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Xufi't
ban nam 2013
La>i
noi dau
Viec
giai
mot bai toan noi chung la mot qua
trinh
tu duy cao do, dua tren
hilu
biet cua nguai
giai
toan. Viec
tinh
mot bai toan nguyen ham hay mot bai
toan
tich
phan cung vay. Co nguai tham chi khong
giai
dugc, c6 nguai
giai
dugc nhimg can qua
trinh
may mo rat lau, thu het
each

mang
tinh
ap dat ma theo huang de hieu, de nha de nguai doe c6 thien cam
han ve cac cong thuc do, phuc vu cho viec van dung
tinh
toan sau nay.
Cuon
sach
viet theo loi dien giang nen kho tranh
khoi
khiem khuyet, rat
mong nhan dugc nhung gop y thiet thuc
ciia
ban doe gan xa.
Xin
chan thanh cam an nhung gop y, chi dan cua quy thay:
-
TS. Nguyen Viet Dong, Truang Bo mon Giao due Toan hoc,
DHKHTN,
DHQG
TP. H6 Chi
Minh.
-
Thhy Nguyen Dinh Do, Pho
Hieu
truong Truang THPT Thanh Nhan -
TP.
Ho Chi
Minh.
-

•
(E>H kh6i A, Ai
- 2013)
Cdch
1: /=
——.lnxdx=
\nxdx
+
Cdch
gidi thong
thir&ng
2
w -I 1
In
xdx. Ta xet:
+
A
=
In
xdx.
Dat
u
= lnx=>
du
=
—dx
;
dv
= dx=>
v

\X
J
f
1
\
,
1
rfX =>
V
= — .
X
1,
=
—
Inx
X
2
^
1
f
1
dx
X
=
—Inx
X
2
I
+
-

, \n 2 In 2
1= ^\\—L-\e'dt=
\[e'-e")dt=
\td(e'+6-)
0 V
= t(e'+e-)
In
2
0
In 2
• '(e'+e-')dt
=
ln2.
V
2,
2-i
2
.l(51n2-3).
Vay
/
=
|(51n2-3).
Cac/i
^w/
nhanh
Cdch
3: cdc ban di y
quan
he
giita

2 2
1 =
x'-\
.
In
xdx
=
1-
1 ^
=
ln
X.
x
+ -
2 V O
-
x
+ -
1
rv
In
xdx
=
In
xd
*
1
X
x
+

hai cdch tren !
Cau
2;
Tinh
tich
phan
/
=
^x^l-x'dx
(DH
kh6i
B - 2013
Cdch
giai thong thu&ng
Cdch
1: Do
dau
hieu
"
nen ta
chon
an
phu
x = V2
sin?.
Dat
X
=
y/l
sin

cos/L//
=
2V2
sin/cos
tdt.
Xet
tich
phan
J = 272
|sin
/
cos^
tdt.
Dat
u =
cos/
^du =
-s'mtdt.
K
72
Doi
can
:
/
=
0 =o
w
= 1;/=
— =>
w

Dat
/
= 72-x^ ^t^ =2-x^
=>tdt
= -xdx.
D6i
can:'
x =
0 =>
/
= 72;x =
1
=i>
/
=
1.
V2 2V2-1
Taco: I = -\t^dt= \t^^t= —
Cdch
gidi
nhanh
Cdch 3: Cdc ban de y quan he giua x vd x^ la:
xdx =
~^d{^x~^
=
-^d{2-x^y
Nen viec ta chon an phu t (0
cdch
2) la hodn todn tu nhien ! khong mang tinh dp dat cua kinh nghiem trong
suy nghi Id :

+
1)^
,
-<lx (DHkh6iD-2013)
x +1
Cdch
gidi
thong
thirong
Cdch 1: Ta c6 :
/
=
(x
+ 1)' + U V. V 2x
x'+\ x'+\
-dx = dx +
0
0
-dx = \
r
2x
-dx
Xet
tich
phan
J =
2x
x'+\
•dx.
Dat / = +1

2x
x'+l
-dx
Dat x = tan/=>(ix =
—^—dt
= {\-^ian-t]dt, ti
cos"/ ^ '
Doican: x = 0 / = 0;x =
1
=^/ = —.
4
2
'2
Tadugc
/=
f^^(l
+ tan^/)^/ = 2 f^c//
tan / +
1
V
1
=
-2 :
of
(cos/) = -2 In
J cos/
1
sin/
<
cos/

0
=
X
+
x^+1
i/(x'+l)
=x
0
+
ln
x^+l
=
l + ln2.
LM
gidi
that
nhanh gon !
D6
CO
each
nhin
"tudng
minh" vh
each
giai
nhanh
Nguyen ham va
Tich
phan, mai ban doc tim hieu nhirng kien giai trong cuon
sach

F(x)
la mot nguyen ham
ciia
ham f(x) thi F(x) + C (C la hang s6) la ho
nguyen ham cua ham f(x) hay
tich
phan hk
dinh
cua ham
f(x).
Ki
hieu :
fix)dx
=
F{x)
+
C
Vi
du 1
a)
J2xdx
=
x^+C vi
(x'+C)'
= 2x.
b)
cosxdx
=
smx
+

''muon
tim nguyen ham
ciia
ham so f(x),
chiing
ta tim ham so md dgo ham bgc nhat
cm no phdi chinh
la f(x)'\i
each
hieu do, chung ta c6 the thanh lap Bang
cong thuc nguyen ham co ban nhu sau :
(1)
Cong thirc 1 :
Qdx
=? Ta suy nghi : ham so nao c6 dao ham bac nhat
bang 0?
Hien
nhien do la hang so
!
Vay ta c6 cong thuc
thii
nhSt:
Qdx =
C
(2)
Cong thii-c 2 :
\dx=l
Ta suy nghi : ham so nao c6 dao ham bac nhat
bang 1? De dang nhan thay do la
X

x" . Vay la ham so c6 dao ham bac nhat bang x". Suy ra
a
+
\
cong thuc thu ba :
a+\
x"dx=-—
+
C
ar
+
1
(a^-1).
(4) Cong thuc 4 : f—c/x =? Ta suy
nghi:
ham so nao c6 dao ham bac nhat
bang — Ta
lien
tuong toi cong thuc (inx) =— thi thu
duac
cong thuc
X
9 X
\-dx^\nx+C
J
V
.
Chiing
ta lay dau gia
tri

\na
(a>0,fl^l).
(6)
Cong thuc 6 : e''dx=? Ta suy
nghi:
ham so nao c6 dao ham bac nhSt
J
bang e"') De dang ta nhan thay do la ham e' vi (e'') =€' , suy ra cong
thiic
thu
sau : e^dx
=
6"
+
C . Cong thuc
thii
sau la truofng hgp rieng
ciia
cong
thiic
thii
nam khi thay "a" bang "e" !
(7) Cong thu-c 7 : jcosxdx =? Ta suy nghi : ham so nao c6 dao ham b$c
nhk bang
cosx?
Tir cong thuc quen thuoc
(sinx)
=cosx, ta c6 ngay cong
thuc thu bay la :
cosxi/x

"cos^x").
Ta c6:
cos
a;
A.
cosx +
sinx.
A
,
ro rang neu chon A = sinx thi
cos X
A.
COS
X
+ sinx.A cos^x + sin^o;
cos^ X
cos
X cos X

Vay ham so c6 dao ham bac
'
^ 1 '
Sill
X
nhat bang —-— la ham so hay tan x. Suy ra cong thuc thu
chin
:
cos X
cos a;
r

. Vay ham so c6 dao ham bac nhat bang ^
sin^
X
sin^
X
ham so
-
cos X
sinx
hay (-
cotx).
Suy ra cong thuc thu muai:
sin^
X
dx = - cot X + C
Vay ta c6
Bang
nguyen ham ca ban sau :
Jodx
= C
f
adx = + C{cy >
0;
Q ^ 1)
In
a
J
dx = X + C
y
cos xdx

tich
phan sau nay. Chinh vi vay, chung ta can
su dung thanh thao cac cong thuc trong bang nguyen ham ca ban.
3.
Tinh
chat thu- hai
J
kf{x)dx
= kj
f{x)dx
Trong
cong thuc nay, dieu ma chung ta can chu y la he s6 "k" (he so k c6
the "ra", "vao" qua dau nguyen ham!), tat nhien k phai la hang so, con bien so
khong
dua ra ngoai dSu nguyen ham dugc.
Vi
du 2. Ap dung
tinh
chSt thu hai va Bang nguyen ham ca ban, ta c6 :
a)
J 6xdx = GJ xdx (dp dung tinh chat thu hai)
=
6 — + C (dp dung bang nguyen ham ca ban)
2
=
3a;' + C .
,
. r cos X , If
1
cos xdx =

tich
phan xuat hien
nhung ham s6 c6 trong bang nguyen ham ca ban. Do vay, viec nam
dugc
Bang
nguyen ham co ban la di^u
kien
rSt quan trong de chung ta
tinh
dugc
nguyen
ham,
tich
phan.
4.
Tinh
chat
thir
ba
J" {J{x) ±
g{x))dx
=
J
f{x)dx
± J
g{x)dx
Chung ta c6 the hieu mot
each
dan gian cong
thiic

(dp dung tinh chat
thu ba)
=
4 J" xdx +3
J
cos xdx (dp dung tinh chdi thu hai)
=
4 h
3
sin
a;
+ C (dp dung bang nguyen ham ca ban)
•
2
=
2a;^ + 3 sin X + C
b)
/ = r
(5e"
^)dx = r
Se'dx-
f dx (dp dung tinh chdt
'
cos' x *^
thif
ba)
cos' x
=
5
r

Sx' 2x 4
+
-
XXX
dx
J
Sxdx - J 2dx + j
:
3Jxdx-2Jdx +
AJ-dx
^ — -2x + A\nx + C.
X
2
Trong
thuc hanh, ta
trinh
bay nhanh nhu sau :
a) I,
=
Ji4x
/(5e
c)
I,
\
J 33;'
3x'
-^)dx
=5e^ - 7 tan x + C.
cos X
-dx

-dx.
42
Gidi
Ta
bien doi ham so dual ddu nguyen ham ve dang ham cd chira cdc ham
trong
bdng nguyen ham co bdn de
tinh.
a) Ta CO :
-I
X
+ 1)^
dx
X
-I
X
dx
X
1
+
2 1
2 1
1
+ —+ -
X
^
dx = X- + 4^2!^ +
In
X + C.
dx

Taco
: =
\—;<ix
=
4'
42
V /
-dx =
ce
-dx-^ dx =
e
v2y
V
^ /
In
'e^
v2/
+ C.
Vi du 5. Cho ham s6 f{x) = xe' va F{x) = {ax + b)e''. Vai gia
tri
nao cua a va
b
thi la mot nguyen ham ciia f(x) 9
Gidi
Tap xac
djnh
cua F(x) va /(x) la R. Ham s6 F(x) la mpt nguyen ham
cua f(x) thi
/^'(^)
= fix) vai Vx e R .

x)e''.
(dpcm)
BAITAP
1.
Tinh
:
25a.''
+122'+ 1991
X
dx;
X
dx; b) 7, = J
15a;
+ 10V^ + 1983
da;.
2.
Tinh
:
a)=
J'^Ssina;
—
4cosa;jda;;
b) 7^ = J"
tan
a;
a;'
-
2a;
+ -
X

c)e'. Vai gia
tri
nao
ciia
a, b va c thi F{x) la mot nguyen ham cua /{x) ?
Bai
2.
BANG NGUYEN HAM
MQ
RONG
Sail
day chung ta se ma
rong
cac cong thuc nguyen ham ca ban de dugc
Bang
nguyen ham ma
rong.
Bang
nguyen ham ma
rong
la cong cu giup chung
ta
tinh nhanh nguyen ham va
tich
phdn.
Truac
tien ta xet dinh li sau :
1. Dinh li
Nh J f(u)du =F{u) + C vau = u(x) la ham s6 c6 dgo ham
lien

bydx
=-F{ax
+
b)
+
C
Tom lai,
ta c6 cong thuc dk mo rong bang nguyen ham ca ban:
'/{ax
+
b)dx
=
-F(ax
+
b)
+
C
a
Cong
thuc nguyen ham ca ban va cong thuc nguyen ham ma rpng
dugc
cho
tuang ung duai bang sau :
Cong
t/iii'c
nguyen
ham
cff
ban
Cong

^
ax +
b
a
J
e'dx = +C
f
e'"'-'dx
= -e'^^'' +C
'J
a
f
adx = — +
C{a>Q]a^
1)
In
a
ra'^^'dx = ±.^
+ C{a
>
0;a
^
1)
^
a \na
j
cos xdx = sin x
-\-
C
/

b)
+ C
^
cos (ax +
b)
a
f
—\ dx
=
—
cot x + C
^
sin X
r
1 1
/
— dx = cot(ax +
6)-1-C
sin (ox+ 6)
a
Trong
thuc hanh
tinh
nguyen ham cung nhu
tich
phan sau nay, a nhieu
truang hop viec ap dung bang nguyen ham ma rong cho ta lai
giai
bai toan
nhanh va "

Sa;" + 8a;' + 4a;^ + a; + C.
16x-'
5
Bay gia chung ta ap dung cong thuc nguyen ham ma rong (cong
thuc r{ax +
bydx
=
^"•'^
^ + C(a ^ -1)) d^
tinh/,
ta c6 :
^
o a +1
«^
^ ^ 2 4 +
1
10
(C/zw
>•
rang,
each
nay va
each
tren
deu cho kit qua
dung,
no chi sai
khdc
nhau
mot

dugc
cong thuc nguyen ham ca ban.
Vi du 1.
Tinh
:
a) = J (2a; + s)'' dx; b) = J'(3 - 2a;)' dx
(3i-l)
^(33;-2)
dx
•
Giai
Ap
dung cac cong thuc
trong
Bang nguyen ham ma rong ta c6 :
/
\13 / vl3
c, xi2 1 2a;+ 3 2a; + 3
a) / = r 2a; + 3 dx = ^.^ = ^ -L-
'
^ J \ 2 13 26
r, v3 1 (3-2a;f
-(3-2a;)'
b) / = / 3-2a; dx = ^ L + =
__V
L
+
C.
r
I ci \-7 1

2a;+
3)
a)/,=/
12
23;+
3 dx =
vl3
26
3
—
2xI dx =
-(3-2.)'
8
dx
1
-5
2 .,,.2_(3x-2)3
18(3a;-l)'
-1
3
-
D6i vai Cdu c) vd cdu d), neu sir
dung
cong
thitc
#x-2)^
+
C.
•dx = —
-1

Dat
M = 2a: + 3 =>
(iti
= 2dx rfx = - du
2
Ta
CO
5:/ = \v}^-du = — + — + C
1 J r, cy -
2
13 26
Thay
u = 2x + 3, ta duac: /, = ^ + C.
•
^ 26
Rd
rang cdch nay to ra khd phuc tap so vai mot bdi todn dan gidn nhu vdy !
Ap
dung cong thuc nguyen ham ma rong ta c6 lai gidi gon vd nhanh han:
\13
12
2a; + 3 dx =
2x + 3
26
+
C
Do
vdy, viec nha vd van dung tot cong thuc nguyen ham ma rong Id can
thiet de chung ta tinh nhanh duac nguyen hdm vd tich phdn sau nay.
Vi

2
—
5a;
+
C =
—In
5
2-5a'
+
C.
b)
7 = f S'^'^dx = 3 r3'^ia; = S —+ C =+ C•
'J J 5 ln3 5 ln3
c)
l3= \e ^''dx = ^e ^''+C = -3e +C.
3
d)
7 =
fe-''dx
=
—e-'+C
=
-e-^+C.
-
Chung ta c6 the trinh bdy nhanh nhu sau :
2-5x
+
C a) 7 = / —-— dx =
—^
In

Bang
nguyen
ham ma
rong
ta c6 :
a) /j =
Jsin(3
-
Ax)dx
= ^.(-cos(3 - 4a;)) + C
cos(3 - 4a;) ^
4
b) h = JScos
\ /
dx =
5.—.sin
+
C 15sin
+
C
\ /
c)
I = f dx = f ^ dx =
-temSx
+ C-
cos\3x)
3
^)
I. =
cos'(3.x)

J
(4x + 2)' dx;
b) ^. = /-^^-;
^
(3-2x)
^(41 + 5)
2. Tinh:
c)/2= |e 3 Jx;
b) = J
4 J
3. Tinh :
r(sin(3 ) +
cos53;)t/x;
b) / f ^
2 ^ cos (4
—
x)
1
.a:
—
h sm
—
sin 3a; 2
;
d)
^4
=/
2cos(3 )rfa;.
Bai
3.

du^ - du^(x)
Buoc 3 : Bieuthi f(x)dx
theo
u va du. Gidsurang
f
(x)dx
=
g{u)du.
Buoc 4 : Tinh I
=
g(u)du. Sau do
thay
u = w(x) de
dime
ket qua can tim.
Chu V
:
chon
an phu u = w(x) sao cho
viec
tinh / = g(u)du phai de hon la
tinh
/= jf(x)dx !
Khi
nhin vao mot bai giai cho bai
toan
tinh
nguyen
ham hay tich
phan

nguyen
ham, tich
phan
cua cac ban.
Truoc
tien cac ban c^n luu y hai ket qua ma chiing ta
thuong
dung
sau day :
(1) df{x) = f\x)dx .
(2) Niu J f(u)du =F{u) + C va u = u(x) la ham sS c6 dqo ham lien tuc
thl:
J
f{u{x)).u\x)du
=
J
i\u{x))du{x)
=F{u{x))
+ C
Vi
du
a)
J cos(2x^ + 3a; +
l)d{2x^
+ 3a; + 1) = sin(2a;^ + 3a,' + 1) + C.
(ta
hieu trong suy nghi "
2x^
+ 3x + 1 " lau)
b)

+
\)x"dx o x"dx
=
—dx"*' '
n
+
\
+
\)
d{ax"^'+b),
trong
do a^O con b tuy y tren R . Vay ta c6 quan he
giira
x" va
x"^\n^-1)
nhu
(ta
hieu cong thicc tren mot each don gidn
sau :
x"dx =
-
1
a{n +
\)
d{ax"''+b)
nhu sau : dua x" vdo trong vi phan thl thdnh
{ax"*^
-^b), voi a ^ Q vd h tuy
ytren
Vidu

+ ifdx =
-(2a;'
+
lfd{2x^
+ 1).
6 6
Do
do, viec chung ta chon an phu la u-2x^
+
\ hodn todn ty nhien, khong
mang tinh dp dat.
L&i
giai
cua bai
todn
J
x'(2x'
+ Ifdx =
J-(2x'
+
lfd{2x'
+ !)•
Dat u = 2x^ +1 du = d(2x^ +1).
Taco: T = - / uMu = + C = + C-
'6-^
6 10 60
Thay u = 2x^+1 ta
duac:
/ +
'

d(x' + l).
Dat u =
Vx'+l
^u'
=x'+l:^du'
=:d(x'+l).
1 1 ^
Taco: = Jiu du' = J-u 2udu =
JuMu
=•— + 0-
/"I—T
t A (Vx^ + 1 )
Thay u ^ Vx +1 ta dugc: i =
'
3
Cach
khac :
'
Dat u = x^+l=:>du =
d(x'+l).
3
Ta c6:
I2
= j-Judu
=
—
ju^du =
+
C
= ^^^^^

nen quan he can xet giila — va ^ la:
X X X
(ta hieu cong thuc tren mot each don gidn nhu sau : dua \ trong vi
phdn thi thanh — +
b,
voi a ^ 0 vdbtuyy tren R)
X
Vi
du 2. Tinh :
l4
re ^
a)
Ii = \—r-dx;
X
r I 1 1
b) / = I — sin — cos — dx.
^ X X X
Gidi
a) Phdn tick bdi todn : Niu chua dugc biet din quan he giua \ — thi
X X
that khong de de chung ta tim ra ngay phep dgt an phu! Cdc ban de y quan he
.,1,11,
-1 ,
giua — va — ; —
dx
= — a
X X 3
1+-
X
nen ta co

Thay u =
1
+ — ta duac: Ii = —e ^ + C.
X
• 3
b) Phdn tick bdi todn : Cdc ban de y quan he giua va —
X X
1
dx = —d
ta
CO
the chon an phu Id u
=
— .
X
.,.1.1 1 , .1 1 ,
nen ta co — sm — cos — dx — - sin — cos —
d
XXX XX
1
. Do do,
L&i gidi cua bdi todn
1 r • 2 J
1
= / sin —d
—
X
2^
X
X

4
1
3. Quan he giua — va line
X
Ta CO (In x) =
—
nen quan he can xet giiia — va In x la :
— dx = — d{a \nx + b)
X
a
(ta hieu
cong
thuc tren mot each dan gidn nhu sau : dua — vdo trong vi
X
phdn
thi thanh {a In x
-\-h),
vai a ^Ovdb tiiyy tren R)
Vi
du 3. Tinh :
2
In
X + 3
-dx;
X
b)I,=/
21n'x + 5ln'x
xlnx
dx.
Gidi:

X + 3).
Ta CO
6:1
=ifuMu=1.^:^+0=^+0.
1 2 10 20
Thay u = 21nx + 3 tadugc: / ^
(2Inx
+ 3)^" ^ ^
' 20
b)
Phdn tich bai todn: Cdc ban dSy quan he giua — va \nx : —dx = d{\n x)
x X
2ln^x + 51n^x , 21n^ x + 51n^ x \ , , ^ ,
yi^n
ta
CO
dx = d[\n x). Do vay, ta chon an phu
xlnx
/fl
w
=
In
X.
Inx
LM
gidi cda bai todn
21n'x + 5ln'x
r21n-'x
In
X

' 3 2
* Nhan xet: Niu da thanh thao trong viec su dung phuang phdp nay, cdc
ban
CO
the trinh bay lai gidi nhanh han nhu sau:
2
In
X + 3 p \ K9
i
^dx=
/•-(21nx
+ 3
d(21nx
+ 3)
a) 1,=/
_(21nx
+ 3r ^ ^
20
., r 2
In^
x + 5
In^
X , r 2
In^
x + 5
In^
x ,,
b) I / dx = / ; d(lnx)
xlnx
^

: Cdc ban de y quan he giua va 2e^ +
1
.•
e'^dx
= ^ d(2e^ + 1) nen ta c6
dx - .e'dx =
.
- d(2e' + 1)
2e" +1 2e" + 1
3 1
2 2e' +1
2e^ + l 2
(i(2e^
+1). Do vgy ta chon dnphu la u
=
2e' +\.
Ld'i
gidi
cua bdi
todn
I
=
r-^-Hl-dx=
r-^.eMx=
.id(2e''+l)
1 J 2e^ +
1
^ 2e^ +1 ^ 2e'' +1 2 '
= r ^^d(2e^+l) = - f—^d(2e^+l).
2 20" +

-dx =
1
1
+
1
-dx = -dx
Cdc
ban de y quan he
giiea
va e'^ +
1
e'^dx
= d(e* + 1) nen ta c6:
•
dx —
1
•
.e'^^dx
1
-d{e'
+!)•
+
1 + 1 + 1
Do
do, ta chon dn phu la u = e"
+
\.
LM
gidi
cua bdi

b)
I2
=
l
+ e
-dx = -dx =
-dx
1
+
e^+1
=
f—^ d(e'' + 1) = ln(e^ + 1) + C
•
c'' + 1
5.
Quan he
giua
sinx
va cosx
Ta
CO
(sinx) =cosx va (cosx) =-sinx nen quan he can xet giua sin a;
va cos X la:
cos xdx =
—
d{a s inx+b)
a
s inxdx =
— —
d(a cos x + b)

cos x(l - sin^ x) sin^ xdx = f{l- sm x)
sin^
xd(sin x).
Dat u = sinx => du = d(sinx).
Taco:
= J (1 -
u')uMu
= J (u'- u')du = y - y+0-
™
• . i T (sinx)^ (sinx)^ „
Thay u
=^
sinjc ta
duac:
I =-^^ ^ -^^ —+C-
' 3 5
b)
Phan tich bai todn : Cdc ban de y quan he giua sinx va
cosx;
cos xdx = d{-3 sin x + 2) nen ta c6
cos = —
e-'^'^'^^di-?,
sin x + 2)-
3
Do vdy, ta chon an phu Id u = -3sinx + 2.
LM
gidi cua bai todn
=
Jcosxe-'''"''^'dx =
J^e-"'"'^+'(i(-3sinx

1
-3
si
3 ^
3sinx+2
5
-3
sin
1+2
+c.
+
c.
dx^j
-1
-38mx+2
c/(-3sinx
+ 2)
6. Quan he gifra sin^x, cos^x va sin2x
Ta CO (sin^ x)
=2sinxcosx
= sin2x va (cos^ x)
=-2cosxsinx
=-sin2x
nen quan he can xet giua sin^x,
cos'^x
va sin2x la :
sin
2xdx =
—
d{a sin^ x -\-b)

E>at u = 3sin^ x+1 ^ du = d(3sin^ x+1).
Taco
: I = - /
udu
= hC= hC-
Thay
u =
3sin'x+l tadugc
: I ^
(3sin
x+1)
'
6
b)
Phan
tick
bai
todn
:
Ta bien doi:
sin
2x sin 2x sin 2x
V2sin^
X
+
3cos^
x
-^2(sin^
x +
cos^ re)

=
V2 + cos^x. r/-o«g truang hap nay
ta
nen chon
u
= V2
+ cos" X
cfe
Z>/ew //zii-c i/j/OT i/aw nguyen ham khong
con can
thuc.
L&i
gidi
cua bdi
todn
:
r
s\n2x
, r —1 2 \
h=J
-/ , dx = J ,
c^(2
+
cos^
x)
•
^
V2sin^x
+
3cos^x

:
,
r
sm2x
, r -1 , ^
/2
= / , dx = i =rf(2 +
cos''x)-
^
V2sin^x
+
3cos^x ^l2
+
cos^x
Dat
u = 2 +
COS"
X
=^ du
= d(2 +
cos^
x).
Ta
CO : \ f
-r=du
=
—2Vu+C
(chu y
cong
thuc

J(Ssin^
x+l)sin2xc?x
=
j^(3sin^ x+l)c/(3sin^
x+1)
_(3sin^x+l)^
6
V2sin^
X
+
3cos^
x
yJ2
+
cos^
x
=
-
'(2
+
cos^x)
2
t/(2
+
cos^x)
=
-2V(2
+
cos^x)
+

-sin^
x+sm^
x + C
•
4
2 2
Chii
y
rang each nay
va
each tren
deu cho ket qua
dung,
no chi sai
khdc
nhau
mot
hang so
xdc
dinh
!
b)
= f ^"^^^ dx =
flpi^dx
(sau do dua
sinx
^
V2sin^x
+
3cos^x

J(2
+
cos^x)
= _
(2±£^ij0i_
^
^
-^1
1
=
-2 (2 +
cos^x)2
+C = -2V(2 +
cos^x)
+ C.
1
1
7.
Quan
hf
giira
— va
tanx
,
cos^x
sin^
X
va
cotx
Ta

dx = — — d{a cot x+b)
a
sin^
x
(ta
hieu cong thuc tren
mot
each
dan
gidn
nhu sau : dua
1
cos^x
vdo trong
vi
phdn thdnh (atanx
+ b), dua
vai
a
^Ovdb tiiyy tren
R)
Vidu
7.
Tinh:
3 tan
X + 4
1
sin^
X
vdo trong

ban de y
quan
he
giira
1
cos X
vd tan
X: dx = - d(3
tan
a;
+ 4)
2 2 o
ODS
a; cos a; o
mn,ac6 0tanx+
4) 1 ^
01'^"^
+
4)irf(3tanx
+ 4). Dov^.
2 cos X 2 3
ta
chon an
phu la M =
3 tan
x + 4.
LM
gidi cua bai todn
/
3tanx

+ c = -^ + c.
6 2 12
Thay
w =
3tanx
+
4tadugc:
T =
tanx+
4)^^^
'
12
b) tick bai todn
:
Ta
biin
doi
cotx
sinx
sin''
x
~
cot^
X.
—\
•
Cdc ban di y
quan
he
sin^

cot
X , r cot" x , r
I
dx = /
—^a;
= I
sm
X
cot
X
sm
X
cot'
X.
—^—dx
sin
X
=
-
J
cot'
a-(i(cot
a;).
Dat
u = cot
X
=> du =
d(cot
x).
Taco:

x + 4
^1 = J T
1
r 3
tan
x
-dx
= I :
^X
+
cos 2x -^2 cos X
=
r(3tanx
+
4)
1
3^^^^^ (3tanx+4f
J 5! 3 ^ 12
(
^2
—
dx = I cot
X.
-—-—dx — — I cot xd(cot x)
sin X sin V ^
3
sm X
= 5—+ c-
Vay
la, chung ta da nghien

doi mot so vi du minh hoa.
Vi
du 8.
Tinh
:
a) j x(x +
12f'Mx
; \
^'('^
+
l)'dx
•
Giai
a) Phan tich bdi toan : Niu khai
triin
(x +12)^'"^ rdi nhdn x vac di tinh thi
khong kha thi rdi ! O day chung ta
cUng
khong nhin
thdy
su xuat hien cua mot
trong
7 moi quan he de dinh huang phep dqt an phu, nhung theo huang giai
tong quat, chung ta chon an phu la u = x + 12 thi
tie
nguyen ham theo biin x
chung ta bieu dien duac nguyen ham do theo bien u rdi! Vi tit u = x + 12 ta
CO
X = u
—

=
x
1 thi tit nguyen ham theo biin x chung ta bieu dien duac nguyen ham do theo
biin
u rdi! Vi tit u = x + \ c6 x = u - \ dx = d(u - 1) = du
(tiic
Id x duac
••
biiu diin theo u vd dx duac bieu dien theo du).
\ giai cua bdi toan
Dat u = x+
1
=>du = dxvax = u- l.
Ta
dugc
:
= J(u -
l)'uMu
= J(u' - 2u +
l)uMu
- /(u' -
2u'*
+ u')du
2n'
u u
10 9
Thay u = x +
1
ta c6 :
(x

X
= u + 2 vd dx = d(u + 2) = du (tuc la x duac bieu dien
theo u vd dx duac bieu dien theo du).
Lcfi
giai cua bdi toan
Dat u = x- 2=>du = dxvax = u + 2.
Ta
dugc
:
r
(u + 2)' +
1
, r u' + 4u + 5 , r, I , 4 , 5 , ,
-9 4 -10
r,
.,M . 11 . ^ , u 4u
-1
-10
+
4u-'^+5u^^)du = —+
+
9 -10 -11
+
C
9u^
lOn'' ' iW
Thay u ~ x - 2 ta c6 :
I.
=
-1

:r +
-5
+
4u + 5
.12
du
T/
1 4 5 ,,
9u'
lOu^" llu^^
Thay
u = x - 2 ta c6 :
+
C.
1.
=
-5
'
9(x-2)' 10(x-2y"
ll(x-2)"
+
C.
b)
Phan
tich bai
todn
:
Ta
cd
bien

-
(2
+
x^)'
{2
+ xy
chon an
phu la u
=
2 +
x\
LM
gidi
cm
bai
todn
1
X
.x'dx
= —
d(2
+ x'), vgy ta
3(2
+
x')'
it
u =:
Ta
dugc
:

Vi
du 10.
Tinh
:
(sinx
+
cosx)
a) I, = f
_
dx; b) I = r sin^ xVcos xdx.
Gidi
a)
Phan
tich bai
todn
:
Ta
bien
doi
cos X cos
X
(sinx
4-cosx)^ ,sina; + cosx., 3 (tanx + 1)^ cos^ x
^
' ( ) cos X '
cos
X
Tai
ddy
chung

= / — = I — dx
(sinx
+
COSx)^
^ ,sinx +
cosx.3
3
^
^ ( ) COS X
cosx
=
f -—i—dx= r -d(tanx + l)
J
(tan X +
1)^
cos^ x <^ (tan x +
1)^
'
Dat
u = tan x +
1
du = d(tan x + 1).
Tadirgc:
T ^ T-^du = f
u-Mu
- — + C - ^ + C•
^
J J -9.
-2
2u'

dat
an
phu
cUng
COS^
X
nhu cdch gidi!
-
Cdc ban cd thi
trinh
bay lai
gidi
gon han, vai chu y
cong thuc
I
=
r——^ix
= f—
(sinx
+
cosx)
^ ,smx
cos X
4-cosx,3
3
f
cos*
x
-dx
cosx

sin x(l
—
cos^
x)Vcosxdx
=
(1 - cos^
x)Vcos
xd(— cos x)
vgy ta chon an phu la u
=
cosx
hoac
u
=
Vcosx
.
Trong
truang hap nay
chung ta nen chon u
=
Vcosx
de bieu thuc duai ddu nguyen hdm khong con
chica
can thuc.
Ldi
giai
cua bai
todn
\
J

Thay u - ta c6 : =
-2{sf^f
^ 2(V^)^
*
Cach
khac : Neu dat
M
= cos
x
ta c6
each
giai
khac
nhu sau :
\—j'^ii^^
xVcosxdx
—
J'sinXsin^
xVcosxdx
= j sin x(l - cos^
x)Vcosxdx
= J{1- cos^
x)Vcosxd(-
cos x)
DSt u = cos X du —
d(cos
x).
Tadugc : = \(l-u')^f7^d{-u)^ \(\-u'yd{-u)
=
-j

Tinh:
a) I, =
rdX
'
b) = J
xVl
+ xdx.
Gidi
a) Phan
tick
bai
todn
: Bai todn ndy sit dung 7 quan he de dinh huang phep
dat dn phu Id khong khd thi. Neu chon an phu Id u = Vl
—
x ihi tit nguyen
hdm theo biin x chung ta bieu diin duac nguyen hdm do theo bien u vi tit
u
= Vl -
X
ta suy ra x =
1
- vd dx = d(l - u^) = -2udu (tuc Id X
duac bieu dien theo u; dx duac bieu dien theo u vd du ).
L&i
gidi
cHa
bai
todn
Dat Li = Vl -X =^ =

l)u.2udu
=.2/(u^-u=)du
= 2(^-^) + C
Thay
+
c.
Vi
du 12.
Tinh
:
dx
e" +1
b)
l, = J
„2x „x
e
—
e
dx.
Gidi
a) P/tan tick bai todn : Neu
chon
an phu la
M
= + 1 thi tir nguyen ham
theo
bien
x chung ta
bieu
dien

u-1
-In
u
+
C
u
+
C.
Thay
u = +1 ta c6: ~ in
e^+1
-)-
C
•
(ta khong lay ddu gid tri
tuyet doi vi
+1
b) Pit
an tick bai todn : Chu y rdng e^'' = (e'')^ tuc Id e^" biiu diin duac
de
qua e"". Lai de y de" = e^dx hay = dx nen ta
chon
dn phu la u = e\
L&i
gidi cua bai todn
E)5t u ^e'' =^du = de''.
r
1 du r 1 , pn^ ~ (u"
Tadugc:
= / — •— = / -du =J

du
-In
u
+
- + C
u
u-1
u
+
i
+
c
u
Thay
u = e'' ta c6 : hi
e" -1
*
Cdch khdc : Cdc ban diyrdng
1
1
_ je-f
vd
d{e"'') = -e~\ix hay ^—^ = dx thi ta
chon
duac
dn phu la u — e
Ta
CO
lai gidi cho bai todn:
=