . . . . . .
15.2 Gia’i phu.o.ng tr`ınh d a.o h`am riˆeng cˆa´p 2 d o.n gia’n nhˆa´t
y to´an co. ba’n . . . . . . . . . .
15.3 C´ac phu.o.ng tr`ınh vˆa.t l´
`en s´ong . . . . . . . . . . . .
15.3.1 Phu.o.ng tr`ınh truyˆ
.

10.1.1 Nguyˆen h`
am v`
a t´ıch phˆ
an bˆ
a´t di.nh . . . . .

4

10.1.2 Phu.o.ng ph´
ap dˆ
o’i biˆe´n . . . . . . . . . . . . 12
`an . . . . . 21
10.1.3 Phu.o.ng ph´
ap t´ıch phˆ
an t`
u.ng phˆ
10.2 C´
ac l´
o.p h`
am kha’ t´ıch trong l´
o.p c´
ac h`
am
.
so cˆ
a´p . . . . . . . . . . . . . . . . . . . . . . 30
10.2.1 T´ıch phˆ
an c´
ac h`
am h˜

D
f (x) trˆen khoa’ng n`ao d´o nˆe´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.i mˆo˜ i diˆe’m trong cu’a khoa’ng v`a F (x) = f (x).
- i.nh l´
`on ta.i nguyˆen h`am) Mo.i h`
`e su.. tˆ
am liˆen tu.c trˆen
D
y 10.1.1. (vˆ
`eu c´
doa.n [a, b] dˆ
o nguyˆen h`
am trˆen khoa’ng (a, b).
- i.nh l´
D
y 10.1.2. C´
ac nguyˆen h`
am bˆ
a´t k`y cu’a c`
ung mˆ
o.t h`
am l`
a chı’
.


C∈R

`an hiˆe’u l`a d˘a’ng th´
u.a
trong d´o C l`a h˘a`ng sˆo´ t`
uy y
´ v`a d˘a’ng th´
u.c cˆ
u.c gi˜
hai tˆa.p ho..p.
C´ac t´ınh chˆa´t co. ba’n cu’a t´ıch phˆan bˆa´t di.nh:
1) d

f (x)dx = f (x)dx.

2)

f (x)dx

3)

df (x) =

= f (x).
f (x)dx = f (x) + C.

T`
u. di.nh ngh˜ıa t´ıch phˆan bˆa´t di.nh r´
ut ra ba’ng c´ac t´ıch phˆan co.

axdx =

ax
+ C (0 < a = 1);
lna

VI.

sin xdx = − cos x + C.

VII.

cos xdx = sin x + C.

VIII.
IX.

X.

XI.

ex dx = ex + C.

dx
π
= tgx + C, x = + nπ, n ∈ Z.
2
cos x
2
dx

dx
+ C, |x| = 1.
= ln
2
1−x
2 1−x

C´ac quy t˘´ac t´ınh t´ıch phˆan bˆa´t di.nh:


10.1. C´ac phu.o.ng ph´ap t´ınh t´ıch phˆan

1)

kf (x)dx = k

2)

[f (x) ± g(x)]dx =

3) Nˆe´u

7

f (x)dx, k = 0.
f (x)dx ±

g(x)dx.

f (x)dx = F (x) + C v`a u = ϕ(x) kha’ vi liˆen tu.c th`ı

`en x > 0 mˆo.t
0 ta c´o e|x| = ex v`a do d´o trong miˆ
Gia’i. V´o.i x
trong c´ac nguyˆen h`am l`a ex . Khi x < 0 ta c´o e|x| = e−x v`a do vˆa.y
`en x < 0 mˆo.t trong c´ac nguyˆen h`am l`a −e−x + C v´o.i h˘a`ng
trong miˆ
sˆo´ C bˆa´t k`
y.
Theo di.nh ngh˜ıa, nguyˆen h`am cu’a h`am e|x| pha’i liˆen tu.c nˆen n´o


Chu.o.ng 10. T´ıch phˆan bˆa´t d .inh

8
`eu kiˆe.n
pha’i tho’a m˜an diˆ

lim ex = lim (−e−x + C)

x→0+0

x→0−0

t´
u.c l`a 1 = −1 + C ⇒ C = 2.
Nhu. vˆa.y


ex
nˆe´u x > 0,

x
x
Nhu. vˆa.y F+ (0) = F− (0) = F (0) = 1 = e|x|. T`
u. d´o c´o thˆe’ viˆe´t:

ex + C,
x
10
5 · 10
5
1 x
1 x
1
=2
dx −
dx
5
5
2
1 x
1 x
1 2
=2 5
−
+C
1
1
5
ln
ln
5
2
2
1
=− x
+
+ C.

3
x+
3 x+
3
3
2
5
2
= x + ln x + + C.
3
9
3
2 x+

V´ı du. 5. T´ınh c´ac t´ıch phˆan sau dˆay:
1)

tg2 xdx,

2)

1 + cos2 x
dx,
1 + cos 2x

3)

√
1 − sin 2xdx.


2

dx
+
cos2 x

dx

1
= (tgx + x) + C.
2
3)
√
1 − sin 2xdx =

sin2 x − 2 sin x cos x + cos2 xdx
(sin x − cos x)2dx =

=

| sin x − cos x|dx

= (sin x + cos x)sign(cos x − sin x) + C.

` TA
ˆP
BAI
.
`ong nhˆa´t, h˜ay du.a c´ac t´ıch phˆan d˜a cho
B˘`ang c´ac ph´ep biˆe´n dˆo’i dˆ

√
dx.
(DS. arc sin x + ln|x + 1 + x2|)
1 − x4
√
√
√
√
x2 + 1 − 1 − x2
√
dx. (DS. ln|x + x2 − 1| − ln|x + x2 + 1|)
x4 − 1
√
1
x4 + x−4 + 2
dx.
(DS. ln|x| − 4 )
3
x
4x

2.
3.
4.
5.

23x − 1
dx.
ex − 1


9.

11

3x

22x − 1
√
dx.
2x

2 22
x
+ 2− 2 )
(DS.
ln2 3

dx
.
x(2 + ln2 x)
√
3
ln2 x
dx.
x

1
lnx
(DS. √ arctg √ )
2


(DS. −x − cotgx)

14.

√
π
.
1 + sin 2xdx, x ∈ 0,
2

15.

ecos x sin xdx.

16.

ex cos ex dx.

(DS. sin ex)

17.

1
dx.
1 + cos x

x
(DS. tg )
2

8
2

dx.

1 − 4 sin x
21.

sin x
1
x−
)
2
2

(DS. −

2
)
2(x + sin x)2

(DS. −

1
2

1 − 4 sin2 x)
√

sin x


dx.

1
arccotg3x
dx.
(DS. − arccotg2 3x)
2
1 + 9x
6
√
x + arctg2x
1
1
2
ln(1
+
4x
arctg3/22x)
dx.
(DS.
)
+
1 + 4x2
8
3

25.

arc sin x − arc cos x

(DS.

1
(arc sin2 x + arc cos2 x))
2

(DS. −

1√
1
1 − 4x2 + arc sin4 2x)
4
8

√
2
(DS. − 1 − x2 − arc cos5/2 x)
5

|x|3
)
3


7
2

− x3 + x2 − 6x + C, x < 2
3
2


- i.nh l´
D
y. Gia’ su’.:

nˆe´u |x|

1
)

nˆe´u|x| > 1


10.1. C´ac phu.o.ng ph´ap t´ınh t´ıch phˆan

13

a.p ho..p gi´
a
1) H`
am x = ϕ(t) x´
ac di.nh v`
a kha’ vi trˆen khoa’ng T v´
o.i tˆ
tri. l`
a khoa’ng X.
2) H`
am y = f (x) x´
ac di.nh v`
a c´


x=ϕ(t)

=

f (ϕ(t))ϕ (t)dt.

(10.2)

u.c (10.2) du.o..c go.i l`a cˆong th´
u.c dˆo’i biˆe´n trong t´ıch phˆan
D˘a’ng th´
bˆa´t di.nh.
u. (10.2) thu
Nˆe´u h`am x = ϕ(t) c´o h`am ngu.o..c t = ϕ−1 (x) th`ı t`
du.o..c
f (x)dx =

f (ϕ(t))ϕ (t)dt

t=ϕ−1 (x)

.

(10.3)

`e ph´ep dˆo’i biˆe´n.
Ta nˆeu mˆo.t v`ai v´ı du. vˆ
√
u.a c˘an a2 − x2, a > 0


Chu.o.ng 10. T´ıch phˆan bˆa´t d .inh

14

´ V´I DU
CAC
.
V´ı du. 1. T´ınh

dx
.
cos x

Gia’i. Ta c´o
cos xdx
dx
(d˘a.t t = sin x, dt = cos xdx)
=
cos x
1 − sin2 x
x π
1 1+t
dt
ln
+
C
=
ln
tg

2
x4
−2 1 − √
2

2

x4
D˘a.t t = √ ta thu du.o..c
2
√
√
2
2 + x4
I=−
ln √
+ C.
8
2 − x4
x2 dx
·
(x2 + a2 )3
adt
Gia’i. D˘a.t x(t) = atgt ⇒ dx =
. Do d´o
cos2 t

V´ı du. 3. T´ınh I =

sin2 t

+ ln|x + x2 + a2| + C.
x2 + a2


10.1. C´ac phu.o.ng ph´ap t´ınh t´ıch phˆan

15

˜e d`ang thˆa´y r˘`ang
Thˆa.t vˆa.y, v`ı sin α = cos α · tgα nˆen dˆ
sin arctg

x
x
=√
·
2
a
x + a2

Tiˆe´p theo ta c´o
x π
1
arctg +
2
a 4
x π
1
cos arctg +
2

Gia’i. D˘a.t x = asht. Khi d´o
a2 (1 + sh2 t)achtdt = a2

I=
= a2
=

a2 1
ch2t + 1
dt =
sh2t + t + C
2
2 2

a2
(sht · cht + t) + C.
2
2

x2 t
x+
1 + 2 . e = sht + cht =
a

1 + sh t =
√
x + a2 + x2
t = ln
v`a do d´o
a


3x + 4
√
dx.
−x2 + 6x − 8


Chu.o.ng 10. T´ıch phˆan bˆa´t d .inh

16
Gia’i. 1) Ta c´o
1+
I1 =

1
x2

x2 − 7 +

d x−
1
x2

dx =
x−

1
x

1

= ln|t +

I2 =

f (x)dx

1
3
(−x2 + 6x − 8)− 2 d(−x2 + 6x − 8) + 13
2
√
= −3 −x2 + 6x − 8 + 13 arc sin(x − 3) + C.

=−

d(x − 3)
1 − (x − 3)2

V´ı du. 6. T´ınh
dx
,
sin x

1)

2) I2 =

sin x cos3 x
dx.
1 + cos2 x

d
d
2
2
x =
x
x
x
sin cos
tg · cos2
2
2
2
2
x
d tg
x
2
x = ln tg 2 + C.
tg
2


10.1. C´ac phu.o.ng ph´ap t´ınh t´ıch phˆan

17

2) Ta c´o
sin x cos x[(cos2 x + 1) − 1]
dx.

ex + 1
dx.
ex − 1

Gia’i
1) D˘a.t ex = t. Ta c´o ex dx = dt v`a
√
√
dt
√
= ln|t + t2 + 5| + C = ln |ex + e2x + 5| + C.
t2 + 5

I1 =

dt
v`a thu du.o..c
2) Tu.o.ng tu.., d˘a.t ex = t, exdx = dt, dx =
t
I2 =

t + 1 dt
=
t−1 t

2dt
−
t−1

dt

18

2.

dx
√
.
ex + 1

√
1 + ex − 1
(DS. ln √
)
1 + ex + 1

e2x
dx.
(DS. ex + ln|ex − 1|)
ex − 1
√
2
1 + lnx
4.
dx.
(DS.
(1 + lnx)3)
x
3
√
1 + lnx

e3x + e2xdx.
(DS. (ex + 1)3/2 )
3
ex/2

e2x

2 +2x−1

(2x + 1)dx.

dx
.
ex − 1

10.

√

11.

e2xdx
√
.
e4x + 1

12.

2x dx
√

(DS. 2[ x + 1 − ln(1 + x + 1)])
1+ x+1
˜
Chı’ dˆ
a n. D˘a.t x + 1 = t2.

14.

x+1
√
dx.
x x−2

15.

√

dx
.
ax + b + m

√
√
(DS. 2 x − 2 + 2arctg
(DS.

x−2
)
2


Chı’ dˆ
a n. D˘a.t x = sin t, t ∈
18.

dx
.
+ a2)3/2

−

π π
,
)
2 2

1
x
sin arctg )
2
a
a
π π
˜
.
Chı’ dˆ
a n. D˘a.t x = atgt, t ∈ − ,
2 2
(x2

19.

a2
20.
)
a2 − x2 dx.
(DS. arc sin +
2
a
2
˜
Chı’ dˆ
a n. D˘a.t x = a sin t.
(DS. −

(DS.

√
x√ 2
a2
a + x2 + ln|x + a2 + x2|)
2
2

˜
Chı’ dˆ
a n. D˘a.t x = asht.
22.
23.

x2
√

ho˘a.c x = atgt, ho˘a.c x = asht.
t
x x√ 2
a2
(DS. arc sin −
a − x2 )
2
a a

x2dx
√
.
a2 − x2
˜
Chı’ dˆ
a n. D˘a.t x = a sin t.
dx
√
.
x x2 − a2

a
1
(DS. − arc sin )
a
x

19



(a2

32.

.

dx
(x2

−

a2)3

(DS.

a2

x
√
)
x2 + a2

√
x2 − 9
)
(DS.
9x

dx
√

− (a2 − x2)3/2 + x x2 − a2 + arc sin )
4
8
8
a

√
x
a+x
dx.
(DS. − a2 − x2 + arc sin )
a−x
a
˜
Chı’ dˆ
a n. D˘a.t x = a cos 2t.
x−a
dx.
x+a
√
√
√
(DS.
x2 − a2 − 2aln( x − a + x + a) nˆe´u x > a,
√
√
√
− x2 − a2 + 2aln( −x + a + −x − a) nˆe´u x < −a)
a
˜

)
x


10.1. C´ac phu.o.ng ph´ap t´ınh t´ıch phˆan

21

˜
Chı’ dˆ
a n. D˘a.t x = sin2 t.
√
√
√
1
+
x2 + 1
x2 + 1
dx.
(DS. x2 + 1 − ln
)
35.
x
x
36.

x3dx
√
.
2 − x2

x√ 2
a2
x − a2 + ln|x + x2 − a2|)
2
2

(x + 1)dx
xex
)
.
(DS.
ln
x(1 + xex)
1 + xex
˜
`oi d˘a.t xex = t.
Chı’ dˆ
a n. Nhˆan tu’. sˆo´ v`a mˆa˜ u sˆo´ v´o.i ex rˆ
dx
ax
1
x
40.
+
.
(DS.
arctg
)
(x2 + a2)2
2a3

am u(x)v (x) c´
o nguyˆen h`
am trˆen
v(x)u (x) c´
D v`
a
u(x)v (x)dx = u(x)v(x) −

v(x)u (x)dx.

(10.4)

`an.
u.c t´ınh t´ıch phˆan t`
u.ng phˆ
Cˆong th´
u.c (10.4) du.o..c go.i l`a cˆong th´
V`ı u (x)dx = du v`a v (x)dx = dv nˆen (10.4) c´o thˆe’ viˆe´t du.´o.i da.ng
udv = uv −

vdu.

(10.4*)

`an l´o.n c´ac t´ıch phˆan t´ınh du.o..c b˘a`ng
Thu..c tˆe´ cho thˆa´y r˘a`ng phˆ
`an c´o thˆe’ phˆan th`anh ba nh´om sau dˆay.
ph´ep t´ıch phˆan t`
u.ng phˆ


u.c du.´o.i dˆa´u t´ıch phˆan. Sau mˆ˜o i
`an bˆa.c cu’a da th´
`an t´ıch phˆan t`
u.c s˜e gia’m mˆo.t do.n vi..
lˆ
u.ng phˆ
`om nh˜
Nh´
om III gˆ
u.ng t´ıch phˆan m`a h`am du.´o.i dˆa´u t´ıch phˆan c´o
`an t´ıch phˆan
da.ng: eax sin bx, eax cos bx, sin(lnx), cos(lnx),... Sau hai lˆ
`an ta la.i thu du.o..c t´ıch phˆan ban dˆ
`au v´o.i hˆe. sˆo´ n`ao d´o. D´o l`a
t`
u.ng phˆ
`an t´ınh.
phu.o.ng tr`ınh tuyˆe´n t´ınh v´o.i ˆa’n l`a t´ıch phˆan cˆ
.
.
.
u a nˆeu khˆong v´et hˆe´t mo.i t´ıch phˆan
Du o ng nhiˆen l`a ba nh´om v`
`an (xem v´ı du. 6).
u.ng phˆ
t´ınh du.o..c b˘`ang t´ıch phˆan t`
`an
u.ng phˆ
Nhˆ
a.n x´et. Nh`o. c´ac phu.o.ng ph´ap dˆo’i biˆe´n v`a t´ıch phˆan t`

dx
x
√
= arc sin + C, a = 0.
2
2
a
a −x
√
dx
√
= ln|x + x2 ± a2| + C.
x2 ± a2


10.1. C´ac phu.o.ng ph´ap t´ınh t´ıch phˆan
´ V´I DU
CAC
.
√
√
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 =


3
3
V´ı du. 2. T´ınh I = arc cos2 xdx.
Gia’i. Gia’ su’. u = arc cos2 x, dv = dx. Khi d´o
2arc cos x
du = − √
dx, v = x.
1 − x2
Theo (10.4*) ta c´o
xarc cos x
√
dx.
1 − x2
u.c thu du.o..c ta d˘a.t u =
Dˆe’ t´ınh t´ıch phˆan o’. vˆe´ pha’i d˘a’ng th´
xdx
. Khi d´o
arc cos x, dv = √
1 − x2
√
√
dx
du = − √
, v = − d( 1 − x2) = − 1 − x2 + C1
1 − x2
√
`an lˆa´y v = − 1 − x2:
v`a ta chı’ cˆ
√
xarc cos x

2
1
x cos 3xdx = − x2 cos 3x + I1.
I = − x2 cos 3x +
3
3
3
3
`an t´ınh I1. D˘a.t u = x, dv = cos 3xdx. Khi d´o du = 1dx,
Ta cˆ
1
u. d´o
v = sin 3x. T`
3
1
2 1
1
x sin 3x −
sin 3xdx
I = − x2 cos 3x +
3
3 3
3
2
2
1
cos 3x + C.
= − x2 cos 3x + x sin 3x +
3
9

1
a
eax cos bxdx.
I1 = − eax cos bx +
b
b