515.076
GI-103N

TRAN TUAN ANH

Gliil NHANH BAI TOAN

HGIIYJNHAH
&TiCH PHAN
D A N H C H O H Q C S I H H L0P11-12


TRAN TUAN ANH

m

r

m

m

m

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1

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-

CONG TY TNHH M^T THANH VIEN SACH VIET

. 391/15A Hajnh T i n Phat, P.T§n Ttw^n Dong, QuSn 7, TP.HCM,
BT: {06} Jf.720.837 • F a x P) 38.726,052 • MST: 03114307135
Email: W i f i a c l R f i e t c o x o m - Website: «»w.sachvie!co.«ni

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.

- ThSy Le Hoanh Sir, Giang vien DHQG TP. HCM
Te Luat.

i '

- Truang DH Kinh

'

- Thhy Nguyen Tat Thu, Giao vien Truang chuyen Luang Th6 Vinh Bien Hoa - Dong Nai.
Tran Tuan Anh
H kh6i A, A i - 2013)

Cdch gidi thong thir&ng
w - I

2

Cdch 1: / =

——.lnxdx=

+ A = In

1

\nxdx +

In xdx. Ta xet:



-1

1

rfX = > V = — .
J
X

\X

f

1\

In xdx =

2 ^f 1
1,
dx
= — Inx 1
X
X

= —Inx

2
1

X


1=

,

\n 2

Cau 2; Tinh tich phan / = ^x^l-x'dx

In 2

^\\—L-\e'dt=

\[e'-e")dt=

\td(e'+6-)
Cdch giai thong thu&ng

0 V

=

t(e'+e-)

In 2
0

In 2

•


X.

x+ -

2

-

V

1
\^2
= Inx. x + -

1^

In xdx = In xd x + *

2

O

T a c o : J = -l4l

x+ -

X

rv

1'

—

Vay / =

272-1

Cdch 2: Theo kinh nghiem thi thay can thuc ta dat can thuc la an phu !
Dat / = 72-x^ ^t^ =2-x^ =>tdt = -xdx.

I,^/ giai that nhanh ggn so v&i hai cdch tren !

D6i can:' x = 0 => / = 72;x = 1 =i> / = 1.


Taco: I = -\t^dt=

\t^^t=

—

V2

2V2-1

Cdch 2:

Ta c6 :
/=


2x
x'+l

-dx

•dx.

x^+1

(0

t

;
dx^
x ^ + 1J

Dat x = t a n / = > ( i x = —^—dt = {\-^ian-t]dt,
cos"/
^
'

ti

2 '2

suy nghi Id : "thay cd can thiic thi dat can thitc la an phu". Chung ta c6 the
gidi nhanh nhie sau:



V.

V

0

0

-dx = dx +

x'+\

x'+\
2x

Xet tich phan J =

2x

-dx = \

r 2x

Cdch 3: Cdc ban de y quan he giua x vd x^ Id:

•dx.
1=

Tadugc J =

X

+

-flX

=

I

,

dx = \dx + f ^"^ dx
0

0

+ ln x ^ + l

i/(x'+l) =x
x^+1

1

= l + ln2.

0

= ln2.
LM gidi that nhanh gon !


cos^x + sin^o;

X

X

—Y~

^ cos

x +C

•dx = ? Ta suy nghi : ham so nao c6 dao ham
0; Q

^

1)

cos xdx — sin x + C
sin xdx = — cos x + C

f — ^ - — dx = tanx + C
^ cos X

+C

X

e'dx = e'


J

Sill X

nhat bang — - — la ham so
hay t a n x. Suy ra cong thuc thu chin :
cos a;
cos X

r

f

=C

(chu y do mau thuc
cos a;

cos^

Jodx

X

1
nhat bang — r — ? Truang hop nay khong de tim nguyen ham hon cac truoTig
cos X

A. COS X + sinx.A


= 6 — + C (dp dung bang nguyen ham ca ban)
2

b) / = r (5e"
'
thif ba)

= 3a;' + C .
, . r cos X ,

I f

h 3 sin a; + C (dp dung bang nguyen ham ca ban)

1
cos xdx = — s inx + C •
3

= r Se'dx- f
dx (dp dung tinh chdt
cos'
x
*^

^)dx
cos' x

= 5 r e^'dx - 7 r — - — dx (dp dung tinh chdt thu hai)
. "J
^ cos' X


\x

I

{x'dx = — + ^ + C = -— +
ln3
1
ln3
3

3x'+C.

J

nhung ham s6 c6 trong bang nguyen ham ca ban. Do vay, viec nam dugc Bang

3x' - 2x + 4

nguyen ham co ban la di^u kien rSt quan trong de chung ta tinh dugc nguyen

[x'dx

dx =

X

ham, tich phan.

I


J 33;'

= J {4x + 3cosa;)(ia; = J Axdx + J 3cosa;c?x- (dp dung tinh chat
thu ba)
= 4 J" xdx + 3 J cos xdx

3x'
(dp dung tinh chdi thu hai)

-dx
X

23; + 4 hi X + C.

i

cix = — +
in 3

3x - 2 + - dx
X

3x'+C.


Tiiy theo kha nang cua nguai lam todn met ta c6 the lucre bo di nhirng buac
gidi khong can thiet.

cua f(x)

a =l

va

Vi du 6, Chung minh rang F(x) - sin xe"" la mot nguyen ham cua ham so
/ ( x ) = (sin X + cos x y .

Gidi
+ 1)^

dx

X

2
1 + — +

-I
CO :

J

7^ =

X

X

-I


+ C.

X

Vay F(x) = sin xe' la mot nguyen ham cua ham so / ( x ) = (sin x + cos

x

•dx

e'x^

3

3

,3
JC

JC

1

2

X

X

dx = — - + In

re
= \—;
b thi

la mpt nguyen ham

vai Vx e R .

CO :
X

b) Ta

va / ( x ) la R . Ham s6 F(x)

< ^ ( o - l ) x + a + Z ) - 0 , V x e R <=>Q]a^
In a

1)
^

ra'^^'dx = ± . ^ + C{a > 0;a ^ 1)
a

\na

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 :

j cos xdx = sin x -\- C

/ cos (ax + b)dx = — sin (ax + b) + C

1. Dinh li

truang hop viec ap dung bang nguyen ham ma rong cho ta lai giai bai toan
nhanh va " sang " han ! Chang han vai bai toan sau :
Tinh nguyen ham : I — J (2x + l^dx.

Neu khong ap dung cong thuc

nguyen ham ma rong thi ta khai trien bieu thuc {2x + 1)^, sau do mai ap dung
cong thuc nguyen ham ca ban de tinh :


I = J{2x + lydx

= J(2x

Giai

+ l)(2x + Ifdx

= J {2x + l)i8x''+12x^+6x

Ap dung cac cong thuc trong Bang nguyen ham ma rong ta c6 :

+ l)dx
a)

= J (16a;' + 24a;' + 12a;' +2x + 8x'' + 12a;' + 6x + l)dx

= J (16x'
16x'


d6i biln s6 (se hoc a bai sau) thi cung c6 lofi giai khong gon bang each nay !
R6 rang cong thuc nguyen ham ma rong to ra uu diem han cong thuc nguyen

dx =

-(3-2a;)'

L + C.

= __V

-7

+ C-

18(3a;-l)
d) /, = {• ,

J^A: = 2

dx=

3'

-5

{3x-2)idx

+1


J

vl3

2a; + 3
= ^ - L 26

(3-2a;f
.-^
L +

ham ca ban ! Cac cong thuc nguyen ham ma rong, neu chung ta cho he so a =

=

/

10

(C/zw >• rang, each nay va each tren deu cho kit qua dung, no chi sai khdc

a)

\13

r
I
ci
\-7
1 (3a;-l]

^(33;-2)

2

-5

.,,.2_(3x-2)3

-1
#x-2)^

3
- D6i vai Cdu c) vd cdu d), neu sir dung cong thitc

+ C.


•dx = —.a (n-l){ax

-1

C{n^l;a^O) se

— +
+ b)

cho

ta lai gidi nhanh han nua (vi gidm duac mot buac biin doi) !
-1


day chung ta dan cu cdu a) duac gidi bdng phuang phdp doi bien so de so
sdnh hai cdch

\e ^''dx = ^ e

l3=

Ta

M =

CO
5

2a; + 3

12

2a: + 3 => (iti = 2dx

:/

=
1

\v}^-du
J

^''+C = -3e

=

^

r3'^'-'dx -

J

c)/3=

Ap dung cong thuc nguyen ham ma rong ta c6 lai gidi gon vd nhanh han:
\13
2x + 3
12
2a; + 3 dx =
+ C
26

+C.

-e-^+C.

=

J

_1

+ C


Gidi
A p dung cac cong thuc trong Bang nguyen ham m a rong ta c6 :

f—^dx;
1
—X
e ^ dx;

C•

+ C.

26

Rd rang cdch nay to ra khd phuc tap so vai mot bdi todn dan gidn nhu vdy !

c) h=

+
5 ln3

3

a) 7 = / — - — dx = —^ In 2-5x
'
J 2-5a;
5

dx



7^

=. J

dx

e-'dx

a)

/j = J s i n ( 3 - Ax)dx
cos(3 - 4a;)
4

= ^ . ( - c o s ( 3 - 4a;)) + C
^


+ C--15sin

dx = 5.—.sin

b) h = JScos

3. Tinh :

\ /

\ /

^
^ cos (4 — x)

d) ^4 = / 2cos(3--)rfa;.

1
-dx = 3.(--cot(2a; - 1)) + C
sin^(22;-l)
2
3

^) I . =

Bai 3. PHl/QNG PHAP D O I B I E N S O

= - | c o t ( 2 a ; - l ) + C-

Bai nay chiing ta se xet hai truong hop Ichi tinh nguyen ham

- Chung ta c6 thi trinh bay nhanh nhu sau :

/{x)dx bang

phuong phap doi bien so :
a) / =

fsin{3-4x)dx=:^^^^^^-^C.
4

J


f (x)dx = g{u)du.

Chu V : chon an phu u = w(x) sao cho viec tinh / = g(u)du phai de hon la

1. Tinh :
=

theo u va du. Gidsurang

Buoc 4 : Tinh I = g(u)du. Sau do thay u = w(x) de dime ket qua can tim.

BAITAP

a)

Buoc 3 : Bieuthi f(x)dx

J (4x + 2)' dx;

tinh / = jf(x)dx
b) ^. =

/ - ^ ^ - ;
^ (3-2x)

!

Khi nhin vao mot bai giai cho bai toan tinh nguyen ham hay tich phan bang
phuong phap dat an phu (hay phuong phap doi bien so), ban doc thuong c6 cau


(2) Niu J f(u)du =F{u) + C va u = u(x) la ham sS c6 dqo ham lien tuc

x^dx = -d(2x^ +1), nen ta c6 x'{2x^ + ifdx = -(2a;' + lfd{2x^ + 1).
6

thl: J f{u{x)).u\x)du

= J i\u{x))du{x)

=F{u{x))

+C

6

Do do, viec chung ta chon an phu la u-2x^
mang tinh dp dat.

Vi du

L&i giai cua bai todn

a) J cos(2x^ + 3a; + l)d{2x^ + 3a; + 1) = sin(2a;^ + 3a,' + 1) + C.
(ta hieu trong suy nghi " 2x^ + 3x + 1 "

J x'(2x'

lau)


nhanh phep dat in phu va dinh huong nhanh each giai cho bai toan nguyen

+
'

ham, tich phan bang phuang phap doi bien so.
l.Quan hegiua x" va

+ \ hodn todn ty nhien, khong

60

b) Phan tich bai todn: Cac ban de y quan he giica x vd x :

x"*\n^-\)

d{ax"^'+b),
Ta CO : dx'"' = (« + \)x"dx o x"dx = —dx"*'
'
n + \ + \)
do a^O con b tuy y tren R . Vay ta c6 quan he giira x" va x"^\n^-1)

xdx = - d(x^ +1) nen ta c6 x^x^ +\dx = - Vx^ +1 d(x^ +1). Do vdy, ta c6 thi
2
2

trong

chon an phu Id w =


each nay da to ra khong hieu qua ! Niu giai bai todn nay bang phuang phap
doi biin so, ta chon an phu la u-lx^

+ \. Tgi sao Igi chon duac an phu nhu

vay? Bay gia cac ban de y quan he giica x^ va x^ nhu sau :

^

= • — + 0-

/"I—T t A
(Vx^ + 1 )
Thay u ^ Vx +1 ta dugc: i =
'
3
Cach khac :

a) Phan tich hai todn: Theo I6i giai thong thuang, cac ban se khai trien

The nhung viec khai trien bieu thicc (2x^ + if

= Jiu

Ta c6: I2 = j-Judu
2

= — ju^du =
2
2


'1'

a) Ii =

^

\—r-dx;

X

X

= — reMu- —e" +C.

X

X

L&i gidi cua bdi todn
X

1
1
b) / = I — sin — cos — dx.
X

X

1


-dx =

f-l

b) Phdn tick bdi todn : Cdc ban de y quan he giua

X

(ta hieu cong thuc tren mot each don gidn nhu sau : dua \ trong vi

Vi du 2. Tinh :

1.1

3
-1
Thay u = 1 + — ta duac: Ii = — e ^ + C.
X
•
3

nen quan he can xet giila — va ^ la:
X

X

L&i gidi cua bdi todn

Ta CO : I

X

Taco: i
X

that khong de de chung ta tim ra ngay phep dgt an phu! Cdc ban de y quan he

=

1 r • 2J 1
/ sin —d —
X
2^
X

i =^ du = d
X

2

_ i f sin(2u) du = - i ( - - c o s 2 u ) + C - - c o s 2 u + C9.J ^
2 2
4

Thay u = — ta duac: I = — cos
X
2
4

+ C-

-1

^of

X

X

X

—

10

/
'

3

/

s m — cos — ax = — i s m — c o s —a

X

"J

1+--

=ifuMu=1.^:^+0=^+0.

— dx
X

=

— d{a
a

\nx

+

2 In^

r 2211nn-''xx + 5 l n ' x dx
In X

Ta CO (In x ) = — nen quan he can xet giiia — va I n x la :

X

+ 5 In^ x

Inx

d(lnx).

Dat u = Inx => du = d(lnx).

b)


tiiyy tren R)

3

V i d u 3. Tinh :
2

In

X +

3

-dx;

b ) I , = /

21n'x + 5 l n ' x
xlnx

X

2

Thay u = Inx ta duac : i ='^i^RJ^ +^A\}12^+ c •
'
3
2
* Nhan xet: Niu da thanh thao trong viec su dung phuang phdp nay, cdc

X

_(21nx + 3r

^dx = ^d(21na, + 3) nen ta c6 i l l ! l f l 2 L a ! x = l(21njc + 3)V(21nx + 3).
vay, ta chon an phu Id u

(2

In

2lnx + 3.

Ij = J

Dat u =

2

p i ,

X +

^dx

In

X +

.,


= ae' nen quan he can xet giira

va ae' + b la:


Cdc ban de y quan he giiea

(a^O)

a
(ta hiiu cong thuc tren mot each dan gicin nhu sau : dua 6

1

• .e'^^dx
+ 1

• dx —
+ 1

vao trong vi

-d{e'
+ 1

+!)•

LM gidi cua bdi todn


Ld'i gidi cua bdi todn
I = r-^-Hl-dx= r - ^ . e M x =
.id(2e''+l)
1
J 2e^ + 1
^ 2e^ + 1
^ 2e'' + 1 2 '

= ln|u|+C.

; Thay u = e" + 1 ta dugc:

* Nhan xet: Neu da thanh thao trong viec su dung phuang phdp nay, cdc
ban CO the trinh bay lai gidi nhanh han nhu sau:
a) I = r — d
'
-'20^+1
b) I2 =

x = r — - — . - d ( 2 e ' ' + l ) = -ln(2e^+1) + C.

= r-.^^d(2e^+l) = - f—^d(2e^+l).

Ta

CO

3 r l ,
' 2 ^ 1 1


f — ^ d(e'' + 1) = ln(e^ + 1) + C •
c'' + 1

cos xdx = — d{a s i n x + b )
a

b) Phdn tich bdi todn : Ta Men doi
h =

1+

2

5. Quan he giua sinx va cosx

va cos

1_

-dx =

^

26" + 1

Thay u = 26" + 1 ta dugc : 1 = - ln(2e"' + 1) + C • (ta khong lay dau gia tri

r

2e^ + 1 2 ^

/ cos xe - 3
1

= J cos^ x sin^ x d x = J cos x cos^ x siii^ x d x
cos x ( l - sin^ x ) sin^ x d x =

sin^ x

nen ta c6

cosx(l -sin^x)sin^xdx = (1 -sin^x)sin^xd(siwc). Do vdy, ta chon an phu la u = sinx.

=J

sin^ x ) sin^ x d ( s i n x) = J* (sin^ x - sin^ x ) d ( s i n x )

- 33sinx+2
si

3 ^

sm x ) sin^ xd(sin x ) .

sin

+ c.
- 1

1+2 dx^j



y+0-

(sinx)^
„
^
-^
—+C5

b) Phan tich bai todn : Cdc ban de y quan he giua sinx va cosx;
cos xdx =

d{-3

sin 2xdx = — d{a sin^ x -\-b)
a

(ta hieu cdng thuc tren mot cdch don gidn nhu sau: dua sin2x vao trong vi
phan thanh (a sin^ x + b) hogc —[a cos^ x + b), voi a ^0 va b tuyy tren M)
V i du 6. Tinh :

sin x + 2) nen ta c6

a)
cos

= — e-'^'^'^^di-?,
3

Do vdy, ta chon an phu Id u = -3sinx

LM gidi cua bai todn
= Jcosxe-'''"''^'dx

= J ^ e - " ' " ' ^ + ' ( i ( - 3 s i n x + 2).

Thay u = - 3 sin x + 2 ta dugc:

ta

c6

(3sin^ x+l)sin2xda;

L&i gidi cua bai todn :

+C.
= —

nen

= •^(3sin^ x + l ) d ( 3 s i n ^ x + 1 ) . Do vdy, ta chon dnphu la u = 3sin^x + I .
o

Dat u = - 3 sin x + 2 =^ du = d ( - 3 sin x + 2 ) .
Taco: I = — f e M u - — e "
'
3 ^
3

sin2xrfa; = - d ( 3 s i n ^ x + 1 )

• * Nhan xet:
- A'ew

trinh bay lai gidi nhanh han nhu sau :
a)

b) Phan tick bai todn : Ta bien doi:
sin 2x

sin 2x

sin 2x

V2sin^ X + 3cos^ x

-^2(sin^ x + cos^ re) + cos^ x

Vs + cos^x

thanh thgo trong viec sir dung phuang phdp nay, cdc ban c6 the

= J(Ssin^

s i n 2a:

_

V 2 + cos^a;

=d{2 + cos^ .x)-
/ew //zii-c i/j/OT i/aw nguyen ham khong con can thuc.

- A'ew chung ta de y den quan he giua

=

s\n2x
,
r
—1
2 \
-/
,
dx = J ,
c^(2 + cos^ x) •
^ V 2 s i n ^ x + 3cos^x
^ V 2 + cos^ x
r

X =>


x4-l)(ix

x + l ) r f ( s i n x ) = y (6sin^ x + 2 s i n x ) ( i ( s i n x )

J2sinx(3sin^

„ sin^
= 6.
4

2 + cos^ x => du^ = d(2 + cos^ x ) .

x + l ) s i n 2 x ( i x = j2sinxcosx(3sin^

„ sin^ x
„
3 . 4
. ,
^
h2.
+ C = -sin^ x+sm^ x + C •
2
2

Chii y rang each nay va each tren deu cho ket qua dung, no chi sai khdc
Tac6:l^=r j d i d u ^ = - J ^ d u

= - 2 j d u = - 2 u + C-

nhau mot hang so xdc dinh !


- 1

r
i

^ V2sin^x + 3cos^x
Dat u = 2 +

=f

^l2 +

, ^
=rf(2 + cos''x)-

(chu y

r

/

-2cosx

r

cos^x

thuc



1/

1

==rd(cosx) = — I — = = = = = r d ( c o s

^ V2 + cos^x

=^ du = d(2 + cos^ x ) .

\ f - r = d u = —2Vu+C

=


1

Dat u = 3 tan x + 4

1

va cotx

7. Quan hf giira cos^x— va tanx , s i n ^ X

Ta CO ( t a n x ) = — ^ - j - va ( c o t x ) = . ]

s i n jc


1

vi phdn thdnh (atanx + b), dua

1

vdo trong

giica — ^
sin

sin^

vdo trong vi phdn thdnh -(acotx
X

sin^ x

sin'' x

c o t x ; -^dx

= -(i(cota;)

sin X

X

M = COtX.



vd

t a n X:

a;

2
cos

a;

dx = - d(3 t a n
o

a; + 4)

o

^ 01'^"^ + 4)irf(3tanx
2
3

cos X

+

4). Dov^.

LM gidi cua bai todn

3 t a n x + 4^^ _

Ir cot' X. — ^ — d x

Dat u = cot X => du = d(cot x ) .

1
cos X

1

r cot
cot" X
x ,
/ — ^ a ; =

= - J c o t ' a-(i(cot a;).

2 cos X

Cdc ban de y quan he giira

,

dx =

sm X

smx



1 4-

T =

cotx
sinx

cos^x

a)/

+ c = - ^ + c.
12

tick bai todn :
Ta biin doi

— — d{a c o t x + b )
a

(ta hieu cong thuc tren mot each dan gidn nhu sau : dua

vai a ^Ovdb

tan x + 4 ) .

'

vd c o t X la :

T + cos 2x
- ^ 2 cos X

3

r ( 3 t a n x + 4) 1

J

5!

3

3^^^^^
^

(3tanx+4f
12


(

^2

—

dx =

I cot


nguyen ham do theo bien u! (tiic Id ta can biiu dien bien "x" theo bien u ,
"dx" theo u vd du)
Mb'i cac ban theo doi mot so vi du minh hoa.
Vi du 8. Tinh :
a)

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).

j x(x + 1 2 f ' M x ;

Dat u = x + 1 =>du = d x v a x = u - l .
Ta dugc :
= J(u - l ) ' u M u = J(u'
u
2n'
10
9
Thay u = x + 1 ta c6 :

1^ =

(x +

ir

- 2u + l)uMu - / ( u ' - 2u'* + u')du

u


-4
+

-5

:r

du

T/ 1

4

-5
10(x-2y"

,sina; + cosx.,
3
(
) cos X
cos X

X

- .x^ nen chung
(2 + x^)^

Ta CO :
r


r

cosx

= I—

^ ,sinx + cosx.3

,
dx

3
) COS X

(

J (tan X
Dat u = tan x + 1

.xMx= r

•-d(2 + x^)

- Trong bai todn nay, phdi thong qua mot sSphep bien doi, chung ta mai dp

u

3


3

-—i—dx=

- d ( t a n x + l)
+ 1)^ cos^ x

d u — d(cos x ) .

-dx

,smx 4 - c o s x ,f3 cos*3 x

Tadugc :

= \(l-u')^f7^d{-u)^

cosx
1

•dx
f( t a—
i i x - + 1)^ cos^ X

=

-1
2(tan X + 1)^

/

•d(tan X + 1)
(tan

X +

5
|

\du-

= -j u^-u

\(\-u'yd{-u)

2 du

3

2w^

Thay w = cos x ta c6 :

+ C-

_2(cosx)2

b) Phan tick bai todn : Ta de y quan he giita sin x vd cos x de dinh
huangphep dgt an phu : sin xdx = —d(cos x). Ta cd
sin^ xVcosxdx = sin x sin^ xVcosxdx

chung ta nen chon u = Vcosx de bieu thuc duai ddu nguyen hdm khong con
chica can thuc.

•

=

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 = V l — x ihi tit nguyen

Ldi giai cua bai todn
\ J sin^ xV cos xdx = J sin x sin^ xVcosxdx
= sin x(l - cos^ x) V cos xdx = (1 - cos" x)Vcosxc/(-cosx)

hdm theo biin x chung ta bieu diin duac nguyen hdm do theo bien u vi tit
vd dx = d(l - u^) = - 2 u d u (tuc Id X

u = Vl - X ta suy ra x = 1 -

duac bieu dien theo u; dx duac bieu dien theo u vd du ) .
L&i gidi cHa bai todn

Dat u = VcosX =4>
Tadugc:

= cosx va d(—cosx) = d(—u^).


X =

1-

dx = d(l - u^)
= -2udu-

Ta duac
gc :

= -Ji-iil2udu

Thay u = Vl - x ta c6:

=2

= 2J(u'

- l)du = 2 u

(Vr^)^

+ c-

u + c

* Cach khac : Neu dat M = cos x ta c6 each giai khac nhu sau :
\—j'^ii^^

xVcosxdx — J'sinXsin^ xVcosxdx

+ c.

Thay

-I

V i du 12. Tinh :
dx

b) l, =

e" + 1

J

Gidi
a) P/tan tick bai todn : Neu chon an phu la M =
theo bien x chung

ta bieu dien

duac

du — rf(e^ + 1 ) = e'^da: ma

nguyen

„2x
„x
e —e


Thay u =

u+ 1

u-1

u'

du

1
du = In u - 1 - I n u + - + C
u

hi

e" - 1

diyrdng

1

1

_

je-f

vd