Bài tập Toán cao cấp A3

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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
’
).