˜
’ 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 chuˆo˜ 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
a´t di.nh
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´t di.nh
10.1.2 Phu.o.ng ph´ap dˆo’i biˆe´n . . . . . . .
`an
u.ng phˆ
10.1.3 Phu.o.ng ph´ap t´ıch phˆan t`
. . . . . . .
4
4
. . . . . . .
4
. . . . . . .
. . . . . . .
10.2 C´ac l´o.p h`am kha’ t´ıch trong l´o.p c´ac h`am so. cˆa´p . . . .
10.2.1 T´ıch phˆan c´ac h`am h˜
u.u ty’ . . . . . . . . . . . .
10.2.2 T´ıch phˆan mˆo.t sˆo´ h`am vˆo ty’ do.n gia’n . . . . .
10.2.3 T´ıch phˆan c´ac h`am lu.o..ng gi´ac . . . . . . . . . .
12
21
11 T´ıch phˆ
an x´
ac di.nh Riemann
11.1 H`am kha’ t´ıch Riemann v`a t´ıch phˆan x´ac di.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
. .
59
. .
. .
61
78
. .
78
11.3.2 T´ınh dˆo. d`ai cung v`a diˆe.n t´ıch m˘a.t tr`on xoay . .
11.4 T´ıch phˆan suy rˆo.ng . . . . . . . . . . . . . . . . . . . .
89
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
2
MU
. C LU
.C
`eu biˆ
12 T´ıch phˆ
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 . . . . . . . . .
.
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117
118
118
118
121
133
133
134
136
136
144
144
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 Chuˆ˜o i Fourier . . . . . . . . . . . . . . . . . . . . . . .
13.4.1 C´ac di.nh ngh˜ıa co. ba’n . . . . . . . . . . . . . .
177
178
178
179
191
191
192
199
199
201
211
211
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MU
. C LU
.C
3
`e su.. hˆo.i tu. cu’a chuˆ˜o i Fourier . . . 212
13.4.2 Dˆa´u hiˆe.u du’ vˆ
14 Phu.o.ng tr`ınh vi phˆ
an
224
14.1 Phu.o.ng tr`ınh vi phˆan cˆa´p 1 . . . . . . . . . . . . . . . 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´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
15.3 C´ac phu.o.ng tr`ınh vˆa.t l´
y to´an co. ba’n . . . . . . . . . .
`en s´ong . . . . . . . . . . . .
15.3.1 Phu.o.ng tr`ınh truyˆ
.
.
`en nhiˆe.t . . . . . . . . . . . .
15.3.2 Phu o ng tr`ınh truyˆ
15.3.3 Phu.o.ng tr`ınh Laplace . . . . . . . . . . . . . .
T`
ai liˆ
e.u tham kha’o . . . . . . . . . . . . . . . . . . . . .
304
306
310
313
314
317
320
327
Chu.o.ng 10
T´ıch phˆ
an bˆ
a´t di.nh
10.1 C´
ac phu.o.ng ph´
ap t´ınh t´ıch phˆ
an . . . . . .
4
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˜
u.u ty’ . . . . . . . . . 30
10.2.2 T´ıch phˆ
an mˆ
o.t sˆ
o´ h`
am vˆ
o ty’ do.n gia’n . . . 37
10.2.3 T´ıch phˆ
an c´
ac h`
am lu.o..ng gi´
ac . . . . . . . 48
10.1
C´
ac phu.o.ng ph´
ap t´ınh t´ıch phˆ
an
10.1.1
Nguyˆ
en h`
am v`
a t´ıch phˆ
an bˆ
a´t di.nh
- i.nh ngh˜ıa 10.1.1. H`am F (x) du.o..c go.i l`a nguyˆen h`am cu’a h`am
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 mˆ˜o 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’
.
`ng sˆ
o.t h˘
a
o´ cˆ
o.ng.
kh´
ac nhau bo’ i mˆ
Kh´ac v´o.i da.o h`am, nguyˆen h`am cu’a h`am so. cˆa´p khˆong pha’i bao
2
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 ,
1 cos x sin x
,
,
,... l`a nh˜
u.ng h`am khˆong so. cˆa´p.
cos(x2), sin(x2),
lnx
x
x
- i.nh ngh˜ıa 10.1.2. Tˆa.p ho..p mo.i nguyˆen h`am cu’a h`am f(x) trˆen
D
khoa’ng (a, b) du.o..c go.i l`a t´ıch phˆan bˆa´t di.nh cu’a h`am f(x) trˆen khoa’ng
(a, b) v`a du.o..c k´
y hiˆe.u l`a
Z
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 di.nh l´
y 10.1.2
Z
f(x)dx = F (x) + C, C ∈ R
`an hiˆe’u l`a d˘a’ng th´
u.a
uy y´ v`a d˘a’ng th´
u.c cˆ
u.c gi˜
trong d´o C l`a h˘a`ng sˆo´ t`
hai tˆa.p ho..p.
C´ac t´ınh chˆa´t co. ba’n cu’a t´ıch phˆan bˆa´t di.nh:
Z
f(x)dx = f(x)dx.
1) d
2)
3)
Z
Z
f(x)dx
df(x) =
′
Z
= f(x).
f ′ (x)dx = f(x) + C.
ut ra ba’ng c´ac t´ıch phˆan co.
T`
u. di.nh ngh˜ıa t´ıch phˆan bˆa´t di.nh r´
ba’n (thu.`o.ng du.o..c go.i l`a t´ıch phˆan ba’ng) sau dˆay:
Chu.o.ng 10. T´ıch phˆan bˆa´t di.nh
6
I.
Z
II.
Z
III.
IV.
V.
0.dx = C.
1dx = x + C.
Z
Z
Z
VI.
xαdx =
dx
= ln|x| + C, x 6= 0.
x
ax
a dx =
+ C (0 < a 6= 1);
lna
x
Z
VII.
xα+1
+ C, α 6= −1
α+1
IX.
Z
XI.
Z
ex dx = ex + C.
sin xdx = − cos x + C.
Z
VIII.
Z
cos xdx = sin x + C.
Z
π
dx
= tgx + C, x 6= + nπ, n ∈ Z.
2
cos x
2
dx
= −cotgx + C, x 6= nπ, n ∈ Z.
sin2 x
Z
arc sin x + C,
dx
−1 < x < 1.
X. √
=
1 − x2 −arc cos x + C
Z
arctgx + C,
dx
=
1 + x2 −arccotgx + C.
√
dx
= ln|x + x2 ± 1| + C
x2 ± 1
u. th`ı x < −1 ho˘a.c x > 1).
(trong tru.`o.ng ho..p dˆa´u tr`
Z
dx
1
1 + x
ln
XIII.
=
+ C, |x| 6= 1.
1 − x2
2 1−x
XII.
√
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)
2)
Z
Z
kf(x)dx = k
f(x)dx, k 6= 0.
[f(x) ± g(x)]dx =
3) Nˆe´u
Z
Z
Z
7
Z
f(x)dx ±
Z
g(x)dx.
f(x)dx = F (x) + C v`a u = ϕ(x) kha’ vi liˆen tu.c th`ı
f(u)du = F (u) + C.
´ V´I DU
CAC
.
V´ı du. 1. Ch´
u.ng minh r˘`ang h`am y = signx c´o nguyˆen h`am trˆen
khoa’ng bˆa´t k`
y khˆong ch´
u.a diˆe’m x = 0 v`a khˆong c´o nguyˆen h`am trˆen
mo.i khoa’ng ch´
u.a diˆe’m x = 0.
Gia’i. 1) Trˆen khoa’ng bˆa´t k`
y khˆong ch´
u.a diˆe’m x = 0 h`am y = signx
l`a h˘a`ng sˆo´. Ch˘a’ng ha.n v´o.i mo.i khoa’ng (a, b), 0 < a < b ta c´o signx = 1
v`a do d´o mo.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`a a < 0 < b. Trˆen khoa’ng (a, 0) mo.i
nguyˆen h`am cu’a signx c´o da.ng F (x) = −x + C1 c`on trˆen khoa’ng (0, b)
nguyˆen h`am c´o da.ng F (x) = x + C2. V´o.i mo.i c´ach cho.n h˘a`ng sˆo´ C1
v`a C2 ta thu du.o..c h`am [trˆen (a, b)] khˆong c´o da.o h`am ta.i diˆe’m x = 0.
Nˆe´u ta cho.n C = C1 = C2 th`ı thu du.o..c h`am liˆen tu.c y = |x| + C
nhu.ng khˆong kha’ vi ta.i diˆe’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. N
V´ı du. 2. T`ım nguyˆen h`am cu’a h`am f(x) = e|x| trˆen to`an tru.c sˆo´.
`en x > 0 mˆo.t
Gia’i. V´o.i x > 0 ta c´o e|x| = ex v`a do d´o trong miˆ
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˘`ang
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 di.nh
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,
F (x) = 1
nˆe´u x = 0,
−e−x + 2 nˆe´u x < 0
l`a h`am liˆen tu.c trˆen to`an tru.c sˆo´. Ta ch´
u.ng minh r˘a`ng F (x) l`a nguyˆen
h`am cu’a h`am e|x| trˆen to`an tru.c sˆo´. Thˆa.t vˆa.y, v´o.i x > 0 ta c´o
`an pha’i
F ′(x) = ex = e|x|, v´o.i x < 0 th`ı F ′(x) = e−x = e|x|. Ta c`on cˆ
ch´
u.ng minh r˘`ang F ′(0) = e0 = 1. Ta c´o
F (x) − F (0)
ex − 1
= lim
= 1,
x→0+0
x→0+0
x
x
−e−x + 2 − 1
F (x) − F (0)
= lim
= 1.
F−′ (0) = lim
x→0−0
x→0−0
x
x
Nhu. vˆa.y F+′ (0) = F−′ (0) = F ′(0) = 1 = e|x|. T`
u. d´o c´o thˆe’ viˆe´t:
Z
ex + C,
x<0
e|x|dx = F (x) + C =
−e−x + 2 + C, x < 0. N
F+′ (0) = lim
`o thi. qua diˆe’m (−2, 2) dˆo´i v´o.i h`am
V´ı du. 3. T`ım nguyˆen h`am c´o dˆ
1
f(x) = , x ∈ (−∞, 0).
x
1
Gia’i. V`ı (ln|x|)′ = nˆen ln|x| l`a mˆo.t trong c´ac nguyˆen h`am cu’a
x
1
h`am f(x) = . Do vˆa.y, nguyˆen h`am cu’a f l`a h`am F (x) = ln|x| + C,
x
`eu kiˆe.n F (−2) = 2, t´
C ∈ R. H˘a`ng sˆo´ C du.o..c x´ac di.nh t`
u. diˆ
u.c l`a
ln2 + C = 2 ⇒ C = 2 − ln2. Nhu. vˆa.y
x
F (x) = ln|x| + 2 − ln2 = ln
+ 2. N
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:
Z
1)
2x+1 − 5x−1
dx,
10x
2)
Z
2x + 3
dx.
3x + 2
Gia’i. 1) Ta c´o
Z x
Z h x
2
1
1 1 x i
5x
I=
2 x−
dx =
2
dx
−
10
5 · 10x
5
5 2
Z x
Z x
1
1
1
dx −
dx
=2
5
5
2
1 x
1 x
1 2
5
=2
−
+C
1
1
5
ln
ln
5
2
1
2
+
+ C.
=− x
5 ln5 5 · 2x ln2
2)
h
2 5i
3
x
+
+
2 dx = 2
3
6 dx
2
2
3
x+
3
3
2
5