1. ˜
ˆ ’
NGUYEN THUY THANH
` ˆ
BAI TAP
.
´ ´
ˆ
TOAN CAO CAP
Tˆp 3
a.
e ınh ıch a y ´
e ˜
Ph´p t´ t´ phˆn. L´ thuyˆt chuˆ i.
o
Phu.o.ng tr` vi phˆn
ınh a
` ´
ˆ ’ ´
ˆ ` ˆ
NHA XUAT BAN DAI HOC QUOC GIA HA NOI
. . .

2. Muc luc
. .
a ´ .
10 T´ phˆn bˆt dinh
ıch a 4
10.1 C´c phu.o.ng ph´p t´ t´ch phˆn . . . . .
a a ınh ı a . . . . . . . 4
a ı a a .´
10.1.1 Nguyˆn h`m v` t´ch phˆn bˆt dinh
e a . . . . . . . 4
10.1.2 Phu.o.ng ph´p dˆi biˆn . . . . . . .
a o e’ ´ . . . . . . . 12
10.1.3 Phu.o.ng ph´p t´ phˆn t`.ng phˆn
a ıch a u `
a . . . . . . . 21
10.2 C´c l´.p h`m kha t´ trong l´.p c´c h`m so. cˆp . . . .
a o a ’ ıch o a a ´
a 30
10.2.1 T´ phˆn c´c h`m h˜.u ty . . . . . . . . . . . .
ıch a a a u ’ 30
10.2.2 T´ phˆn mˆt sˆ h`m vˆ ty do.n gian . . . . .
ıch a . ´
o o a o ’ ’ 37
10.2.3 T´ phˆn c´c h`m lu.o.ng gi´c . . . . . . . . . .
ıch a a a . a 48
11 T´ phˆn x´c dinh Riemann
ıch a a . 57
’ ıch
11.1 H`m kha t´ Riemann v` t´ch phˆn x´c dinh . . .
a a ı a a . . . 58
-.
11.1.1 Dinh ngh˜ . . . . . . . . . . . . . . . . . .
ıa . . 58
- ` e e e a
. ’ ’ ı
11.1.2 Diˆu kiˆn dˆ h`m kha t´ch . . . . . . . . . . . . 59
a ınh a ´
11.1.3 C´c t´ chˆt co ’ . ban cua t´ch phˆn x´c dinh
’ ı a a . . . 59
11.2 Phu.o.ng ph´p t´ t´ phˆn x´c d inh . . . . . . .
a ınh ıch a a . . . 61
.ng dung cua t´ch phˆn x´c d inh . . . . . .
. ´
11.3 Mˆt sˆ u
o o´ . ’ ı a a . . . 78
. ’ ’
11.3.1 Diˆn t´ h` ph˘ng v` thˆ t´ch vˆt thˆ . .
e ıch ınh a a e ı a. e’ . . 78
11.3.2 T´ dˆ d`i cung v` diˆn t´ m˘t tr`n xoay . .
ınh o a
. a e ıch a o
. . 89
11.4 T´ phˆn suy rˆng . . . . . . . . . . . . . . . . . . . .
ıch a o
. 98
11.4.1 T´ phˆn suy rˆng cˆn vˆ han . . . . . . . . . 98
ıch a o
. a o .
.
ıch a o
. ’ a
11.4.2 T´ phˆn suy rˆng cua h`m khˆng bi ch˘n . . 107
o . a .

3. 2 MUC LUC
. .
12 T´ phˆn h`m nhiˆu biˆn
ıch a a `e e´ 117
12.1 T´ phˆn 2-l´.p . . . . . . . . . . . . . .
ıch a o . . . . . . . . 118
.`.ng ho.p miˆn ch˜. nhˆt . . .
12.1.1 Tru o `
e u a . . . . . . . . 118
. .
.`.ng ho.p miˆn cong . . . . . .
12.1.2 Tru o `
e . . . . . . . . 118
.
12.1.3 Mˆt v`i u
o a ´ .ng dung trong h` hoc
ınh . . . . . . . . . 121
. .
12.2 T´ phˆn 3-l´
ıch a o.p . . . . . . . . . . . . . . . . . . . . . . 133
12.2.1 Tru.`.ng ho.p miˆn h`nh hˆp . . .
o . ` ı
e o
. . . . . . . . . 133
.`.ng ho.p miˆn cong . . . . . .
12.2.2 Tru o `
e . . . . . . . . 134
.
12.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 136
12.2.4 Nhˆn x´t chung . . . . . . . . . .
a e
. . . . . . . . . 136
12.3 T´ phˆn d u.`.ng . . . . . . . . . . . . .
ıch a o . . . . . . . . 144
12.3.1 C´c dinh ngh˜a co. ban . . . . . .
a . ı ’ . . . . . . . . 144
12.3.2 T´ t´ phˆn du o
ınh ıch a .`.ng . . . . . . . . . . . . . . 146
12.4 T´ phˆn m˘t . . . . . . . . . . . . . .
ıch a a. . . . . . . . . 158
12.4.1 C´c dinh ngh˜a co. ban . . . . . .
a . ı ’ . . . . . . . . 158
12.4.2 Phu.o.ng ph´p t´ t´ch phˆn m˘t
a ınh ı a a
. . . . . . . . . 160
12.4.3 Cˆng th´
o u.c Gauss-Ostrogradski . . . . . . . . . 162
12.4.4 Cˆng th´.c Stokes . . . . . . . . .
o u . . . . . . . . 162
y ´
13 L´ thuyˆt chuˆ i
e ˜
o 177
13.1 Chuˆ i sˆ du.o.ng . . . . . . . . . . . . . . . . . . . . . .
˜ o
o ´ 178
13.1.1 C´c dinh ngh˜a co. ban . . . . . . . . . . . . . .
a . ı ’ 178
˜ o
o ´
13.1.2 Chuˆ i sˆ du .o.ng . . . . . . . . . . . . . . . . . . 179
˜ o . . ´ . o e o
. ´
13.2 Chuˆ i hˆi tu tuyˆt d ˆi v` hˆi tu khˆng tuyˆt d ˆi . . .
o . e o a o . 191
13.2.1 C´c dinh ngh˜a co. ban . . . . . . . . . . . . . .
a . ı ’ 191
˜
o ´
a a a ´
13.2.2 Chuˆ i dan dˆu v` dˆu hiˆu Leibnitz . . . . . .
e
. 192
˜ u
13.3 Chuˆ i l˜y th`
o u.a . . . . . . . . . . . . . . . . . . . . . . 199
13.3.1 C´c dinh ngh˜a co. ban . . . . . . . . . . . . . .
a . ı ’ 199
13.3.2 Diˆu kiˆn khai triˆn v` phu.o.ng ph´p khai triˆn
- `e e
. ’
e a a ’
e 201
˜
13.4 Chuˆ i Fourier . . . . . . . . . . . . . . . . . . . . . . .
o 211
13.4.1 C´c dinh ngh˜a co. ban . . . . . . . . . . . . . .
a . ı ’ 211

4. MUC LUC
. . 3
13.4.2 Dˆu hiˆu du vˆ su. hˆi tu cua chuˆ i Fourier . . . 212
´
a e
. ’ ` . o . ’
e . ˜
o
14 Phu.o.ng tr` vi phˆn
ınh a 224
14.1 Phu.o.ng tr` vi phˆn cˆp 1 . . . . . . . . . . . . . . . 225
ınh a a ´
14.1.1 Phu.o.ng tr` t´ch biˆn . . . . . . . . . . . . . . 226
ınh a ´
e
14.1.2 Phu .o.ng tr` d ang cˆp . . . . . . . . . . . . . 231
ınh ˘ ’ ´
a
14.1.3 Phu.o.ng tr` tuyˆn t´ . . . . . . . . . . . . . 237
ınh ´
e ınh
14.1.4 Phu.o.ng tr` Bernoulli . . . . . . . . . . . . . . 244
ınh
14.1.5 Phu .o.ng tr` vi phˆn to`n phˆn . . . . . . . . 247
ınh a a `a
14.1.6 Phu.o.ng tr` Lagrange v` phu.o.ng tr` Clairaut255
ınh a ınh
14.2 Phu .o.ng tr` vi phˆn cˆp cao . . . . . . . . . . . . . . 259
ınh a a ´
14.2.1 C´c phu
a .o.ng tr` cho ph´p ha thˆp cˆp . . . . 260
ınh e ´ ´
. a a
14.2.2 Phu.o.ng tr` vi phˆn tuyˆn t´ cˆp 2 v´.i hˆ
ınh a ´
e ınh a´ o e .
´ `
sˆ h˘ng . . . . . . . . . . . . . . . . . . . . . . 264
o a
14.2.3 Phu.o.ng tr` vi phˆn tuyˆn t´nh thuˆn nhˆt
ınh a ´
e ı `
a ´
a
cˆp n (ptvptn cˆp n ) v´.i hˆ sˆ h˘ng . . . . . . 273
a´ ´
a o e o `
. ´ a
.o.ng tr` vi phˆn tuyˆn t´ cˆp 1 v´.i hˆ sˆ h˘ng290
´ ´ o e o `
14.3 Hˆ phu
e
. ınh a e ınh a . ´ a
15 Kh´i niˆm vˆ phu.o.ng tr`
a e
. `
e ınh vi phˆn dao h`m riˆng
a . a e 304
15.1 Phu.o.ng tr` vi phˆn cˆp 1 tuyˆn t´ dˆi v´.i c´c dao
ınh a a ´ ´ ´
e ınh o o a .
h`m riˆng . . . . . . . . . . . . . . . . . . . . . . . . .
a e 306
15.2 Giai phu.o.ng tr` d ao h`m riˆng cˆp 2 d o.n gian nhˆt
’ ınh . a e ´
a ’ ´
a 310
15.3 C´c phu.o.ng tr` vˆt l´ to´n co. ban . . . . . . . . . .
a ınh a y a
. ’ 313
15.3.1 Phu.o.ng tr` truyˆn s´ng . . . . . . . . . . . .
ınh ` o
e 314
15.3.2 Phu .o.ng tr` truyˆn nhiˆt . . . . . . . . . . . .
ınh `
e e 317
.
15.3.3 Phu .o.ng tr` Laplace . . . . . . . . . . . . . .
ınh 320
a e . ’
T`i liˆu tham khao . . . . . . . . . . . . . . . . . . . . . 327

5. Chu.o.ng 10
ıch a ´
T´ phˆn bˆt dinh
a .
10.1 C´c phu.o.ng ph´p t´
a a ınh t´ phˆn . . . . . .
ıch a 4
e a a ıch a a . ´
10.1.1 Nguyˆn h`m v` t´ phˆn bˆt dinh . . . . . 4
10.1.2 Phu.o.ng ph´p dˆi biˆn . . . . . . . . . . . . 12
a o e’ ´
10.1.3 Phu.o.ng ph´p t´ phˆn t`.ng phˆn . . . . . 21
a ıch a u `
a
10.2 C´c l´.p h`m kha t´ trong l´.p c´c h`m
a o a ’ ıch o a a
. cˆp . . . . . . . . . . . . . . . . . . . . . . 30
so a ´
10.2.1 T´ phˆn c´c h`m h˜.u ty . . . . . . . . . 30
ıch a a a u ’
10.2.2 T´ phˆn mˆt sˆ h`m vˆ ty do.n gian . . . 37
ıch a . ´
o o a o ’ ’
10.2.3 T´ phˆn c´c h`m lu.o.ng gi´c . . . . . . . 48
ıch a a a . a
10.1 C´c phu.o.ng ph´p t´
a a ınh t´ phˆn
ıch a
10.1.1 a a ıch a ´
Nguyˆn h`m v` t´ phˆn bˆt dinh
e a .
Dinh ngh˜ 10.1.1. H`m F (x) du.o.c goi l` nguyˆn h`m cua h`m
-. ıa a . . a e a ’ a
’ ´ ’ o a ’
f (x) trˆn khoang n`o d´ nˆu F (x) liˆn tuc trˆn khoang d´ v` kha vi
e a o e e . e

6. 10.1. C´c phu.o.ng ph´p t´ t´ phˆn
a a ınh ıch a 5
˜ ’ ’ ’
tai mˆ i diˆm trong cua khoang v` F (x) = f(x).
. o e a
Dinh l´ 10.1.1. (vˆ su. tˆn tai nguyˆn h`m) Moi h`m liˆn tuc trˆn
-. y ` . ` .
e o e a . a e . e
. ` o
e e a e ’
doan [a, b] dˆu c´ nguyˆn h`m trˆn khoang (a, b).
-. ´
a y ’ u
Dinh l´ 10.1.2. C´c nguyˆn h`m bˆt k` cua c`ng mˆt h`m l` chı
y a e a o a a ’
.
.i mˆt h˘ng sˆ cˆng.
kh´c nhau bo
a ’ . `
o a ´ .
o o
Kh´c v´.i dao h`m, nguyˆn h`m cua h`m so. cˆp khˆng phai bao
a o . a e a ’ a ´
a o ’
gi`. c˜ng l` h`m so. cˆp. Ch˘ng han, nguyˆn h`m cua c´c h`m e−x ,
2
o u a a ´
a ’
a . e a ’ a a
1 cos x sin x
cos(x2), sin(x2), , , ,... l` nh˜.ng h`m khˆng so. cˆp.
a u a o ´
a
lnx x x
D.nh ngh˜ 10.1.2. Tˆp ho.p moi nguyˆn h`m cua h`m f (x) trˆn
-i ıa a
. . . e a ’ a e
’ .o.c goi l` t´ phˆn bˆt dinh cua h`m f (x) trˆn khoang
khoang (a, b) du . . a ıch a a . ´ ’ a e ’
(a, b) v` du.o.c k´ hiˆu l`
a . y e a .
f(x)dx.
´ a o . a e a ’ a e ’
Nˆu F (x) l` mˆt trong c´c nguyˆn h`m cua h`m f(x) trˆn khoang
e
(a, b) th` theo dinh l´ 10.1.2
ı . y
f(x)dx = F (x) + C, C∈R
trong d´ C l` h˘ng sˆ t`y y v` d˘ng th´.c cˆn hiˆu l` d˘ng th´.c gi˜.a
o a `
a ´
o u ´ a a ’ u ` a ’
e a a ’ u u
hai tˆp ho.p.
a
. .
C´c t´ chˆt co. ban cua t´ phˆn bˆt dinh:
a ınh a ´ ’ ’ ıch a a . ´
1) d f (x)dx = f (x)dx.
2) f (x)dx = f (x).
3) df(x) = f (x)dx = f(x) + C.
T`. dinh ngh˜ t´ phˆn bˆt dinh r´t ra bang c´c t´ch phˆn co.
u . ıa ıch a ´
a . u ’ a ı a
ban (thu.`.ng du.o.c goi l` t´ phˆn bang) sau dˆy:
’ o . . a ıch a ’ a

7. 6 Chu.o.ng 10. T´ phˆn bˆt dinh
´
ıch a a .
I. 0.dx = C.
II. 1dx = x + C.
xα+1
III. xαdx = + C, α = −1
α+1
dx
IV. = ln|x| + C, x = 0.
x
ax
V. axdx = + C (0 < a = 1); ex dx = ex + C.
lna
VI. sin xdx = − cos x + C.
VII. cos xdx = sin x + C.
dx π
VIII. 2x
= tgx + C, x = + nπ, n ∈ Z.
cos 2
dx
IX. = −cotgx + C, x = nπ, n ∈ Z.
sin2 x

dx arc sin x + C,
X. √ = −1 < x < 1.
1 − x2 −arc cos x + C

dx arctgx + C,
XI. =
1 + x2 −arccotgx + C.
dx √
XII. √ = ln|x + x2 ± 1| + C
x2 ± 1
(trong tru.`.ng ho.p dˆu tr`. th` x < −1 ho˘c x > 1).
o . ´
a u ı a
.
dx 1 1+x
XIII. 2
= ln + C, |x| = 1.
1−x 2 1−x
´ ınh ıch a a .´
C´c quy t˘c t´ t´ phˆn bˆt dinh:
a a

8. 10.1. C´c phu.o.ng ph´p t´ t´ phˆn
a a ınh ıch a 7
1) kf(x)dx = k f(x)dx, k = 0.
2) [f(x) ± g(x)]dx = f (x)dx ± g(x)dx.
´
3) Nˆu
e ’
f(x)dx = F (x) + C v` u = ϕ(x) kha vi liˆn tuc th`
a e . ı
f (u)du = F (u) + C.
CAC V´ DU
´ I .
V´ du 1. Ch´.ng minh r˘ng h`m y = signx c´ nguyˆn h`m trˆn
ı . u `
a a o e a e
’ ´
khoang bˆt k` khˆng ch´
a y o u .a diˆm x = 0 v` khˆng c´ nguyˆn h`m trˆn
’
e a o o e a e
moi khoang ch´.a diˆm x = 0.
. ’ u ’
e
Giai. 1) Trˆn khoang bˆt k` khˆng ch´.a diˆm x = 0 h`m y = signx
’ e ’ ´
a y o u ’
e a
` ´ ’
l` h˘ng sˆ. Ch˘ng han v´
a a o a .i moi khoang (a, b), 0 < a < b ta c´ signx = 1
’
. o . o
o . e a ’ o e
v` do d´ moi nguyˆn h`m cua n´ trˆn (a, b) c´ dang
a o .
F (x) = x + C, C ∈ R.
e ’ a e ’
2) Ta x´t khoang (a, b) m` a < 0 < b. Trˆn khoang (a, 0) moi .
e a ’ o . o e ’
nguyˆn h`m cua signx c´ dang F (x) = −x + C1 c`n trˆn khoang (0, b)
nguyˆn h`m c´ dang F (x) = x + C2. V´.i moi c´ch chon h˘ng sˆ C1
e a o . o . a . `
a ´
o
v` C2 ta thu du.o.c h`m [trˆn (a, b)] khˆng c´ dao h`m tai diˆm x = 0.
a . a e o o . a ’
. e
Nˆu ta chon C = C1 = C2 th` thu du.o.c h`m liˆn tuc y = |x| + C
e´ . ı . a e .
nhu.ng khˆng kha vi tai diˆm x = 0. T`. d´, theo dinh ngh˜a 1 h`m
o ’ . e’ u o . ı a
signx khˆng c´ nguyˆn h`m trˆn (a, b), a < 0 < b.
o o e a e
V´ du 2. T` nguyˆn h`m cua h`m f (x) = e|x| trˆn to`n truc sˆ.
ı . ım e a ’ a e a . o ´
’
Giai. V´
o.i x |x| x `
0 ta c´ e = e v` do d´ trong miˆn x > 0 mˆt
o a o e o
.
trong c´c nguyˆn h`m l` ex . Khi x < 0 ta c´ e|x| = e−x v` do vˆy
a e a a o a a
.
e o a e a a −x
trong miˆn x < 0 mˆt trong c´c nguyˆn h`m l` −e + C v´ `
` .i h˘ng
o a
.
´ ´
sˆ C bˆt k`.
o a y
Theo dinh ngh˜ nguyˆn h`m cua h`m e|x| phai liˆn tuc nˆn n´
. ıa, e a ’ a ’ e . e o

9. 8 Chu.o.ng 10. T´ phˆn bˆt dinh
´
ıch a a .
’ ’ a `
phai thoa m˜n diˆu kiˆn
e e
.
lim ex = lim (−e−x + C)
x→0+0 x→0−0
t´.c l` 1 = −1 + C ⇒ C = 2.
u a
Nhu. vˆy
a
.

ex
 ´
nˆu x > 0,
e


F (x) = 1 ´
nˆu x = 0,
e


 −x
−e + 2 nˆu x < 0
´
e
l` h`m liˆn tuc trˆn to`n truc sˆ. Ta ch´.ng minh r˘ng F (x) l` nguyˆn
a a e . e a . o ´ u `
a a e
’ |x| ´
h`m cua h`m e trˆn to`n truc sˆ. Thˆt vˆy, v´
a a e a .i x > 0 ta c´
. o a a
. . o o
F (x) = ex = e|x|, v´.i x < 0 th` F (x) = e−x = e|x|. Ta c`n cˆn phai
o ı o ` a ’
ch´.ng minh r˘ng F (0) = e0 = 1. Ta c´
u `
a o
F (x) − F (0) ex − 1
F+ (0) = lim = lim = 1,
x→0+0 x x→0+0 x
F (x) − F (0) −e−x + 2 − 1
F− (0) = lim = lim = 1.
x→0−0 x x→0−0 x
Nhu. vˆy F+ (0) = F− (0) = F (0) = 1 = e|x|. T`. d´ c´ thˆ viˆt:
a
. u o o e e ’ ´

ex + C, x<0
e|x|dx = F (x) + C =
−e−x + 2 + C, x < 0.
V´ du 3. T` nguyˆn h`m c´ dˆ thi qua diˆm (−2, 2) dˆi v´.i h`m
ı . ım e a o ` .
o ’
e ´
o o a
1
f (x) = , x ∈ (−∞, 0).
x
1
’
Giai. V` (ln|x|) = nˆn ln|x| l` mˆt trong c´c nguyˆn h`m cua
ı e a o . a e a ’
x
1
a
. e a ’
h`m f(x) = . Do vˆy, nguyˆn h`m cua f l` h`m F (x) = ln|x| + C,
a a a
x
C ∈ R. H˘ng sˆ C du.o.c x´c dinh t`. diˆu kiˆn F (−2) = 2, t´.c l`
`
a ´
o . a . u ` e e
. u a
ln2 + C = 2 ⇒ C = 2 − ln2. Nhu a. vˆy
.
x
F (x) = ln|x| + 2 − ln2 = ln + 2.
2

10. 10.1. C´c phu.o.ng ph´p t´ t´ phˆn
a a ınh ıch a 9
V´ du 4. T´ c´c t´ phˆn sau dˆy:
ı . ınh a ıch a a
2x+1 − 5x−1 2x + 3
1) dx, 2) dx.
10x 3x + 2
’
Giai. 1) Ta c´
o
2x 5x 1 x 1 1 x
I= 2
x
− x
dx = 2 − dx
10 5 · 10 5 5 2
1 x 1 1 x
=2 dx − dx
5 5 2
1 x 1 x
1 2
=2 5 − +C
1 5 1
ln ln
5 2
2 1
=− x + + C.
5 ln5 5 · 2x ln2
2)
3 2 5
2 x+ x+ +
I= 2 dx = 2 3 6 dx
2 3 2
3 x+ x+
3 3
2 5 2
= x + ln x + + C.
3 9 3
V´ du 5. T´ c´c t´ phˆn sau dˆy:
ı . ınh a ıch a a
1 + cos2 x √
1) tg2 xdx, 2) dx, 3) 1 − sin 2xdx.
1 + cos 2x
’
Giai. 1)
2 sin2 x 1 − cos2 x
tg xdx = dx = dx
cos2 x cos2 x
dx
= − dx = tgx − x + C.
cos2 x

11. 10 Chu.o.ng 10. T´ phˆn bˆt dinh
´
ıch a a .
2)
1 + cos2 x 1 + cos2 x 1 dx
dx = 2x
dx = + dx
1 + cos 2x 2 cos 2 cos2 x
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.
` ˆ
BAI TAP
.
B˘ng c´c ph´p biˆn dˆi dˆng nhˆt, h˜y du.a c´c t´ch phˆn d˜ cho
`
a a e ´
e o `’ o ´ a
a a ı a a
vˆ t´ phˆn bang v` t´ c´c t´ch phˆn d´1
` ıch a
e ’ a ınh a ı a o
dx 1 x−1 1
1. . (DS. ln − arctgx)
x4 − 1 4 x+1 2
1 + 2x2 1
2. dx. (DS. arctgx − )
x2 (1 + x2 ) x
√ √
x2 + 1 + 1 − x2 √
3. √ dx. (DS. arc sin x + ln|x + 1 + x2|)
1 − x4
√ √
x2 + 1 − 1 − x2 √ √
4. √ dx. (DS. ln|x + x2 − 1| − ln|x + x2 + 1|)
x4 − 1
√
x4 + x−4 + 2 1
5. 3
dx. (DS. ln|x| − 4 )
x 4x
23x − 1 e2x
6. dx. (DS. + ex + 1)
ex − 1 2
Dˆ cho gon, trong c´c “D´p sˆ” cua chu.o.ng n`y ch´ng tˆi bo qua khˆng viˆt
1’
e . a ´
a o ’ a u o ’ o ´
e
`ng sˆ cˆng C.
c´c h˘
a a ´ .
o o

12. 10.1. C´c phu.o.ng ph´p t´ t´ phˆn
a a ınh ıch a 11
3x
22x − 1 2 22 x
7. √ dx. (DS. + 2− 2 )
2x ln2 3
dx 1 lnx
8. . (DS. √ arctg √ )
x(2 + ln2 x) 2 2
√
3
ln2 x 3 5/3
9. dx. (DS. ln x)
x 5
ex + e2x
10. dx. (DS. −ex − 2ln|ex − 1|)
1 − ex
ex dx
11. . (DS. ln(1 + ex))
1 + ex
x 1 sin x
12. sin2 dx. (DS. x− )
2 2 2
13. cotg2 xdx. (DS. −x − cotgx)
√ π
14. 1 + sin 2xdx, x ∈ 0, . (DS. − cos x + sin x)
2
15. ecos x sin xdx. (DS. −ecos x )
16. ex cos ex dx. (DS. sin ex)
1 x
17. dx. (DS. tg )
1 + cos x 2
dx 1 x π
18. . (DS. √ ln tg + )
sin x + cos x 2 2 8
1 + cos x 2
19. dx. (DS. − )
(x + sin x)3 2(x + sin x)2
sin 2x 1
20. dx. (DS. − 1 − 4 sin2 x)
1 − 4 sin x 2 2
sin x √
21. dx. (DS. −ln| cos x + 1 + cos2 x|)
2
2 − sin x

13. 12 Chu.o.ng 10. T´ phˆn bˆt dinh
´
ıch a a .
sin x cos x 1 sin2 x
22. dx. (DS. arc sin √ )
3 − sin4 x 2 3
arccotg3x 1
23. 2
dx. (DS. − arccotg2 3x)
1 + 9x 6
√
x + arctg2x 1 1
24. dx. (DS. ln(1 + 4x2) + arctg3/22x)
1 + 4x2 8 3
arc sin x − arc cos x 1
25. √ dx. (DS. (arc sin2 x + arc cos2 x))
1 − x2 2
x + arc sin3 2x 1√ 1
26. √ dx. (DS. − 1 − 4x2 + arc sin4 2x)
1 − 4x2 4 8
x + arc cos3/2 x √ 2
27. √ dx. (DS. − 1 − x2 − arc cos5/2 x)
1 − x2 5
|x|3
28. x|x|dx. (DS. )
3
29. (2x − 3)|x − 2|dx.

− 2 x3 + 7 x2 − 6x + C, x < 2

(DS. F (x) = 3 2 )
2 3 7 2
 x − x + 6x + C, x 2
3 2

1 − x2, |x| 1,
30. f(x)dx, f(x) =
1 − |x|, |x| > 1.
 3
x − x + C
 ´
nˆu |x|
e 1
(DS. F (x) = 3 )
x − x|x| + 1 signx + C
 ´
nˆu|x| > 1
e
2 6
10.1.2 Phu.o.ng ph´p dˆi biˆn
a o’ ´
e
Dinh l´. Gia su.:
-. y ’ ’

14. 10.1. C´c phu.o.ng ph´p t´ t´ phˆn
a a ınh ıch a 13
1) H`m x = ϕ(t) x´c dinh v` kha vi trˆn khoang T v´.i tˆp ho.p gi´
a a . a ’ e ’ o a .. a
’
tri l` khoang X.
. a
a . a o e a e ’
2) H`m y = f (x) x´c dinh v` c´ nguyˆn h`m F (x) trˆn khoang X.
a
o a a e a ’
Khi d´ h`m F (ϕ(t)) l` nguyˆn h`m cua h`m f(ϕ(t))ϕ (t) trˆn
a e
’
khoang T .
T`. dinh l´ 10.1.1 suy r˘ng
u . y `
a
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 nˆn d˘ng th´.c (10.1) c´ thˆ viˆt du.´.i dang
e a ’ u ’ ´
o e e o .
f(x)dx x=ϕ(t)
= f (ϕ(t))ϕ (t)dt. (10.2)
D˘ng th´.c (10.2) du.o.c goi l` cˆng th´.c dˆi biˆn trong t´ phˆn
’
a u . . a o u o e ’ ´ ıch a
´
bˆt dinh.
a .
Nˆu h`m x = ϕ(t) c´ h`m ngu.o.c t = ϕ−1 (x) th` t`. (10.2) thu
´
e a o a . ı u
.o.c
du .
f(x)dx = f (ϕ(t))ϕ (t)dt t=ϕ−1 (x)
. (10.3)
e o a ı . ` e o e
Ta nˆu mˆt v`i v´ du vˆ ph´p dˆi biˆn.
. e ’ ´
√
i) Nˆu biˆu th´.c du.´.i dˆu t´ phˆn c´ ch´.a c˘n a2 − x2, a > 0
´
e e’ u ´
o a ıch a o u a
π π
th` su. dung ph´p dˆi biˆn x = a sin t, t ∈ − ,
ı ’ . e o e’ ´ .
2 2 √
ii) Nˆu biˆu th´.c du.´.i dˆu t´ phˆn c´ ch´.a c˘n x2 − a2, a > 0
´
e e’ u ´
o a ıch a o u a
a π
e o e ’ ´
th` d`ng ph´p dˆi biˆn x =
ı u , 0 < t < ho˘c x = acht.
a
.
cos t 2 √
´ .´.i dˆu t´ phˆn ch´.a c˘n th´.c a2 + x2, a > 0
iii) Nˆu h`m du o a ıch a
e a ´ u a u
π π
’ .
th` c´ thˆ d˘t x = atgt, t ∈ − ,
ı o e a ho˘c x = asht.
a
.
2 2
´ .´.i dˆu t´ phˆn l` f (x) = R(ex , e2x, . . . .enx ) th`
iv) Nˆu h`m du o a ıch a a
e a ´ ı
c´ thˆ d˘t t = ex (o. dˆy R l` h`m h˜.u ty).
o e a ’ . ’ a a a u ’

15. 14 Chu.o.ng 10. T´ phˆn bˆt dinh
´
ıch a a .
CAC V´ DU
´ I .
dx
V´ du 1. T´
ı . ınh .
cos x
’
Giai. Ta c´
o
dx cos xdx
= (d˘t t = sin x, dt = cos xdx)
a
.
cos x 1 − sin2 x
dt 1 1+t x π
= = ln + C = ln tg + + C.
1 − t2 2 1−t 2 4
x3 dx
V´ du 2. T´ I =
ı . ınh .
x8 − 2
’
Giai. ta c´
o
√
1 2 x4
d(x4 ) d √
4 4 2
I= =
x8 − 2 x4 2
−2 1 − √
2
x4
D˘t t = √ ta thu du.o.c
a
. .
2
√ √
2 2 + x4
I=− ln √ + C.
8 2 − x4
x2 dx
V´ du 3. T´ I =
ı . ınh ·
(x2 + a2 )3
adt
’
Giai. D˘t x(t) = atgt ⇒ dx =
a
. . Do d´
o
cos2 t
a3tg2t · cos3 tdt sin2 t dt
I= = dt = − cos tdt
a3 cos2 t cos t cos t
t π
= ln tg + − sin t + C.
2 4
x
V` t = arctg nˆn
ı e
a
1 x π x
I = ln tg arctg + − sin arctg +C
2 a 4 a
x √
= −√ + ln|x + x2 + a2| + C.
x2 + a2

16. 10.1. C´c phu.o.ng ph´p t´ t´ phˆn
a a ınh ıch a 15
. . ı e ˜ a
e ´ `
Thˆt vˆy, v` sin α = cos α · tgα nˆn dˆ d`ng thˆy r˘ng
a a a a
x x
sin arctg =√ ·
a x 2 + a2
´
Tiˆp theo ta c´
e o
1 x π x π x
sin arctg + 1 − cos arctg + 1 + sin arctg
2 a 4 = a 2 = a
1 x π x π x
cos arctg + sin arctg + − cos arctg
2 a 4 a 2 a
√
x + a2 + x2
=
a
v` t`. d´ suy ra diˆu phai ch´.ng minh.
a u o `
e ’ u
√
V´ du 4. T´ I =
ı . ınh a2 + x2 dx.
’
Giai. D˘t x = asht. Khi d´
a
. o
I= a2 (1 + sh2 t)achtdt = a2 ch2 tdt
ch2t + 1 a2 1
= a2 dt = sh2t + t + C
2 2 2
a2
= (sht · cht + t) + C.
2
√
2 x2 t x+ a2 + x2
V` cht =
ı 1 + sh t = 1 + 2 . e = sht + cht = nˆn
e
√ a a
x + a2 + x2
t = ln v` do d´
a o
a
√ x√ 2 a2 √
a2 + x2 dx = a + x2 + ln|x + a2 + x2| + C.
2 2
V´ du 5. T´
ı . ınh
x2 + 1 3x + 4
1) I1 = √ dx, 2) I2 = √ dx.
x6 − 7x4 + x2 −x2 + 6x − 8

17. 16 Chu.o.ng 10. T´ phˆn bˆt dinh
´
ıch a a .
’
Giai. 1) Ta c´
o
1 1
1+ d x− dt
I1 = x2 dx = x = √
1 1 2 t2 − 5
x2 − 7 + x− −5
x2 x
√ 1 1
= ln|t + t2 − 5| + C = ln x − + x2 − 7 + 2 + C.
x x
2) Ta viˆt biˆu th´.c du.´.i dˆu t´ phˆn du.´.i dang
´ e
e ’ u ´
o a ıch a o .
3 −2x + 6 1
f (x) = − · √ + 13 · √
2 −x2 + 6x − 8 −x2 + 6x − 8
v` thu du.o.c
a .
I2 = f(x)dx
3 1 d(x − 3)
=− (−x2 + 6x − 8)− 2 d(−x2 + 6x − 8) + 13
2 1 − (x − 3)2
√
= −3 −x2 + 6x − 8 + 13 arc sin(x − 3) + C.
V´ du 6. T´
ı . ınh
dx sin x cos3 x
1) , 2) I2 = dx.
sin x 1 + cos2 x
’
Giai
1) C´ch I. Ta c´
a o
dx sin x d(cos x) 1 1 − cos x
= dx = = ln + C.
sin x sin2 x cos2 x − 1 2 1 + cos x
C´ch II.
a
x x
dx d d
= 2 2
sin x x x = x x
sin cos tg · cos2
2 2 2 2
x
d tg x
2
= x = ln tg 2 + C.
tg
2

18. 10.1. C´c phu.o.ng ph´p t´ t´ phˆn
a a ınh ıch a 17
2) Ta c´
o
sin x cos x[(cos2 x + 1) − 1]
I2 = dx.
1 + cos2 x
Ta d˘t t = 1 + cos2 x. T`. d´ dt = −2 cos x sin xdx. Do d´
a
. u o o
1 t−1 t
I2 = − dt = − + ln|t| + C,
2 t 2
trong d´ t = 1 + cos2 x.
o
V´ du 7. T´
ı . ınh
exdx ex + 1
1) I1 = √ , 2) I2 = dx.
e2x + 5 ex − 1
’
Giai
1) D˘t ex = t. Ta c´ ex dx = dt v`
a
. o a
dt √ √
I1 = √ = ln|t + t2 + 5| + C = ln |ex + e2x + 5| + C.
t2 + 5
dt
2) Tu.o.ng tu., d˘t ex = t, exdx = dt, dx =
. a . v` thu du.o.c
a .
t
t + 1 dt 2dt dt
I2 = = − = 2ln|t − 1| − ln|t| + C
t−1 t t−1 t
= 2ln|ex − 1| − lnex + c
= ln(ex − 1)2 − x + C.
` ˆ
BAI TAP
.
T´ c´c t´ phˆn:
ınh a ıch a
e2x 4
1. √
4
dx. (DS. (3ex − 4) 4 (ex + 1)3 )
ex+1 21
’ ˜
Chı dˆ n. D˘t ex + 1 = t4.
a a
.

19. 18 Chu.o.ng 10. T´ phˆn bˆt dinh
´
ıch a a .
√
dx 1 + ex − 1
2. √ . (DS. ln √ )
ex + 1 1 + ex + 1
e2x
3. dx. (DS. ex + ln|ex − 1|)
ex − 1
√
1 + lnx 2
4. dx. (DS. (1 + lnx)3)
x 3
√
1 + lnx
5. dx.
xlnx
√ √
(DS. 2 1 + lnx − ln|lnx| + 2ln| 1 + lnx − 1|)
dx x x
6. . (DS. −x − 2e− 2 + 2ln(1 + e 2 ))
ex/2+e x
√
arctg x dx √
7. √ . (DS. (arctg x)2)
x 1+x
√ 2
8. e3x + e2xdx. (DS. (ex + 1)3/2 )
3
2 +2x−1 1 2x2+2x−1
9. e2x (2x + 1)dx. (DS. e )
2
dx √
10. √ . (DS. 2arctg ex − 1)
ex − 1
e2xdx 1 √
11. √ . (DS. ln(e2x + e4x + 1))
e4x + 1 2
2x dx arc sin 2x
12. √ . (DS. )
1 − 4x ln2
dx √ √
13. √ . (DS. 2[ x + 1 − ln(1 + x + 1)])
1+ x+1
’ ˜
Chı dˆ n. D˘t x + 1 = t2.
a a
.
x+1 √ √ x−2
14. √ dx. (DS. 2 x − 2 + 2arctg )
x x−2 2
dx 2 √ √
15. √ . (DS. ax + b − mln| ax + b + m| )
ax + b + m a

20. 10.1. C´c phu.o.ng ph´p t´ t´ phˆn
a a ınh ıch a 19
dx √ √
16. √ √ . (DS. 3 3 x + 3ln| 3 x − 1|)
3
x( x − 1)
3
dx
17. . (DS. tg(arc sin x))
(1 − x2)3/2
π π
’ a˜
Chı dˆ n. D˘t x = sin t, t ∈
a
. − , )
2 2
dx 1 x
18. . (DS.sin arctg )
(x2 + a2)3/2 a2 a
π π
’ a˜
Chı dˆ n. D˘t x = atgt, t ∈ − ,
a
. .
2 2
dx 1 1
19. . (DS. − , t = arc sin )
(x2 − 1)3/2 cos t x
1 π π
’ a˜
Chı dˆ n. D˘t x =
a
. , − < t < 0, 0 < t < .
sin t 2 2
√
√ a2 x x a2 − x2
20. a2 − x2 dx. (DS. arc sin + )
2 a 2
’ ˜
Chı dˆ n. D˘t x = a sin t.
a a
.
√ x√ 2 a2 √
21. a2 + x2dx. (DS. a + x2 + ln|x + a2 + x2|)
2 2
’ ˜
Chı dˆ n. D˘t x = asht.
a a
.
x2 1 √ 2 √
22. √ dx. (DS. x a + x2 − a2ln(x + a2 + x2) )
a2 + x2 2
√
dx x2 + a2
23. √ . (DS. − )
x2 x2 + a2 a2x
1
’ ˜
Chı dˆ n. D˘t x =
a a
. ho˘c x = atgt, ho˘c x = asht.
a
. a
.
t
x2dx a2 x x√ 2
24. √ . (DS. arc sin − a − x2 )
a2 − x2 2 a a
’ ˜
Chı dˆ n. D˘t x = a sin t.
a a
.
dx 1 a
25. √ . (DS. − arc sin )
x x2 − a2 a x

21. 20 Chu.o.ng 10. T´ phˆn bˆt dinh
´
ıch a a .
1 a
’ a˜
Chı dˆ n. D˘t x = , ho˘c x =
a. a
. ho˘c x = acht.
a
.
t cos t
√ √
1 − x2 1 − x2
26. dx. (DS. − − arc sin x)
x2 x
dx x
27. . (DS. √ )
(a2 + x2)3 a2 x2 + a2
√
dx x2 − 9
28. √ . (DS. )
x 2 x2 − 9 9x
dx x
29. . (DS. − √ )
(x2 − a2)3 a2 x2 − a2
√
30. x2 a2 − x2dx.
x a2 √ a4 x
(DS. − (a2 − x2)3/2 + x x2 − a2 + arc sin )
4 8 8 a
a+x √ x
31. dx. (DS. − a2 − x2 + arc sin )
a−x a
’ a˜ n. D˘t x = a cos 2t.
Chı dˆ a
.
x−a
32. dx.
x+a
√ √ √
(DS. ´
x2 − a2 − 2aln( x − a + x + a) nˆu x > a,
e
√ √ √
´
− x2 − a2 + 2aln( −x + a + −x − a) nˆu x < −a)
e
a
’ a˜
Chı dˆ n. D˘t x =
a
. .
cos 2t
√
x − 1 dx 1 x2 − 1
33. . (DS. arc cos − )
x + 1 x2 x x
1
’ a˜
Chı dˆ n. D˘t x = .
a
. t
dx √
34. √ . (DS. 2arc sin x)
x − x2

22. 10.1. C´c phu.o.ng ph´p t´ t´ phˆn
a a ınh ıch a 21
Chı dˆ n. D˘t x = sin2 t.
’ a˜ a.
√ √
x2 + 1 √ 1 + x2 + 1
35. dx. (DS. x2 + 1 − ln )
x x
x3dx x2 √ 4√
36. √ . (DS. − 2 − x2 − 2 − x2)
2 − x2 3 3
(9 − x2)2 (9 − x2 )5
37. dx. (DS. − )
x6 45x5
x2dx x√ 2 a2 √
38. √ . (DS. x − a2 + ln|x + x2 − a2|)
x2 − a2 2 2
(x + 1)dx xex
39. . (DS. ln )
x(1 + xex) 1 + xex
Chı dˆ n. Nhˆn tu. sˆ v` mˆ u sˆ v´.i ex rˆi d˘t xex = t.
’ ˜ a a ’ o a ˜ o o
´ a ´ ` a
o .
dx 1 x ax
40. . (DS. 3 arctg + 2 )
(x2 + a2)2 2a a x + a2
’ ˜
Chı dˆ n. D˘t x = atgt.
a a.
10.1.3 Phu.o.ng ph´p t´ phˆn t`.ng phˆn
a ıch a u `
a
Phu.o.ng ph´p t´ phˆn t`.ng phˆn du.a trˆn dinh l´ sau dˆy.
a ıch a u `
a . e . y a
D.nh l´. Gia su. trˆn khoang D c´c h`m u(x) v` v(x) kha vi v` h`m
-i y ’ ’ e ’ a a a ’ a a
v(x)u (x) c´ nguyˆn h`m. Khi d´ h`m u(x)v (x) c´ nguyˆn h`m trˆn
o e a o a o e a e
D v`
a
u(x)v (x)dx = u(x)v(x) − v(x)u (x)dx. (10.4)
Cˆng th´.c (10.4) du.o.c goi l` cˆng th´.c t´nh t´ phˆn t`.ng phˆn.
o u . . a o u ı ıch a u `
a
V` u (x)dx = du v` v (x)dx = dv nˆn (10.4) c´ thˆ viˆt du.´.i dang
ı a e o e e’ ´ o .
udv = uv − vdu. (10.4*)
Thu.c tˆ cho thˆy r˘ng phˆn l´.n c´c t´ phˆn t´nh du.o.c b˘ng
. e ´ ´ `
a a ` o
a a ıch a ı . `
a
ph´p t´ phˆn t`.ng phˆn c´ thˆ phˆn th`nh ba nh´m sau dˆy.
e ıch a u ` o e a
a ’ a o a

23. 22 Chu.o.ng 10. T´ phˆn bˆt dinh
´
ıch a a .
Nh´m I gˆm nh˜.ng t´ch phˆn m` h`m du.´.i dˆu t´ phˆn c´ ch´.a
o `
o u ı a a a ´
o a ıch a o u
.a sˆ l` mˆt trong c´c h`m sau dˆy: lnx, arc sin x, arc cos x, arctgx,
u ´
th` o a o a a a
.
(arctgx)2, (arc cos x)2, lnϕ(x), arc sin ϕ(x),...
Dˆ t´ c´c t´ phˆn n`y ta ´p dung cˆng th´.c (10.4*) b˘ng c´ch
’
e ınh a ıch a a a . o u `
a a
. ` o
. a a a ’
d˘t u(x) b˘ng mˆt trong c´c h`m d˜ chı ra c`n dv l` phˆn c`n lai cua
a a o a ` o . ’
a
biˆu th´.c du.´.i dˆu t´ phˆn.
e’ u ´
o a ıch a
Nh´m II gˆm nh˜.ng t´ phˆn m` biˆu th´.c du.´.i dˆu t´ phˆn
o `
o u ıch a a e ’ u ´
o a ıch a
c´ dang P (x)e , P (x) cos bx, P (x) sin bx trong d´ P (x) l` da th´.c, a,
o . ax
o a u
a `
b l` h˘ng sˆ.
a ´
o
’
Dˆ t´ c´c t´ phˆn n`y ta ´p dung (10.4*) b˘ng c´ch d˘t u(x) =
e ınh a ıch a a a . `
a a a
.
P (x), dv l` phˆn c`n lai cua biˆu th´.c du.´.i dˆu t´ phˆn. Sau mˆ i
a ` o . ’
a e’ u ´
o a ıch a ˜
o
lˆn t´ phˆn t`.ng phˆn bˆc cua da th´.c s˜ giam mˆt do.n vi.
` ıch a u
a `
a a ’ . u e ’ o
. .
Nh´m III gˆm nh˜
o `o u .ng t´ch phˆn m` h`m du.´.i dˆu t´ch phˆn c´
ı a a a o a ı ´ a o
ax ax ` ıch a
dang: e sin bx, e cos bx, sin(lnx), cos(lnx),... Sau hai lˆn t´ phˆn
. a
t`.ng phˆn ta lai thu du.o.c t´ch phˆn ban dˆu v´.i hˆ sˆ n`o d´. D´ l`
u `a . . ı a ` . ´
a o e o a o o a
phu .o.ng tr` tuyˆn t´ v´.i ˆn l` t´ phˆn cˆn t´
ınh ´ ’
e ınh o a a ıch a ` ınh. a
Du .o.ng nhiˆn l` ba nh´m v`.a nˆu khˆng v´t hˆt moi t´ch phˆn
e a o u e o e e ´ . ı a
t´ du.o.c b˘ng t´ phˆn t`.ng phˆn (xem v´ du 6).
ınh . ` a ıch a u `
a ı .
Nhˆn x´t. Nh` a
a e o. c´c phu.o.ng ph´p dˆi biˆn v` t´ phˆn t`.ng phˆn
’ ´
a o e a ıch a u `
a
.
ta ch´.ng minh du.o.c c´c cˆng th´.c thu.`.ng hay su. dung sau dˆy:
u . a o u o ’ . a
dx 1 x
1) = arctg + C, a = 0.
x2 +a 2 a a
dx 1 a+x
2) = ln + C, a = 0.
a2 −x 2 2a a − x
dx x
3) √ = arc sin + C, a = 0.
a2 − x2 a
dx √
4) √ = ln|x + x2 ± a2| + C.
x2 ± a2

24. 10.1. C´c phu.o.ng ph´p t´ t´ phˆn
a a ınh ıch a 23
CAC V´ DU
´ I .
√ √
V´ du 1. T´ t´ phˆn I =
ı . ınh ıch a xarctg xdx.
’
Giai. T´ phˆn d˜ cho thuˆc nh´m I. Ta d˘t
ıch a a o
. o a
.
√
u(x) = arctg x,
√
dv = xdx.
1 dx 2 3
Khi d´ du =
o · √ , v = x 2 . Do d´
o
1+x 2 x 3
2 3 √ 1 x
I = x 2 arctg x − dx
3 3 1+x
2 3 √ 1 1
= x 2 arctg x − 1− dx
3 3 1+x
2 3 √ 1
= x 2 arctg x − (x − ln|1 + x|) + C.
3 3
V´ du 2. T´ I = arc cos2 xdx.
ı . ınh
Giai. 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
I = xarc cos2 x + 2 √ dx.
1 − x2
Dˆ t´ t´ phˆn o. vˆ phai d˘ng th´.c thu du.o.c ta d˘t u =
’
e ınh ıch a ’ e ´ ’
’ a u . a
.
xdx
arc cos x, dv = √ . Khi d´
o
1 − x2
dx √ √
du = − √ , v = − d( 1 − x2) = − 1 − x2 + C1
1 − x2
√
’ ` a
a ´
v` ta chı cˆn lˆy v = − 1 − x2:
a
xarc cos x √
√ dx = − 1 − x2arc cos x − dx
21 − x2
√
= − 1 − x2arc cos x − x + C2 .

25. 24 Chu.o.ng 10. T´ phˆn bˆt dinh
´
ıch a a .
Cuˆi c`ng ta thu du.o.c
´
o u .
√
I = xarc cos2 x − 2 1 − x2arc cos x − 2x + C.
V´ du 3. T´ I =
ı . ınh x2 sin 3xdx.
’
Giai. T´ phˆn d˜ cho thuˆc nh´m II. Ta d˘t
ıch a a o
. o a
.
u(x) = x2,
dv = sin 3xdx.
1
Khi d´ du = 2xdx, v = − cos 3x v`
o a
3
1 2 1 2
I = − x2 cos 3x + x cos 3xdx = − x2 cos 3x + I1.
3 3 3 3
` ınh
Ta cˆn t´ I1. D˘t u = x, dv = cos 3xdx. Khi d´ du = 1dx,
a a
. o
1
v = sin 3x. T`. d´
u o
3
1 2 1 1
I = − x2 cos 3x + x sin 3x − sin 3xdx
3 3 3 3
1 2 2
= − x2 cos 3x + x sin 3x + cos 3x + C.
3 9 27
Nhˆn x´t. Nˆu d˘t u = sin 3x, dv = x2dx th` lˆn t´ phˆn t`.ng
a e
. ´ .
e a ı ` ıch a u
a
phˆn th´. nhˆt khˆng du.a dˆn t´ phˆn do.n gian ho.n.
`
a u ´
a o ´
e ıch a ’
V´ du 4. T´ I =
ı . ınh eax cos bx; a, b = 0.
Giai. Dˆy l` t´ phˆn thuˆc nh´m III. Ta d˘t u = eax, dv =
’ a a ıch a o
. o a
.
1
cos bxdx. Khi d´ du = aeaxdx, v = sin bx v`
o a
b
1 a 1 a
I = eax sin bx − eax sin bxdx = eax sin bx − I1 .
b b b b
’
Dˆ t´ I1 ta d˘t u = eax, dv = sin bxdx. Khi d´ du = aeaxdx,
e ınh a. o
1
v = − cos bx v`
a
b
1 a
I1 = − eax cos bx + eax cos bxdx.
b b

26. 10.1. C´c phu.o.ng ph´p t´ t´ phˆn
a a ınh ıch a 25
Thˆ I1 v`o biˆu th´.c dˆi v´.i I ta thu du.o.c
´
e a e’ u o o´ .
1 a a2
eax cos bxdx = eax sin bx + 2 cos bx − 2 eax cos bxdx.
b b b
Nhu. vˆy sau hai lˆn t´ phˆn t`.ng phˆn ta thu du.o.c phu.o.ng
a
. ` ıch a u
a `
a .
tr` tuyˆn t´ v´.i ˆn l` I. Giai phu.o.ng tr` thu du.o.c ta c´
ınh ´
e ınh o a a ’ ’ ınh . o
a cos bx + b sin bx
eax cos bxdx = eax + C.
a2 + b2
V´ du 5. T´ I =
ı . ınh sin(ln x)dx.
1
’
Giai. D˘t u = sin(lnx), dv = dx. Khi d´ du = cos(lnx)dx,
a. o
x
v = x. Ta thu du.o.c
.
I = x sin(lnx) − cos(lnx)dx = x sin(lnx) − I1.
’
Dˆ t´
e ınh I1 ta lai d˘t u = cos(lnx), dv = dx. Khi d´ du =
. a . o
1
− sin(lnx)dx, v = x v`
a
x
I1 = x cos(lnx) + sin(lnx)dx.
Thay I1 v`o biˆu th´.c dˆi v´.i I thu du.o.c phu.o.ng tr`
a e’ u o o´ . ınh
I = x(sin lnx − cos lnx) − I
v` t`. d´
a u o
x
I= (sin lnx − cos lnx) + C.
2
Nhˆn x´t. Trong c´c v´ du trˆn dˆy ta d˜ thˆy r˘ng t`. vi phˆn d˜
a e
. a ı . e a a a `
´ a u a a
´
biˆt dv h`m v(x) x´c dinh khˆng do
e a a . o .n tri. Tuy nhiˆn trong cˆng th´.c
e o u
.
a ’
o e . a a ´
(10.4) v` (10.4*) ta c´ thˆ chon v l` h`m bˆt k` v´
a y o .i vi phˆn d˜ cho
a a
dv.

27. 26 Chu.o.ng 10. T´ phˆn bˆt dinh
´
ıch a a .
V´ du 6. T´
ı . ınh
xdx dx
1) I = ; 2) In = , n ∈ N.
sin2 x (x2 + a2)n
Giai. 1) R˜ r`ng t´ phˆn n`y khˆng thuˆc bˆt c´. nh´m n`o
’ o a ıch a a o ´
o a u
. o a
dx
trong ba nh´m d˜ nˆu. Thˆ nhu.ng b˘ng c´ch d˘t u = x, dv =
o a e ´
e `
a a a
.
sin2 x
v` ´p dung cˆng th´.c t´ phˆn t`.ng phˆn ta c´
aa . o u ıch a u `
a o
cos x
I = −xcotgx + cotgxdx = −xcotgx + dx
sin x
d(sin x)
= −xcotgx + = −xcotgx + ln| sin x| + C.
sin x
2) T´ phˆn In du.o.c biˆu diˆn du.´.i dang
ıch a . ’
e ˜
e o .
1 x2 + a2 − x2 1 dx x2 dx
In = 2 2 + a2 )n
dx = 2 2 + a2 )n−1
−
a (x a (x (x2 + a2)n
1 1 2xdx
= 2 In−1 − 2 x 2 ·
a 2a (x + a2)n
Ta t´ t´ phˆn o. vˆ phai b˘ng phu.o.ng ph´p t´ch phˆn t`.ng
ınh ıch a ’ e ´ ’ ` a a ı a u
2xdx d(x2 + a2)
`
phˆn. D˘t u = x, dv = 2
a a
. = 2 . Khi d´ du = dx,
o
(x + a2 )n (x + a2 )n
1
v=− v`
a
(n − 1)(x2 + a2)n−1
1 2xdx −x 1
x = 2 + 2 In−1
2a2 (x2+a 2 )n 2a (n − 1)(x2 + a2 )n−1 2a (n − 1)
T`. d´ suy r˘ng
u o `
a
1 x 1
In = I
2 n−1
+ 2 2 + a2 )n−1
− 2 In−1
a 2a (n − 1)(x 2a (n − 1)
hay l`
a
x 2n − 3
In = + 2 In−1 . (*)
2a2 (n − 1)(x2 + a2 )n−1 2a (n − 1)

28. 10.1. C´c phu.o.ng ph´p t´ t´ phˆn
a a ınh ıch a 27
Ta nhˆn x´t r˘ng t´ phˆn In khˆng thuˆc bˆt c´. nh´m n`o trong
a e `
. a ıch a o . ´
o a u o a
a ’
ba nh´m d˜ chı ra.
o
Khi n = 1 ta c´o
dx 1 x
I1 = = arctg + C.
x2 + a2 a a
Ap dung cˆng th´.c truy hˆi (*) ta c´ thˆ t´nh I2 qua I1 rˆi I3 qua
´ . o u `
o ’
o e ı `
o
I2,...
V´ du 7. T´ I =
ı . ınh xeax cos bxdx.
Giai. D˘t u = x, dv = eax cos bxdx. Khi d´ du = dx,
’ a
. o
a cos bx + b sin bx
v = eax
a2 + b2
(xem v´ du 4). Nhu. vˆy
ı . a
.
a cos bx + b sin bx 1
I = xeax 2 + b2
− 2 eax(a cos bx + b sin bx)dx
a a + b2
a cos bx + b sin bx a
= xeax 2 + b2
− 2 eax cos bxdx
a a + b2
b
− 2 eax sin bxdx.
a + b2
T´ phˆn th´. nhˆt o. vˆ phai du.o.c t´nh trong v´ du 4, t´ phˆn
ıch a u ´
a ’ e ’ ´ . ı ı . ıch a
th´. hai du.o.c t´ tu.o.ng tu. v` b˘ng
u . ınh . a ` a
a sin bx − b cos bx
eax sin bxdx = eax ·
a2 + b2
Thay c´c kˆt qua thu du.o.c v`o biˆu th´.c dˆi v´.i I ta c´
a e ´ ’ . a ’
e u o o´ o
eax a
I= x− (a cos bx + b sin bx)
a2 + b2 a2 + b2
b
− (a sin bx − b cos bx) + C
a2 + b2
` ˆ
BAI TAP
.

29. 28 Chu.o.ng 10. T´ phˆn bˆt dinh
´
ıch a a .
2x (x ln 2 − 1)
1. x2x dx. (DS. )
ln2 2
2. x2 e−x dx. (DS. −x2e−x − 2xe−x − 2e−x )
2 1 2
3. x3 e−x dx. (DS. − (x2 + 1)e−x )
2
1 5x 3 3 2 31 31
4. (x3 + x)e5xdx. (DS. e x − x + x− )
5 5 25 125
√
5. arc sin xdx. (DS. xarc sin x + 1 − x2 )
1 1 √
6. xarc sin xdx. (DS. (2x2 − 1)arc sin x + x 1 − x2)
4 4
x3 2x2 + 1 √
7. x2 arc sin 2xdx. (DS. arc sin 2x + 1 − 4x2)
3 36
1
8. arctgxdx. (DS. xarctgx − ln(1 + x2))
2
√ √ √
9. arctg xdx. (DS. (1 + x)arctg x − x)
x4 − 1 x3 x
10. x3 arctgxdx. (DS. arctgx − + )
4 12 4
x2 + 1 1
11. (arctgx)2xdx. (DS. (arctgx)2 − xarctgx + ln(1 + x2))
2 2
√
12. (arc sin x)2dx. (DS. x(arc sin x)2 + 2arc sin x 1 − x2 − 2x)
arc sin x √ √
13. √ dx. (DS. 2 x + 1arc sin x + 4 1 − x)
x+1
√
arc sin x arc sin x 1 + 1 − x2
14. dx. (DS. − − ln )
x2 x x
xarctgx √ √
15. √ dx. (DS. 1 + x2arcrgx − ln(x + 1 + x2))
1 + x2

30. 10.1. C´c phu.o.ng ph´p t´ t´ phˆn
a a ınh ıch a 29
√
arc sin x √ √ √
16. √ dx. (DS. 2( x − 1 − xarc sin x))
1−x
17. ln xdx. (DS. x(ln x − 1))
√ 2 3/2 4 8
18. x ln2 xdx. x
(DS. ln2 x − ln x + )
3 3 9
√ √ √
19. ln(x + 16 + x2)dsx. (DS. x ln(x + 16 + x2) − 16 + x2 )
√
x ln(x + 1 + x2) √ √
20. √ dx. (DS. 1 + x2 ln(x + 1 + x2) − x)
1 + x2
x
21. sin x ln(tgx)dx. (DS. ln tg − cos x ln(tgx))
2
(x3 + 1) ln(x + 1) x3 x2 x
22. x2 ln(1 + x)dx. (DS. − + − )
3 9 6 3
1 − 2x2 x
23. x2 sin 2xdx. (DS. cos 2x + sin 2x)
4 2
1
24. x3 cos(2x2)dx. (DS. (2x2 sin 2x2 + cos 2x2))
8
ex (sin x − cos x)
25. ex sin xdx. (DS. )
2
sin x + (ln 3) cos x x
26. 3x cos xdx. (DS. 3 )
1 + ln2 3
e3x
27. e3x(sin 2x − cos 2x)dx. (DS. (sin 2x − 5 cos 2x))
13
28. xe2x sin 5xdx.
e2x 21 20
(DS. 2x + sin 5x + − 5x + cos 5x )
29 29 29
1 2
29. x2ex sin xdx. (DS. (x − 1) sin x − (x − 1)2 cos x ex)
2

31. 30 Chu.o.ng 10. T´ phˆn bˆt dinh
´
ıch a a .
2 x (x − 1)2 sin x + (x2 − 1) cos x x
30. x e cos xdx. (DS. e )
2
[3 sin x(ln x) − cos(ln x)]x3
31. x2 sin(ln x)dx. (DS. )
10
32. T` cˆng th´.c truy hˆi dˆi v´.i mˆ i t´ phˆn In du.o.c cho du.´.i
ım o u ` o o
o ´ ˜
o ıch a . o
dˆy:
a
1 n ax n
1) In = xn eaxdx, a = 0. (DS. In = x e − In−1 )
a a
2) In = lnn xdx. (DS. In = x lnn x − nIn−1 )
xα+1 lnn x n
3) In = xα lnn xdx, α = −1. (DS. In = − In−1 )
α+1 α+1
√
xn dx xn−1 x2 + a n − 1
4) In = √ , n > 2. (DS. In = − aIn−2 )
x2 + a n n
n cos x sinn−1 x n − 1
5) In = sin xdx, n > 2. (DS. In = − + In−2 )
n n
sin x cosn−1 x n − 1
6) In = cosn xdx, n > 2. (DS. In = + In−2 )
n n
dx sin x n−2
7) In = nx
, n > 2. (DS. In = n−1 x
+ In−2 )
cos (n − 1) cos n−1
10.2 C´c l´.p h`m kha t´ trong l´.p c´c
a o a ’ ıch o a
h`m so. cˆp
a a´
10.2.1 T´ phˆn c´c h`m h˜.u ty
ıch a a a u ’
1) Phu.o.ng ph´p hˆ sˆ bˆt dinh. H`m dang
. ´ ´
a e o a . a .
Pm (x)
R(x) =
Qn (x)

32. 10.2. C´c l´.p h`m kha t´ trong l´.p c´c h`m so. cˆp
a o a ’ ıch o a a ´
a 31
trong d´ Pm (x) l` da th´.c bˆc m, Qn (x) l` da th´.c bˆc n du.o.c goi l`
o a u a . a u a . . . a
h`m h˜
a u.u ty (hay phˆn th´.c h˜.u ty). Nˆu m
’ a u u ’ ´
e n th` Pm (x)/Qn (x)
ı
du.o.c goi l` phˆn th´.c h˜.u ty khˆng thu.c su.; nˆu m < n th`
. . a a u u ’ o . . ´
e ı
Pm (x)/Qn (x) du ..o.c goi l` phˆn th´.c h˜.u ty thu.c su..
u u ’ . .
. a a
´
Nˆu R(x) l` phˆn th´ u ’
e a a u.c h˜.u ty khˆng thu.c su. th` nh`. ph´p chia
o . . ı o e
da th´.c ta c´ thˆ t´ch phˆn nguyˆn W (x) l` da th´.c sao cho
u ’
o e a `a e a u
Pm (x) Pk (x)
R(x) = = W (x) + (10.5)
Qn (x) Qn (x)
trong d´ k < n v` W (x) l` da th´.c bˆc m − n.
o a a u a.
T`u. (10.5) suy r˘ng viˆc t´ t´ch phˆn phˆn th´.c h˜.u ty khˆng
`
a e ınh ı a a u u ’ o
.
.c su. du.o.c quy vˆ t´nh t´ phˆn phˆn th´.c h˜.u ty thu.c su. v` t´ch
thu . ` ı
e ıch a a u u ’ . . a ı
. .
phˆn mˆt da th´
a o u.c.
.
Dinh l´ 10.2.1. Gia su. Pm (x)/Qn (x) l` phˆn th´.c h˜.u ty thu.c su.
-. y ’ ’ a a u u ’ . .
v`
a
Q(x) = (x − a)α · · · (x − b)β (x2 + px + q)γ · · · (x2 + rx + s)δ
trong d´ a, . . . , b l` c´c nghiˆm thu.c, x2 + px + q, . . . , x2 + rx + s l`
o a a e
. . a
nh˜.ng tam th´.c bˆc hai khˆng c´ nghiˆm thu.c. Khi d´
u u a . o o e
. . o
P (x) Aα A1 Bβ Bβ−1
= + ··· + + ··· + + + ···+
Q(x) (x − a)α x−a (x − b)β (x − b)β−1
B1 Mγ x + Nγ M1 x + N1
+ + 2 + ··· + 2 + ···+
x − b (x + px + q)γ x + px + q
Kδ x + Lδ K1 x + L1
+ 2 δ
+ ··· + 2 , (10.6)
(x + rx + s) x + rx + s
trong d´ Ai, Bi , Mi , Ni , Ki v` Li l` c´c sˆ thu.c.
o a a a o . ´
a a u.c o. vˆ phai cua (10.6) du.o.c goi l` c´c phˆn th´.c do.n
C´c phˆn th´ ’ e ’ ’´ . . a a a u
gian hay c´c phˆn th´.c co. ban v` d˘ng th´.c (10.6) du.o.c goi l` khai
’ a a u ’ a a ’ u . . a
’ u.c h˜.u ty thu.c su. P (x)/Q(x) th`nh tˆng c´c phˆn th´.c
triˆn phˆn th´ u ’ . .
e a a o’ a a u
. ban v´.i hˆ sˆ thu.c.
co ’ . ´
o e o .
’ . ´ ’
Dˆ t´ c´c hˆ sˆ Ai , Bi , . . . , Ki , Li ta c´ thˆ ´p dung
e ınh a e o o ea .