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8/9/2019 PHN LOI V PHNG PHP GII TON TCH PHN V CC BI TON NG DNG - L MU THO, L MU THNG

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L MU THO - L MU THNG

PHN LOI V PHNG PHP GII TON

V

CC BI TON NG DNG

N H X U T B N H N I

WWW.FACEBOOK.COM/DAYKEM.QU

WWW.FACEBOOK.COM/BOIDUONGHOAHOCQU

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W.DAYKEMQUYNHON.UCOZ.COM

ng gp PDF bi GV. Nguyn Thanh T


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3/146

M C X C

Chuyn 1: Nguyn h m ....................................................... .......................Chuyn 2 : Tch phn ...................................................................................

Chuyn 3 : Phng php i bin s .........................................................

Chuyn 4 : Tch phn ca hm s' hu t .................................................

Chyn 5 : Tch phn ca hm slng gic .........................................

Chuyn 6 : Tch ph n ca hm s c cha cn thc ..............................

Chuyn 7 : Php lng gic ha trong vic tnh tch phn ................

Chuyn 8 : Tch phn hm s c du tri tuyt i ..................................

Chuyn 9 : Tch phn tng phn .................................................................

Chuyn 10 : Cng thc truy hi ...................................................................

Chuyn 11 : Tch ph n v cng thc nhi thc N ew tn .......... ................

Chuyn 12 : Din tch v th tch ...............................................................



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5/146

Chuyn 1 NGUYN HM

I . Kin thc c bn

1) nh ngha

Cho hm s' F(x) v f(x) cng xc nh trn khong (a, b).

F(x) l nguyn hm ca f(x) trn khong (a; b)

F (x) = f (x), Vx 6 (a, b)

2) nh l

Nu hm s f (x) c mt nguyn hm l F(x) th f(x) c v s nguynhm v cc nguyn hm ny c dng F(x) + c (C l h ng s ty )

K hiu f(x)dx ch nguyn hm ca f(x).

Vy |f( x )d x = F(x) +' c o F(x) = f(x)

3) S tn ti ca nguyn hm

Mi hm s lin tc trn on [a, b] u c nguyn hm trn oi .

4) Tnh ch't

a) a = hng s' => Ja.f(x)dx = jf(x )d x

b) j[f(x) g(x)]dx = jf(x)x Jg(x)dx

5) .Bng cn g thc tnh nguyn hm

L oi 1

1/ a = hn g s => adx = ax + c . 2/ fxmdx = + c (m 1)J J m + 1

H qu

a/ dx = -------------------r + c (n 1) . . b/ c ~ d x = - - + cxn (n - Dx11 1 X X

, c/ T-^d x= 2-Jx + c d/ fVxdx = xVx + C

- \X J 3 L o i 2

1/ jcosxdx = sinx + c 2/ jsinxd x = -c o s x + C

3/ I" dx = tgx + c . 41 f^ dx = - cotgx + cJcos X sin X



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6/146

Lo i 3

1/ fexdx = ex+ c 2/ faxdx = + c (0 < a * 1)J J ln a

3/ [dx = In IX + cJ X

* Cn nh (quan trng)

V Nu jf(x )dx = F(x) + c th |f(a x + b)dx = F (a x + b )+C (a *0)

V du 1 fx3dx = + c f(7x - l)3dx = l . (7 x~ 1)4 + c J 4 1 . 7 4

V du 2 fsinxdx = - cosx + c * fsin(2x + )dx = - cos(2x + ) + cJ J 6 2 6

V du 3 f d x = In 1XI + c f - dx = =In 13 - fx I + cX J 3 - 5 x - 5 ^

Cn nh

1. Chng minh F(x) l mt nguyn hm ca F(x), X e (a, b).

Ta ch cn chng minh F(x) = f(x) , X. (a, b).

2. Xc nh cc h s cha- bit ca hm s F(x) Sao cho F(x) l mt n g u y n hm ca f (x ) , X (a, b).

Tnh F(x) ri cho F(x) = f(x) , Vx (a, b) (ng nh t cc h s cng bc)

II. Cc dg

Dng 1 [Chng minh F(x) l mt nguyn hm c a f(X)

B i 1. Ch ng t rng.-hm s F(x) = ln(x + y f s F + 1 ) l mt nguyn

hm trn R ca hm s f (x) = p =

* H ng d n Chng minh F(x) = f(x), Vx R .

GII

Hm s F(x) v f(x) xc nh tr n R

. T a c F ( x ) = I n u => F ( x ) = v i u = X + J x 2 + 1u

X X + A'Jx2 + 1 _ u

^/x2 + ^/x2 + yx2 + 11 +



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7/146

\ p j J Vy F(x) = ln (x + J x 2 + ) l

f(x) = 1 (pcm)


8/146

GII

Ta c F(x) = (2ax + b )j2 x + 1 + (ax2 + bx \+ c) .: L=\ j 2 x + 1

hay F(x) = 5ax2 + (2a + 3b>x + b + C

/2x + 1

* F(xj) mt nguyn hm ca f(x) trn khong (- ; + 00)

F(x) = f(x), Vx e (- ; + 00 ) o

5a = -15

2a + 3b = - 3 o

b + c = 2

a = -

b = 1

c 1

Bi 4. Tnh A v B d F(x) = Ax + ln |3c os x - sinx l l __> t/ \ _ B(cosx - sinx)ngu yn hm ca f(x) = . -------------------------.

3cosx - sinx

* Hng dnT n h F ( x ) , c h o F ( x ) = f ( x ) v c n b n g h s c a c o s x v c a s i n x

GII

Ta c F(x) = A +' (- 3sin X - c o s x ) (3A - l) COS X - ( A + 3) sin X

3 cos X - sin X 3cos X - sin X

[3 A -1 = B [AF(x) l mt nguyn hm ca f(x) nn F(x) = f(x) J o

A + 3 = B B

T g ca h

3i 5. Tnh a v b F(x) = e '3x(acosx + bsinx) l mt nguy

hm trn R ca f (x) = e3x(- llc o sx + 13sinx)

Hng dn ,

. T n h F ( x ) = . .. v b i n i F ( x ) v d n g F ( x ) = e~ S x[ ( . . . ) C 0S X + ( . . J s i n x

Cho F(x) = f(x), X e R, ta c h phng trnh hai n s l a v b.

p s a = 2, b = - 5

i 6. Chng minh Inlco sx - sin xll mt ngu yn hm ca hmsinx + cosxsinx - cosx

* Hng dn D.ng nh ngha nguyn hm v ch (n/l ly - .,



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9/146

Bi 7. Tm nguyn hm F(x) ca hm s f(x) =X 3 + 3x2 + 3x - '

X 2 + 2 x + 1

b i t r n g F (l) =o

(e thi TNTIIPT - 2003)

* Hng dn

X3 + 3x2 + 3x - 1 = (x + 1$ - 2 v X2 + 2x.+ 1 = (x + l f

Vy f(x) = ...

Tnh F(l) = .

p s F(x) =

F(x) =(.....' 1

+ c = - o C3

+ c

Bi 1. Hm s' y = lnsinx -3cosx l mt nguyn hm ca hm s no sau

y ?

A) y = cosx + 3sinx

- cos X - 3 sin XC) y:

sin X - 3 cos X

B) y =

D) y =

sin X - 3 cos Xcosx + 3sinx

cos X + 3sin Xs inx-3cosx

_ _ 5 __Bi 2. Tm nguyn hm F(x) ca f(x) = - - bit F(3) = 1

(x + 2)5 . - x + 8A) F(x) =

C) F(x):

X + 2

2x -1X + 2

B) F(x) =

D) F(x)

Bi 3. Tm A v B sao cho -- =+X + 2x X X + 2

A> A = 2 v B = - 1

C) A = 1 v B = 2 ,

B) A = 2 v B = 1

D ) A = l v B = - 2

i 4. t A = Jsin2 xdx v B = Jcos2 xdx, tnh A- B ?

A) A -B = sin 2 x + C B) A - B = -cos2 x + c2 2

C) - B = - sin 2x + c D) A -B = - cos2x + c

6 I I 4 S' i . ' / S T



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10/146

Bi 5. Tm F(x) = J1 - tgx.tg2xtg2x + tgx

+ 1 dx

A) F(x) = cotg 3x + c3

C) F(x) = tg 2x + c D) F(x) = - tg 2x + c

Bi 6. Tm F(x) = j|^3co s--4 co s3| j d x

A) F(x) = sin - - sin4 - + c B) F(x) = 6 sin I - 1 sin4X + c

G) F(x) = - cosx + c D) F(x) = - sinx + c

Bi 7. Tnh o hm ca x(lnx - 1), t suy ra nguyn hm F(x) ca

1A) F(x) = (x + l)lnx - X + c B) F(x) = ln x - A- +c

C) F(x) = X + (x + l)lnx + c ' D) F(x) = In X + - + c

2

Bi 8. Bit sin6 X + COS6 X = 1 - sin22x v sin2X =1 - COS2x

Tnh F(x) = jcos6 xdx + jsin 6 xdx .

A) F(x) = - x - - s in 4 x + C B) F(x) = -^x-J-sin4 x + c8 8 8 32

C) F(x) = x + sin4x.+ c D) F(x) = X.+ sin4x + c8 32 8 8

Bi 9.Tm nguyri hm ca

(2X +

A) - + cXT

B) +c

X3 1C) + - + 2X + C

3 X

_ X 1D) - - + 2X + C

3 X

10



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11/146

A) - ( -icos6x + ic os2 x) + C B) ico s3 x + cosx + C. 1 6 2 ; 3

C) cos6x + cos2x + c D) - cos3x + cosx|.+ c6 2 [3 J

Hg d

n-M I. I , cosx +3sinxBi 1. y = In sin X - 3 COS X => y = ---------------

s inx-3cosx

* . Vy y = In |sin X - 3 COS x| l mt nguyn hm ea hm s

cos X+ 3 sin Xy=

s i n X - 3 co s X

Bi 2. F(x) = ---- + c m F(3) = 1 n n ---- + c = i o c = 2X+ 2 3 + 2 .

* Vy F(x) = ---- + 2 =X + 2 X + 2

Bi 3. = - j - - Xrr = - +

Bi 10. Tm |2sin4x.cos2xdx

2 - X _ 2 - X _ BX2 +2x ~ x (x + 2). ~ X X + 2-

/ A A + B = -1 A = 1o (A +B)x + 2A = -X + 2 r 1

v . [2A = 2 [B = - 2

B i 4. A -B = J"(sin2 x -c o s2 xjdx = J(-cos2x)dx = - sin2x + C

Bi 5. F(x) = Jcotg23x + ljdx = J-r~ 2 = - - ctg3x + Gsin 3x 3

r-1. t o _ w o \ _ ^ 2* + tx * o _ 1 _ 1 - tgx.tg2xCh tg3x = t s( 2x +X) = - -- => C0tg3x= - = ---- -1 - tgx.tg2x tg3x tg2x + tgx

Bi 6. F(x) = J- cos Xdx = - sin X+ c

Bi 7. [x (ln x - 1)]' = ln x -1 + x ^~j = lnx => [x( ln X-1 ) + lnx] = lnx +

* Vy nguyn hm ca lnx + l F(x) = x(lnx - 1) + lnx + cX

hay F(x) = (x + l)lnx - X+ c

11



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12/146

* Vy F(x) =, J(cos6 X+ sin 6 xjx = j + cos4 xjdx = X+ sin4x

Bi 9. j [ ^ j d x = / ( x + x ) dx = | x 2 + -i- + 2jdx = - i + 2x + C

Bi 10. |2sin4x.cos2 xdx = J(sin6x + sin2x)x = - cos6x + cos.2x j +

Bi 1 2 3 4 5 6 7 .8 9 10

p s' D c D c B D A c D A



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13/146

Chuyn 2 T C II P II N

(^)nfi ngfia v tnPi cht)

I. Kin th c c bn1) nh ngh

Cho hm s f(x) lin tc trn on [a, b] v F(x) l mt nguyn hmca f(x) trn on [a; b]

Hiu F(b) - F(a) gi l tch phn t a n b ca hm s f(x)

b

K hiu Jf(x)dxa

bVy jf(x)dx = F(x)| = F(b) - F(a) (Cng thc Newton - Leibniz).a *

2) Tnh chat

f(x) v g(x ) l cchm s lin tc trn on [a ; b]

k, c, m, M l cc hng s".

a/ Jf(x)dx = 0a

a h -c/ Jf(x)dx = - Jf(x)dxb a

b b , b

e/ J[f(x) g(x)]dx= Jf(x)dx |g(x)dxa a a

b c b

f/ c e [a, b] =5 Jf(x)dx = jf(x)dx + Jf(x)dxa a c

b b

g/ f(x) > g(x), Vx s [a, b] => Jf(x)dx > Jg(x)dxa c

b .H qu 1 f(x) > 0, Vx e [a, b] =>. |f(x)dx > 0

a

b

I qu 2 m < f(x) < M, Vx e [a, b] m(b - a) < |f(x)dx < M(b - a)

b

b/ Jcdx = c (b - a)a

b b

dI |kf(x)dx = k |f(x)dx, a a

13



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14/146

II. Bi tp p dng

Cc bi tp sau y nhm gip hc sinh thuc cc cng thc nguyn hmv vn dng nh l sau:

b b ,Nu Jf(x)dx. = F(x) th Jf(ctx + P)dx = F(ax + p)| , a * 0

a a a

xm + 1

m + (m * - 1)

3S _ 2435 5

Si 1. Jx^d x =a r;:'

"l %' I s;,'fea) fx4d';H!y~^| | J 0 - ^ I f

'i 'b) ' f (4 x^J ' 2x?4 j l )4 x- X* - - X 3 - xl *J ~ ' 3 . ;1l

2 - 21.c) J(1 - 3x)3dx =

0

2 _ 283

1 (1 - 3x)43 ' 4

0 0d) J(4x2 - 4x + l) 3dx = J(2x - l) 6dx =

-1 -1

1 1e) J(x3 +; 3x2 + 3x + l)3dx = J(x + l)9 dx' =

- 1 - I

1 (2x - 1)72 ' 7

1093

(x + 1),10

Bi 2. j * i = 4 - mj x n (n - l )x n_1

10

b . 3(n * 1)

3. -I 1 3 . 11a) f-g dx = - 2 = - b) f------

{ X 2x 1 9 0(2x -

1 1 1%

Tdx =(2x - 3) 2 3.(2x - 3)

-'I

1381

c) j. i - g dx = .. -I-_J1( l - x ) 5 4(1. ~ xT

IS64

dx =

/

- ;I (' 1

3(x + 3)

1 1 1 1

d) f--

=--- -

----;--

dx = f----

3-1.0J(x2 + 2x +DM 0J (x + l ) ^h*

Bi 3. J~ =dx = 2\/xj

a) Usx\+ dx = - j5x+ l| = (^ 6 1 )0 : 5 10 5 - ,

? a(*4

375184



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15/146

b)

c)

d)

2 ,2 .j - Xdx = - 2^4:- x| = - 2('j2 - -JE)

1 !j n/1 - 2xd x = -% /l -2x | = n/7 - ^

2 1 , ____ ;

f - - dx = J*4x + 10JV47T 2 = 1

Bi 4. f^rdx = - - i x x

a)

b)

c)

JJ fOxr_(2x - 3)

JJ n_

2 2x - 3

0 /

-v/ '/!

i d - x r-dx =

1 - X

f - i d - 2:

J _ 1 1rdx = -f.-_2(1 - 2x)z 2 l - 2 x

Bi 5. j id x = lnxa

1 1f- dx = ln|2x + 3|I 9

y 4- 3 9 1 1

-1

2

_1_15

' ~ ' .1'

A/,v /1 '

a) = -r (ln 9 -ln 3 ) = 4 In 3

= - | l n | l - 3 x f = - i ( ln | -1 l - In|-2j) = |( l n 2 - l n l l )

1 *' r< . 1 . 1 . 1 0 1I dx = [ - =:. = dx = l ----- 5-dx = f---- - dx-1 4x 7 1 { 2 x - 1 )2 _ { l-2 x

= " l n | - 2 * |1 1 /

= - (lnl - i n 3) = ln3 v-f 2- 2

{Ch X e [-1; 0] => 2x - 1 < 0 => 12x - 11 = 1 2x )

3 V1f- dx = -4 1 nJ , X2 1 - -

1 - = - 4 ( l n - l n )4 2

= -4 (l n 2 - ln4) = '41n2.

15

r



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16/146

Bi 6

a)

b)

c)

d)

e)

Bi 7.

a)

b)

c)

d)

cosxdx = sinx

> 1 l 1 .^r+ In ' \v~Jcos2-xdx = sin2-x = J t..!

2 - 2 ^ X~T^-' ' 'vr -y. \ ;? )h ^ yy '-_

I"cos(- 4x)dx - - sin(- 4x)12' 6 4 6 2

r A.f& Ca^i \ r j | -. (


17/146

b .Bi 8. fV -d x = tgxf

COS X !

. ^ sx . -

A/ 1.,.

c ^ J J

a )

71 -t tt

f i dx = 4 tg y = 4(tg~ - tgO) = 40cos 4 0 r4

4. f .

n9

b) Iit

1

cos 3x-dx = tg3x

= = * 1 - \r

. f , Jr V. /\

12 I/ 'v

%

1 } . 1 , 3 + X ; dx = r t g - J X 2 3

3 ) , L.2 0cos2 *

} vT -?

1 2f 1 J X 2- - dx = t g -2 J 2 X &2 . cos ,3 2 3

~u

.2

h X

1 1

3 4tg4x

V ( V - ? u M : l )

' h ( U ; c ' ^ ; '

*2 1 , , 71 . JI, 1 /* = j(tg i - tg^f) = W3 - 1)

J a)

16

0~Cn nh 1 + cos2a = cos2a ,/1' COS-Ci = oLct>i

- m ok - c t r u ^

f - A d x = - * * * - sin X a_ Sin X ^ // \

* - V. ft

f ~-dx = -co tg (x + )|2 = - (cotg - cotg ) = 20 s in 2/Y + - ) ^ lo 4 4sin (x + )

4

b)^ 1 1f JIdx = - cot g.3xsin 3x a

1 , 371 TC. 1 (cdtg - co tg ) = ~

3 4 2 3

V 1 J_ l' r 1 J---------- dx = ------dx = -c otg, ; l - c o s x 2 ' 2XX Sin -r2 2 2

4 7X,.4 .

hs=r2__

17



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18/146

3 f \ J

Cn nh 1 - cos2a = 2sin2a

bJexdx = .Bi 10.

e4, - e

01ii2 , ln2 , 1

f e2xdx = e2x = ( e 21n2 - e) = ( e ln4- 1) = (4 - 1)

0 2 io 2 2

b) ex +1dx = e*'+1f = e3 - e2 .1

c) fe13xdx = e1- 3x = - ( e - e 4)_1 3 1-1 3

1 2 4 4 2I d) j j dx = J 2dx = -2e 2V y o J 0 c

T c gh71 X

Bi 1. Cho I =cos2 dx vJ = fsin2 dx . Tnh I - J ? 20 0

- 2(e - 1) = 2 (1 - -)e

>A) 1 B) - 1

Bi 2. Tnh A = 'J\/l - cosx dx ?0

C)

A) A = V2 B) A = C) A = 2/2

Bi 3. Tnh I = j |x + j^ - x jd x

11 11A ) I = ,B) I = zr

6 618

D)

D) A = + 1

C) 1 =10

D) I = -10

coI


19/146

n a 2X 1 C a - n 2Bi 4. Cho A = f--------dx (a > 1) v B = -------

0

, ' 0)

So vi gi th it ta c ln(a + 1) = ln3 o a = 2

20



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21/146

Bi 9. I ^ r - ) sin X cos XJ

2

dx 'Vsin2xJ ' sin 2x6 6

dx = - 2cotg2x

of + 71 + 71 _

2*"'J 371 ' 7t12 1^ 1

Bi 10. J = f cos2xsin2xdx =Jsin 4xdx = -co$4x0 2 0 8

12 _ 1

0 16

Bi 1 2 -3 4 5 6 7 8 9 10

p s A c B c D c A D B A

21



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22/146

Chuyn 3 P H l lC ^ G P H P i B l N s

I. L h

1) Dng 1

a) n h l G s hm s' f(x) lin tc tr n on [ a ; b|

* Ham s X = u(t) c ao hm lin tc trn on [a; p]

Nu


23/146

II. Cc dg

Tinh tch phn bng phng php i bin s (cc bi tp sau ytp trung vo dng 2 ni phn kin thc c bn, cn dng 1

hc sinh s gp trong chuyn 7 Php lng gic ho trong vic

tinh tch phn)

hc sinh s gp trong chuyn 7 p

tinh tch phn)71

2

Bi 1. Tnh tch ph n I = Jsinx.ecbs!dx0

GI

t t .'= cosx, =>dt = -s in xd x

n

X 0 J

t 1 0

0 ,Vy I = -j.eMt = e|

1

T a1

Bi 2. Tnh J(2x - 3 ). e f -- 3x +/ d x0

* Hng dn t, t = X2 3x + 2 p s J = 1 - e2

Bi 3. A = ) x ~ 2oV* 2 - 4x + 5

dx

GII

t t = Vx2 - 4x + => t2 = X 2 - 4x + 5

2t.dt = 2(x - 2)dx => (x - 2)dx = t.d t

0_____2

Ta c A

1

T ' * - - 1 - 1

2 ______________

Bi 4. Tnh I = j(x - 2)%jx2 - 4x + 8 dx

^^^[t n g dn t t = \x2 - 4x + 8p s I = (f~- 4)

Bi 5.



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24/146

GII

TC

T a '

7C

Bi 8. Tnh tch phn I =. 4[sinx ,COSX dx0J 1 + sin2x

* Hng dn 1 + siil2x = (sinx + co.sx)2

24



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25/146

1 =sinX - cosX - 1

rdx . (t t = smx + cosx) p so T. = -4rr,2 I f r Q(sin X + cos x)

Bi 9.

71

2

1 = ]0

sinx

2 + 5cosx-d x

t u = 2 + 5cosx =>

GII

du __ 5 sin71

X 0 2

u 7 2

1 2 - 1^1 1Vy I = - 4 f-du = - f-d u = -InIII

5 7Ju 5 "U 5 1= (ln7 - ln2)

2 5

Bi 10. T nh tch

T a

ircos22x - sin22xX

ch p h n A = fco -- 2x0 1 + 3sin4x

dx

" 2 2 Hng dn COS a - sin a = cos2a

A = f-os^xd x, t t = 1 + 3sin4xg l + 3sin4x

, p s' A = ln26

Bi 11.

. i A

T _ 6f ln x - 2 J

GII

t t = ^/ln Xr1 => t = lnx - 1

1 */+2 _ 1 1 /^ Vy A = 2 I; dt = 2 f(t2 - t)dt = 2

0 + k I' >

-2tdt = dxX'

X e e2 't 0 1

t2' 1

2 j 3

K fl 2-t l . =25



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26/146

T a

vs 1Bi 12. Tnh tch ph n J = f------- -------5-

~ / x ( l + lnx)

* Hng d n t t = 1 + lnx

1; :',::y-!'':'1' ';

I . I0

dx

Bi 13.f(5 + tg x )

COS2X

t t = tgx =>

d t

GII

dXi

v \

COS2 X71

0X 0 4

t 0 1/ T 3 1

Vy I|= j(5 + t 2)t) = 5t + =V o. , 0

163

T a

a 1- ) Tnh tch phn I a [- ,c 2 x {

s in X

1 .* Hng d n t t = cotgx ,( U0 \ p s

-dx

A 11

Bi 15. A . f e - i d x Q COS X

GXI

t t = tgx => 71

. x | 0 - 4t o 1

^ t s V y A = |(e2t + l)dt = -e ^ /+ t|J = li fe 2 :1) - ~ = ~Ce2 + 1)I Q /i i\ f ~iCi

>~~\ X *' . 1, . . . .

26 '

>dt =1

cos2Xdx



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27/146

T a

71

2-1 Ocotg x I

i 16. Tnh tch phn I = f------%------. . s in X

* Hng dn t t = cotg X

sinx .cosx

p s' 1 = 2

Bi 17.

* Hng dn

Ta c I

T

2r sinx .cosx1 + 2sin2x

dx

sina.cosa = sin2a v (sin x) = sin2x

GII

l ri -* sin 2x dx+ 2sin2x

t t = 1 + 2sin X =>

dt = 2sin2xdx => sin2xx = -rdt2

7t

0 9

=> I 3fdt = 1 ' , 3 l ln34 t 4 1 1,1 4

T g ca h

Bi 18. Tnh

t .4f l - 2sinzx

in h tch ph n = f---------0 1 + sisin2xu

Hng dn

S dng cng thc cos2a = 1 - 2sin a

t t = 1 + sin2xGII

dx (H - Khi B - 2003)

+ sin 2x

dt = 2cos2xdx

COS2x dxsin 2x

t t = 1 + sin2x

4rl - 2s in2 x , % cos26 ------ dx = - -

' 1 + sin2x J 1 + si

V y 1= i = l n t 2 t 2

n0_____L ,

- n 22

27



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28/146

T a

%2

Bi 19. Tnh tch phn A = [-----0-3 - si

sin2x

3 - sin X + 4cs X-dx

2 2in X + 4cos X

Bi 20.1 2

I= J-------- -0J 1 + 2x +

1Ta c I = J-

sin2x , (cos x ) = -s n 2 x

-dx

GII

1 2 , d x , t u = 1 + X3

( 1 + X 3 ) 2

Ta c d u = 3x d x = > X d x = dU '

X 0 1

u 1 2

Vy I , I f i d u

d 1U

1Bi 21. A = J

.1 1

3 u

T a

xdxq(4x + 4x2 + l )2

Hng dn

A = J. xdx0( 2 x 2 + l ) 4 '

t t = 2x + 1 p s A =

T g ca h

Bi 22. I = f-I s

2. 2X - 1X3 + 4x2 + X

dx

28



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29/146

M\ .. GII ,1 - - L/ - 2 ---- 2 1Ta c I = f------- ---------- dx t t = X + - + 4

1. (x + ) + 4X

dt = f l - - i - 1d x 3f v x ' Vy I = f -d t = ln|t||g2 = l n l ^ -

1 1 * fi

lnl2

Bi 23.

2

* Hng dn J = J

T a

T r X + 1 JJ = -----5---------- -dx'x (x - 1) + 2x

1 1

2 2 1 2 1 +t= Jh V .--d* = J ^ - '

J X + 2x - X I X _ 1 + 2,X

t t = X - + 2X

-dx.

p s J = ln7 21n2

i 24.21n2

A . I 0 e2x + 4ex + 4

dx

GII V/vl'V i M ^21n2Ta C A = J

n

t t = ex + 2 ^

0 . (ex + 2)2

dt = exdx

X 0 21n2t 3 6

dx

T a

Bi 25. B = J -rr---- ---- ----------------- dxln 3

B + 3e2* + 3ex + 1

* Hng dn

ln3 - X*:1f . t t = ex + 1

0 (ex + l )3



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30/146


31/146

Hng dna a

...... = + jf(x)dx = 0- a

t n

, _ 4 2 x 3 - X ^ - 1

p dng I = dx + J dx COS X - COS X

Xt hm s' f (x) =

f(-x)

2x3 - X

COS2 X

lin tc trn onK n

4 4v ta c:

2(-x) - ( - X ) s

cos2(-x)

31



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32/146

Hm s f(x) lin. tc v l trn on 71 7

' V 42x - X

COS2 X

dx

7C4 1

Vy I = J =2dx = tgx|


33/146

Bi 6i 6. Tnh 1= f, = .....=--dxXV1 + In X

A) I = V2 B) I = 2( n/2 - 1) C) I = 2 V - 1 D) p s khc .

73 , ____

Bi 7. Cho' A = -x t t = V1 +X2 th A tr thnh :1 xvl + x2 ,)L

A ) A = C> A = f r r D ) A = F 1- 1 1 1

* c 71

4 - 2Bi 8. Bng cch chn bin s t nh th no yJ3 + tgx. ^dx = 2 I"t2t

0 cos x i

A) t = 7tgx + 3 B) t = 3 +yftgx C) t = tgx D) t = 1 V3 + tgx

Bi 9. Bng cch t t = \/x +1, hy bin i : $ -0.0

1= f/x+(x + 2.)dx = ff(t)dt) a'----Q~ .t = ; x = e=>t = 2

:''jl 2 2* Vy 4 =;J t.2t.dt = 2 J t2dt = 2 J x2x = 2C

is/3 ' J Sn4

Bi 4. I = %/cos X.+ sin X (cos X + s i n x ) ( c o s x - sin.x)dx0

(v cos2x = cos2x - sin2x = (cosx + sinx).(cosx - sinx))

4-2t 2 = cos X + sin X => 2tdt = (cos X - sin x) dx

x = 0=>t = l ; x = => t = Vi4

t t = Vcosx + sin x

i* Vy I = 2 J t 4dt

1

2 xBi 5. A= fVcosx.sinxdx .(v COSX = l- 2 s i n )

0 2

t t= \Jcosx (t >0)

0 1* Vy A= jt.( -2 td t) = 2 j t 2dt

1 0

Bi 6. t t = x/r+lnx => ( X[ x = l=> t = l ; x = e=>t = >/2

72. J2* V y = j i .2 t .d t = 2 j d t = 2(V2- l )

1 * 1

t 2 = cos X => 2tdt = - sin X dx

x = 0= >t = l ; x = =>t = 02

t2 = 1+ InX => 2tdt =

Bi 7. t t = vl + X2 ( t > 0 ) :

t2 = 1 + X2 => 2tdt = 2xdx

X2 = t 2 - 1

x = l=i>t = V2;x = V3=>t = 2

34



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35/146

Bi 8. t t = - 3 + tgx (t > 0)t = 3 + tgx =? 2td t =

COS2 Xdx

X = 0 = > t = \ / 3 : - x = = > t = 24

71

4 1 2 2

Vy t/3 + tgx.-dx = j t.2 td t = 2 J t 2dt0cos x

Sccurica U

Bi 9. t t = %/x +

t 3 = X + 1 => 3t2dt = x .

X = t 3 - 1

x = 0=> t = l ; x = 7=>t = 2 x = u=> t = i ; x = 7=>t =

* Vy I = Jt(t3 + l).3t2dt = 3 J(t6 + t3)dt1 1 .

Bi 10. t u = 2 = e* - 1 => 2udu = eMx[ x = 0=3>u = 0 ;x = l n 2 = > u = l

1 19 1 9Vy I = |u .2udu = 2ju2du = u3 =

0 03 10 3

Bi 1 2 ,3 4 5 6 7 8 9 10

p s D B 'B D B B c A D A

-1 '

' 35



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36/146

Chuyn 4

TCH PIIN CA IIM s HU T

I. L h

Trong chuyn ny, tt c cc hm s di du tch phn'utc trn on ch ra.1) Cng thc

a/Jdx = lnx||P Tng qut - dx = n|ax + b||p (a * 0)a x

p 1

b/ dx! * n (n - l)xn

1f------------------ d x =

(ax + b)n

(n * 1)

1

a(n - l)(ax + b)n3

a V /

p 1 1

t bit (T-dx = ~ j x 2 X

p 1 Tng qut f----- - d x =

j(ax b)2

(a * 0 v n * 1)

1 1a ax + b

(x - x0)n X - x0 (x - x0)2

3x - 5x + 2 A B

P(x)2) Phn tch hm s hu t y = v dng tng

* iu k i n Bc ca P(x) < bc ca Q(x)

* Dng 1 p(x) A B

V d

* Dng 2

V d

Tng qut

(x - x0)n

( x - l ) 3 X - 1+ ----

( x -- 1 )2 ( x - l ) 3P(x) A B c

(x - x1)(x - x2)(x -- x 3 ) . .. X - X - x2 X - Xg

2x2 - X A B c(x - l)(x + 3)(x + 8) X - 1 X + 3 X + 8

P(x) A B

,)n.(x - XjKx - x2).... X - X 0 (x - x0c D E

----------- + -- ----- + -------(x X q ) n X -X i x - x 2

V dX3____________ A B c D

(x + 2)2(x - l)(x + 3) x + 2 (x + 2)2 X - 1 X + 3

36



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37/146

t bit = a ( - A =(x - x0)(x - xx) Vx _ x0 x _ x i J V x0 " X1

. V d (v cch tnh A, B, ...)

, 2x2 - 9x + 10 ,Phn tch - - v dang tng.

X - 3x + 2

' Mu s' X3- 3x + 2 - (x - l)2.(x + 2)

v 2x2 - 9x +10 = 2x2 - 9x + 10 = A B cX3 - 3x + 2 (x -1)2.(x+ 2) x - 1 (x - 1)2 x + 2

o 2x2 - 9x + 10 = \A(x - l)(x + 2) + B(x + 2) + C(x - l)2 (*)

tnh A, B, c

Cch 1 (Phng php ng nht cc h s cng bc)

T (*) ta c 2x2 - 9x + 10 = (A + C)x2 + (A+ B - 2C)x - 2A+ 2B +c

A + c 2 A = - 2

o A + B - 2C = -9 B = 1

-2A + 2B + c = 10, c = 4

. Cch 2 (Cho X cc gi tr thch hp)

T (*), cho X cc gi tr thch hp.

X= 1=>3= 3B o B = 1 X = - 2 = > 3 6 = 9 C < = > C = 4

X= 0=>10= -2A + 2B + C-vi B = l , C = 4 o A = - 2

2x2 - 9x + 10 2 1 4Vy ; ----= --------- + ---- - + -

X - 3 x + 2 x - 1 ( x 1) X + 2

3) Tnh tch phn hm s'hu t I =JQ(X)

a/ NuC(5mt trong cc dang , hoc - , ------------- -----1 th taQ(x) X xn 1 ax + b (ax + b)nJ

p dng trc tip cc cng thc, ni mc 1 tnh I

P(x)b/ Nu cha c dang ni trn (3a) v bc ca P(x) < bc ca Q(x) th

Q(x)ta tin hnh nh sau :

I g BC 1 Phn tch mu s' Q(x) v dng tch so._______________________



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38/146

BC 2 Phh tch -- v dang tng (ty theo mu s Q(x) chnQCx)

'; mt trcing'jcac dng ni mc 2).

- Tnh cc h s",B ...

Bc 3 Tnh I bng cch s dng cc cng thc mc 1.

S(x)Cn nh Nu gp hm s hu t y - - m bc ca s (x) > bc ca Q (x)Q(x)

th ta phi thc hin php chia a thc bin di hm s y v dngP(x) .

y = T(x) + (trong bc ca P(x) < bc ca Q(x) ri tip tucQ(x) ' ' '

nh ni mc 2). -

3x4 - X3 - 7x2 +8 'V d . y =

X 3 - 3x+ 2

2x2 '-9X+-10Thc hin php chia a thc, ta c y = 3x - 1 + 3 3x + 2

9 2 1 4 y= 3x- 1 --------- + ----- f + -----x - 1 ( x - 1 ) x + 2

II. Cc dg

Bi 1. Tnh

a) I = I 22x + 6 dx b ) J = J-0J X - 2x - 3 0J j

l l x + 6 ,dxX - 2x - 7x - 4

* Hng dn

Phn tch hm s' (di du tch phn) v dng tng s' (ch, dng,tch s ca mu s - xeiii mc 2 - kin thc c bn) -

Tnh tch phn.GII

2x + 61a) I = j

X 2 - 2x - 3dx

9v + fi A R Mu s - 2x - 3 = (x + 1) (x - 3) Ta c - = ------ +

X 2 - 2x - 3 X + 1 X - 3 2x + 6 = A(x - 3) + B (x + 1) (*)

Tnh A v B

Cch 1 T (*) ta c 2x + 6 = (A + B)x + (B - 3) O- ^ ^ ) 1[B 3A = 6 [B =3

38



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39/146

- Cho X = - 1 => 4 -4ACch 2 T (*)

A = -1

- Cho X 3 => 4B = 12 => B = 3

2 x + 6 _ - 1 3x 2 - 2 x - 3 ~ x + 1 X - 3

3 - 3 x + 1 X - 3 J

hay I = 31n IX - 3 I - ln |x + |* = ' 21n2 - 31n3.

1b) J =

0 J

l l x + 6 2 x l - 7 x - 4

- d x

Mu s X3- 2x2 - 7x - 4 = (x + l)2(x - 4)

Ta c l lx + 6 _ A BX3 - 2x2 - 7 x - 4 ~ x + 1 ( x + l )2 X - 4

ll x + 6 =A(x + 1)(x 4) + B(x - 4) + C(x + l)2 (*)

Cho x =- l = > - 5 = - 5B. => B = 1

Gh X = 4 => .50 = 25C => c = 2 'Cho X = 0 - 4A - 4 B + C = 6 => A = - 2

T (*)

Vl l x + 6

X3 - 2x2 - 7x - 4

-2

V X + 1 ( x + l ) 2 X - 4 ,

2 1 2------------ - - --- --- --- - --------- -+ --- --- --- --- --- -

x + 1 (x + l r ' x - 4

dx

- 2Inx + 1 + 2In[x - 4||fx + 1 1 1,0 3

T.g ca h

'2 2 2(ln3 + ln 2 ) = - - 2 1n 63

Bi 2.1

a) Tnh I = j0

2x + 1

2 2X

x z + 3 x + 2

7 x + 2 4

d x

j ' x 3 - 5 x 2 + 3 x + 9dx

* Hng d n Gii g i n g b i 1

2 x + 1 3 1a )

.X2 + 3 x + 2 X + 2 X + 1 p s I = 3 1n 3 - 4 1n 2

b) X2 - 7x + 24' 2 1 3 , T , 3 o --------- = T - + 5- p s J = 2 1n 3 - l n 2 + 5 x + 3 x + 9 X + 1 X - 3 ( x - 3 )2 2

39'



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40/146

2 1I = ) 2 -

, x z + 3sBi 3. 1 = -----1-------- dx

I X + 3 x + 2

* Hng dr ------------------ = AI - - 1 (A = ---------- -( x - x 0 ) ( x - x x ) [ x - x 0 x - x i ) Xq -

GIIm 1 1 . _ 1 1Ta C: ------- = --------------------------------------------- ----- =

X + 3x + 2 (x + l) (x + 2) x + 1 X + 2

X + l | l n | x + 2 ||^

X + 12

- In - ln -! 4 3X + 2

= In

T a12 . 1 ^

Bi 4. a) I = -=r-----dx b )J = I- -Jx - 1 J 4 -0* 1 -1*

* Hng dn = - f - v . = - f - ^ 7 + yX - 1 2 V X - 1 - x + 1J 4 - x 4 . 2

p s a) I ; = - ln3 b) J = ln32 . 2

Bi 5.

a)Tnh M v N sao cho f (x) = r3x2+ 5x - 1 = -M _ + N(2x ~X -2 x + 2 x \- l X -1 x - x

0b) Tnh J f(x). d x .

-1

Hng dn

a ) - 3x2 + 5x - 1 = M(x2 - X + 1) + N(2x - l)(x - 1) r i cho X = 1, X =

b) p dng kt qu cu a v t t = X 2 - X + 1 i vi J ------ d

GII

^ PM ~3%2 + 5x - 1 -3 x2 + x - 1 Ma) f(x) = - : " ------- = ------ --------- =

X - X + 1

-3x2 + 5x - 1 = -3x2 + 5x - 1 _ M . N(2x -

X3 - 2 x 2 + 2 x - 1 "" (x - 1 ) ( x 2 - X + 1 ) ~ X - 1 X 2 - X +

- 3x2 + 5x - 1 = M(x2 X + 1) + N(2xl)(x - 1)

40



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41/146

Cho X = 1 => M = 1

Cho X = 0 => ' - 1 = M + N => N = - 2

Vy f(x) = L ------------ y 2x. r (M = 1,N = -2 )X - 1 X - X + 1

b) Jf(x)dx = ln |x - l|| - 2 J. 2^x ~ dx = - ln2 - 2J

- \ T ' : T f 2 x - 1Vi J = T----------dxX 2 - X + 1

dt = (2x - l)dx

X + 1 ^ 'X - 1 0

.t 3 1

1

Vy J = j = lntg = - ln3 p s I = - ln.2 - 21n3..3 t

T a

Bi 6. a) Tnh A v B sao cho - , + 2x -X + X + 2

b) Tnh I = 2f* 2 + 2x - - -3dx0J X 3 + X + 2

Hng d n Gii ging bi 5

a) A = - l v B = l

Bi 7. Tnh A = 2*2 ~ x ~ 1 dx v B = v - dx' X + 1 2 - X

GII

A = J- - - -d x - j2 x -3 + . 2 -ld x = X2 - 3x + 21n|x + l||2 = - 2 + 21n3

0 X+ 1 A . X + 1J B = J^ 3d x = j^ -3 + - ^ - j d x = - ( 3 x + 71 n | 2 - x | ) | = 7(ln3 - ln2) - 3

T g ca h

2 fBi 8. Tnh tch phn I = f x ' * I dx ( tlii TNPT - 1998)

- iV x + 2 J

A B (2x - 1)x + 1 X 2 - X + 2

b) I = ln2 - li3.

41



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42/146

* Hng dn

Phn tch ----- - = A +X + 2

, Tnh tch phn I

Ta c

X + 2

GII

X + 2,

X - 1

X -t- 2= 1 -

2 'Vy I = 1 + -

-l i t = 0 ; x = e=> t = lBi 2. t t = lnx =>

* Vy 1 j r f r 11 l1- 14 f ) d* = j - * * M t * 1)

Bi 3. ~r = + A(x + 3).+ B(x + 1) = - X+ 3X +4x + 3 x + 1 X + 3 /

A + B = -1 | A = 2

. 13A + B = 3 ] B = -3

, V 1 dt = exdxBi 4. t t = e {

= 21n 2

o ( + B)x + (3A + B) = -;X + 3 o * o 2(3A + B = 3

, V [ dt = exdxt t = ex => {[ x = 0=>t = l ; x = ln 2=> t = 2

IT* T-11? 6* -1 X, V , 2 V.Vy 1= ------ -.e dx = - d t = 1 dt0 ex +1 . Jj t + 1 /1 t + 1J

\= [t-2 1 n (t + l)]|2 = l + 2 (ln2 -ln3 )

lYx4 + 2x 2 + l) + l] xd x 1 (x2 + l ) 2 +1i 5. . I = P ---------- 52 ------= fi1-i------- .xdx

0J X + 1 0J x 2 + l

t t = X + 1 = *dt = 2xdx

x = 0 ^ . t = l ; x = l=>t = 2

y 4 I . l A d t. i ^ ] dt

dt = dxBi 6. t t = lnx=>- X >

I x = l => t = 0 ; x = e=>t = l



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45/146

Vy A = J:t 3 - l

r +1 +1

Bi 7. t-t = X - 2 x + 2 =>

1 t 2dt = J(t l) dt = - t

0 2 .

d t = 2 ( x - l ) d x

12

x = 0 = > t = 2 ; x = l = > t = l

, 1 Vdt 1. f 1. Vy A = = nt = - ln22 t 2 |2 2

B ' o 4. 3/]t 3=X=>dx=3t2dtBi 8. t t = /x=>{[ x = 0=>t = 0; x = l=>t = l

1t 3 + l 3t2dt 1 1Vy J = p ----- --------- = 3 j t 2 (t 2 - 1 + l) d t = 3 J(t4 - 13 + t 2)dt

0 t +1 0 0

f(t) = t4 - 13 + 12

Bi 9. f(x) = 2x - 3 x - 1

Bi 10. t t =yfx.=>

Vx = t 2 ; t/x = t3

x = t4 =>dx = 4t3dt

X = 1 = > t = l ; x = 16=>t = 2

* Vy 1 = 2( t 7 ) = 4 ( t - 1)2dt = ! ( t 1)5=

1 = 3

Bi 1 2 3 4 5 6 7 8 9 10

p s c A D B D D B A c B .

45



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46/146

Chuyn 5

TCH P S p p C l ll M S LI\G GIC

I. L thuyt

(Trong chuyn ny, tt c cc hm s di du tch phri u lin tc trn on ch ra)

1) Cng thcp p

a/ jcosxdx = sinx|P b/ js inxdx = -cosx| ' a

c/ ^ dx = tgx p d/ \ - dx = -co tgx' cos X sin X

L d n Hc'sinh phi thuc t t c cc cng thc.p .

2) Tnh tch phn ca hm s lng gic l= Jf(x)dxa

(Trong f (x) l hm s lng gic lin tc trn on [ a ; p ])

Vguyn tc c,bn

hoc Bin i hm s f (x) v mt trong cc dng c cng thc tnh tchphn nh ni mc 1,

hoc S dng phng php i bin s bin i:15 t, .

I = J"f(x)dx = g(t)dt trong g (t) c nguyn hm l G(t).

to

II. Cc dg

* Dng 1

p _I = P(x).Q(x) trong p (x) v Q (x) l c c hm s lng gi c c dng

acos(a x + b )sin (cx + d) (a # O v c ? 0)

Bin i cc hm s lng gic t dng tch th nh dng tng.

S dng cc cng thc a v b ni mc 1.JT n2 1 2

V d = jcos Sx.cosxdx = j(cos4x + cos2x)dx0 2 0

46



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47/146

sina.sinb = [cos(a - b) - cos(a+ b)]

sina.cosb = [ sin(a + b) + s in(a - b)]

i 1. Tnh cc tch phn sau :. JI T .

12 8a) I = j c os4 x . cos2x dx b) J = Js in 3x .COSX dx

0 0

GII

^ 12 1 1 1

) I = 4- f(c os 6x + cos2x )dx = ( sin.6x+ -r sin 2x)2 0 2 6 - 2 0 ^

X

' .

j = (sin4x + s i n 2 x ) d x == - ( c s x + c o S 2 x ) 8 = ~ ^2 0 2 4 2 . 0 . 8

T an 71

8 3

a) A = Jcos3x .CSX dxb) B =Jsinx . sin2x dx0 0

p so a) A = ----- b) B = 8 4

= + +

12_ 1 1 1. _ _5__ 2 6 + 4 24

Dng 2

S 1 P

I = Jcosnx dx v J = |s in nx dx (n 6N, n l v khng qu ln, n > 3)a . .0

T nh I - Bin i cosnx dx = cos" 1X. cosx dx = (1 - sin2x)k. cosx dx

* [ d t = c o s x d x 9 ,- i bin s t = sinx =>J , Vy I = |(1 - t ) dt

[ i cn tJ

/ Tnh J (tng t tch phn I)

sinnxdx = ....... = (1 - cos2x)k.sinxdx t t= cosx => - d t =s i nxdx

i 2. Tnh cc tch phn sau : 7 . Tt

4 - 2

a) I = |c o s5xd x v b) J = |s in 7xdx 0

47



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48/146

GII71 3T TC

4 4 . 4

a) I = Jcos5 xdx = co s4 x.cosxd x = J(1 - sin 2 x)2.cos xdx0 , 0 0

t t = sinx =>

' dt = COSxdx7t

X 0 7

0 JL

1

Vy I = ' l (1 - t2)Mt = ' l (1 - 2t2 + t4)dt = t - +0 0 3

7 . l 71 V

2 2 2

b) J = jsn7 xdx = J(sin2x)3sinxdx = J(1 - COS2x)3sinxdx0 0 0

J = 43

n ~ 6 0 V

dt = - sin xdx => sin xdx = - dt

t t = cosx =>< X 0_____2

1 0

Vy J = J(1 ~'-t2)3(-dt) =/ ( 1 - 3t2 + 3t4 - t 6)dt = t - t3 + - 1 0 5 7

1635

T g ca h1C

2

Tnh tch phn J = jsin 5xd x0

( thi TNPT - 1994

* Hng dn Phn tch sin5x = (1 - cos2x)2.sinx ri t t = cosx

GII

* Ta C sin5x = (sin2x)2. sin x = (1 - cos2x)2.~:

t . 2 2

Vy J = J(l-cos2x .sinxdx t t = cosx => 0

Ta c J = j(.l - t 2j .(-dt) = JYl - 2t2 + t4 jd t - t - t 3 + t 1 0 3

smx

dt = - sn xdx7t

0- 9 .

15

48



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49/146

T a- .6 4

a) A = jc o s5x d x b) B = Js in 3 2 xdx 0

9 f ) 0 1

p s' a) A = - b) B =

480 3.Ch sin32x = sin22x. sin2x = (1 - cos22x) sin2x.

t t = cos2x .dt = - 2sin2xdX

Dng 3p p

I = Jcosnxdx v J = Js innxdxa a

(ne N*, n chn v khng qu ln)

a/ Tnh I Bin i cosnx = (cos2x)n/2 = cosfe j

(Khai trin (1 + cos2x) n / 2v tip tc bin i sau cng cosnx c dngtng cc hm s lng gic m ta c th s dng c cc cng thc ni mc I)

Cn nh cd x = c(p - a) COS2a = cos^ai 2

b/ Tnh J Bin i sinnx = (sin2x)n/2 = cos2xj

(Sau tip tc ging vic tnh tch phn I)Cn nh . 2 1 - Cs2asin a = ------------

n % -4 4

Bi 3. a) Tnh I = Jcos2xdx v J = Jsin2xdx.t 6 . 6

7 t . J t

12 8

b) Tnh A = Jsin6xd x v B = Jcos4xd x0 0

GIIn ' ..-* 71

a) I = fcos2x d x = f(l + cos2x)dx = ( x + s iil 2 x )4 = I i - 2 2 2 * 2*' E" ' C ^6 6% X

- / J = Jsin2xdx = J (1 - cos2 x)dx = ( x - s i n 2 x ) 4 =

_ 2 _ 2 2 2K K \6 ' 6 6

f j L12 + 2 4

lf_7__ 1 V312 2 + 4

49



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50/146

Ch Ta c th tnh I v J cch khc.

n K6

71

4,

I + J = l.dx = - 46

. 54 4 1 '

I - J = J(cos2X - s i n 2x)dx= Jcos 2'xdx= sin 2 x5 7 t ^

6 6

T (1) v (2) ta tnh c I v J ,

12

b) A = sinexd x0

Ta c sin6x = (sin2x) 3= f - cos2x j _ 2: ( 1 _ 3cos2x + 3cos22x - cos32x)

1 - 3cos2x + 3 1 + cosx^ f3c os2x + cosx

2

-(10 - 15cos2x + 6cos4x - cos6 x)32

Vy A = (lOx - sin2x + -sin 4x - - s i n 6 x)J 32 2 2 6

12_ I O t i 4- 9^3 - 47384

__.

3

3cosa + cos3a. 3

3sina - sin3aCn nh COS a = --------- ---------- , s in a = -----------------

us

B = |c o s4x dx0

Ta C cos4x = (cos2x)2 = ( - - i = - (1 + 2cos2x + cos22x)

= ( 1 + 2 cos2 x +

4

2 J 4

1 + COS4x. 1 , n . A -----------)=-r(Z +4cos2x + eos4x)

2 8

Vy B = [3x + 2sin2x + s in 4x ] 8 1 'E + A + 0

000

50



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51/146

T g ca h

a. Tnh cc tch phn sa u :2

a) I = Jcos24xdx0

71

b) b) J = fsin2dxJ 2

0

( thi TNPT - 1999)

* Hng dn

i vi I dng cng thc cos2a =

i vi J dng cng thc sin2a =

GII

1 + COS2a

1 - cos2 a

-2 1 ^ . 1 1 "12 7) = fcos24xdx = fil + cos8 x)dx = X + sin 8 x = 0J 2 0J 2- 8 Jo 4n 71

) J = Js in2|d x = | ) ( 1 cosx)dx = [x - sinx = - - i j

T tra

b. Tnh tch ph n K = js in 4xdx p s K = - 2j

Dng 4

p ./ Tnh =: fsm * dx (i b i n s t = c g s x )

J co sxa

pKt qu Jtgxdx = -In cos x|p

up

/ T n h J = [COS- dx .(i bin s t = sinx)Jsnxa

Kt qu cotgxdx = ln |sin x||a

51



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52/146

c) A

a) I

t t

Vy I

b) J

t t

Vy. j

c) A

Bi 4.

a) I

52

Tnh cc tch phn sau :7[

= 2[ t g - d x0J 2

tg 3x

1 . X2 J sin

= )tg |dx = ^ f0 0 COS-Z

l. 4

b) J = j cotg2x dx TC

12

71

d) B = J tgx' ' t g X

GII

dx

Idt = -.sin dx => sindx = -2

dt2 2 2=;>

X 0

7 .2

t 1 1

. . 7 2

d t1

= 2 d t = lnltlfi .

t 1 t V2

JI 7C4 4

J cotg2 x dx = JJt -.TC-

12 2

cos2 xsin2 x

dx

sin2 x =>

' t = 2 cos 2xdx => COS 2xdx = dt2

t . . ' 7112 4 ' '

i VdtJ

In = In 22 2 2

18f l - tg 2 3xI tg 2 3x24

71 ' 7C

dx = 2 fcotg6 xdx = 2 f cos6x dxJ J sin 6 x ''

TC Tt

24 24



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53/146

t t = sin6x

'd = 6 cos6 xdx => cos6 xdx = -rdt6

7t It. x 24 18

t 1

V1 2r d t6 J T

J2

N * - f i n - l n 31 2 J2,. Vy A = 2 . )

J2

n ' . 7Z n . T

) B = J tgx dx = - I f . 2tgx dx = - ] t g 2 x d x = ' - j0V x - l 2 Q 1 - t g X 2 0J 6 2 0J

(ln3 - ln2)

1 rsin2 xdx

dt = - 2 sin 2 xdx => sin 2 xdx = -dt

" 2t t = cos2x => - LJL _0________8x 1 J _

Vy B = 1 ' ] ^ = _ I j i d t = - - l n | t | | 1- .= - . - ln = - - in 24 t 4 J t 4 '_i_ 4{ J 8

T a

7C4a) I = jtgxdx

0

p s, a) I = ln2

TC

_ 8f 1 tg 2 2 x ,b) J = -------2----dx

n tg 2 x16T

- 8 . 1

b) J = 2 cotg 4x dx = ln2

16

Cn nh tg2a = 2ts a1 - tg a

Dng 5

a)p. 1 ,p. t

I = [,------ dx = f--------- dxj l + cosx A c o s 2 ^

2

p 1(Sau dng cng thc, frdx = tgxf )

a COS X

Cn nh 1 + cos2a = 2cos2a

53



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54/146

p 1 1 p, 1b ) J = ! _ _ _ i ---- d x = J d x

J 1 c o s x 2 a s i n 2 -2

e 1(Sau dng cng thc f^x = - cotgxf )

sin X

Cn nh 1 - cos2a = 2sin2a

M rng dng 5

Lc I = -V

'Snx v d ng tch acosx bs inx = i/a + b .cos(x cp)

1 1= = ' I-------------------dx (sau tip tc nh trn)+ b2 i 1 cos(x


55/146


56/146

Dng 6p 1 p 1

I = f- dx v J = ; f . dxcosx 1sinx .a ,a

aJ Tnh I = ' c ^ - d x = I C O S * 2 dxJ c o s X J 1 - s i n X

i bin so t = sirix Ltfu ^-77= ------- ------ = 1 I1 - t 2 ( l - t) ( l + t) . 2 U - t 1

b/ Tnh J = f slnx dx = f S1I~- dx (t. t = cosx)J s in X J 1 - c o s X

* M rng dng 6

p !K =; f---------- ----------- dx (a2 + b2 > 0)

Jacosx bsinxa

Bin i a cosx b sinx v dng tch s

a cosx b sinx = Va2+ b2 COS (x (|j) hoc a COSX b si nx = Va2+ b 2 sin (x

K

hoc K =

- ==.,/ ------- ----- -d xVa2 + b2 jcos(x cp)

Va2 + b2 'jsinCx- : Cp)

I 1Bi 6. a / J = * dx; sin3x

12

71

6

c/ A = s

p 1f------------ dx , sau tip tG nh trn.sin(x :cp)

T

? 1b/ K = f - ----- -------*sinx + cosx

dx

sinx + V3cosx

GIIa 71

a/ J = J - ^ d x = ) - js in 3x l12 12

sin3xco s 3x

-dx

t t = cos3x =>

dt = - 3 sin 3xdx

X 7t

. x 1 2 6.t 4 2

0

d l3

56



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57/146

Vy J = -

J2 ' . J2

{ 1 dt = 1 f ---- dt = - \

42

= -In61 + t1 - t

0

1 "r 1

-In

1 +

42

7t4

b/ K = f -o si

dxsin X + cos X Xn/ 2 sin(x + )

4 4

- d x

du = dx I

71 I t u = x + J => { u 0 4 \

t I - - /, I 4 2 .

7t K_ Tr 2r d 1 2f sinu , 1 2f sinu ,

Vy K = -T=-- = -J= : du; - -?r |- - 2 du^ %/2 sin u ' s/2 , sin u - V2 1 - COs u4 4 4

dt = - sin udu => sin u du ~ - dt7T 71

t t = cosu =>< u 2

. * 0I >/2

Vy K = - - L J - ^ d t = 1 yf i - J _ r + r - ^ - r i d t7 2 J 1 - t2 2 V2 0J u + t 1 - t j

71 : _1_

. J O 11* .1 a . = ^ 1 . ( 3 , * f .

71

i - --------1 e i n V J _ c/ A

Jsmx + %/3cosx

Bin i

sinx +%/ cosx = 2 ( cosx + - sinx) = 2 ,(cosx.cos-+ . sinx. s in ^ ) = 2 cos(x -2 2 6 6 D

57



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58/146

Jt6 ^

Vy A = - f----- ------ dx = f------ du (u = X - ) (xem cu b)2 / n\ i cosu 6

0 cos(x - _6 s

COSu du

- sin2u~6 6

* t cosudu

= sinu => u7t6 0

t 1 0

Vy A == ) - L ~,2 1 1 - t 2

~2

- 1 i 1 + 14 I U + t 1 -

2in l1 . tffi = 4 l n

~21 + t1 - t

dt

0

= n3 1 4

It* 1

1 = H :*cos:

T a

a)

*Hng n

_ _

'l

I =

-dx

COS X COS X

cosx

dt = ln2

, t t = sinx, ta c:

p s' I = ln(3 + 2 \ 2 )2

b) -dxcos2x + sin2x

Hng dn cos2 x + sin2 x = \2 cosi...) = \2cosu (u = 2 x - ....)

n nX

8 4

u = 2x - - 0 14 4

T1 * 1

Vy T = f-du, sau tip tc gii ging bi 6 .2 V2 Qcos u

58



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59/146

p s T = _ ln2v2

1 + t1 - 1

2V2ln(3 + 2J2).

2itT t

B, = J * V3.sinx -

-dx.smx - cosx

3

Hng d n %/3 sinx - cosx = 2sin (x - ....)

p s B = ln4

& + 2

x/3 - 2= ln

42 + >/3

2 - s= ln ( 2 + S )

Dng 7

p

I = / acos2x + bsin2x + csinx.cosx + d dx

Bin i mu s :

acos2x + bsin 2x + csinx.cosx + d = cos2x(a + btg 2x + ctgx + ^ )COS2 X

= cos2x[a + btg2x + ctgx + d( + tg 2x)]

i bin s t = tgx => dt = dx, ta c :COS X

I = J d t , trong f(t) l mt hm s hu t theo t. .f I \ V /

Sau tip tc tnh tch phn I bng cch phn tch f( t) v dngtng s.

Cn nh

* t = t g x d t = d x COS X

C O S 2 X

714

i 7. I = J0

5c o s 2x + 3sin2x + 4sinx.cosx - 2dx

59



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60/146

Ta c

GII

5 cos2 X+3 sin 2X + 4sin X.C O SX-2 c o g 2x ( 5 + g t g 2 x + 4tgx 2c o s 2 X

_________________ 1_____________ , _ ._________1c o s 2 x | 5 + 3 t g 2 x + 4'tgx - 2 ( 1 f COS2 x)J cos2 x |t g 2x + 4tgx + 3j.

71

Vy I = j-0c,cos2x(tg2x + 4tgx + 3 )

.11dt

t t = tgx '=>Cos2 X

dx

7t

X 0 4t ,1 sL u X s

Vy I = ) - ---------- -dt = f---------------dt = - f i-^ - idfc0 t + 4t+ 3 J (t + 1 ) (t + 3) 2 0\ t 4 - l t + 3 J

- [ In 11 + l| - l n | t + 3|] 1 = iln

= IIn InI= (ln3 - ln2)

2l 2 3 2

T a

t + 1t + 3

1

02sin2x - cos2x COS X + 3

-d x

*Hng dn Mu s = cos2x(2tgx + i f

* Ch Cos2x = cos2x - ... v sin2x ... v 3 = 3(cos2x + ... )

1 ^ 1 1t t= 2tgx + 1, ta c J = - p s J = -2 11 3

Dng 8p

I = jR(sin2x, cos2x).sin2x dxa

R(sin2x, cos2x) l mt biu thc theo sin2x, cos2x.30



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61/146

Bi 8.

t t .= R (sin2x, cos2x) .=> dt = a.sin2xdx (a = hng so)

i cn, ta s c I = f(t)d tto

cn nh (sin2x ) = sin2x ; (cos2x ) = - s i n2x

%2

I sin2x

0 v/2 c o s 2x + sin2x + 2

GII

i cn

a / 2 c o s 2X + s i n 2X +

(- 2 sin 2 x + sin2 x) dx

TX 0 2t 2 . s

' 'ftdt 2c ,- 2 = 2 d t =

J * i

7C

2

8 a. T = J - p =

T a

sn x . cosx rdx0yjssin2x + cos2x + 3

* Hng dn sinx. cosx = sin2x p s T = - 2)

8b.

t2

J = dt =

Ch (cos2x) '= - sin2x

* Dng_9_________ __________p 1 (5 1

1 = ' d x v J = f-* d x (n e N* v n c h n )^ c o s " x ' S i n n x

61



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62/146


63/146

T a.4 1...

1 - T dxQCOS X

Hng dn = . = (1 + tg2x). -

cos X COS X COS X COS X4

t t = tgx (Gii ging bi 9) p s' I = 3

Dng 10

Cp tch phn lng gi c c lin quan c bit

Ta thng gp cp tch phn m hai hm s ing gic di du tch phn c lin quan c bit.

Sau y l mt s' v d minh ha.

dxX

TC

d 1 I = f ------ csnx -. Qcosnx + s in 11:

X2

J = ------0 COSn

71

2ch gii - Bc 1 I + J = fdx =

0 2

v J = I------ --------------dx (n e N * )cosnx + sinx

- Bc 2 Chng minh I = J bng cch di bin s X = - 1.

Ta c sinx = cost cosx = sint (x = - dt

0 2 , + J = *Kt qu < 2 I = J

n

d 2 I = J -C-S- ^ -----dx0J{cosx + sinx

K2

J = J 0

v J = -sin y _ dx (n e N*)cosx + sinx

63



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64/146

Cch gii

- Bc 1 Chng minh I = J bng cch i bin s X = - t.

- BC 2 Tnh I + J . Sau cng ta c h phng trnh * ( ; - JII + J =

(T y ta tnh c I = J = ..... )

sinx------ dxcosx

p ___ pXT/ J O T _ f cosx J X 1- _ f sinV d 3 I = --------------dx v J = -----

J co sx +s inx s inx +a ap

Cch gi i - Bc 1 Tnh I + J = jdx = p - a

3 ' ' m' 1 T T rcosx - sinx ,- Bc 2 Tnh I - J = -------------- -dx

J s i n X + c o s xa

* i bin s t = sinx' + cosx, ta c dt = (cosx - sinx) dx

tf l t I + J = p* Vy I - J = I dt = ln |t |r . Sau cng ta c h phng trnh

-7 [l-J =..

(T y ta s tnh c I v J)

Ghi ch

1) Trong v d 3 nu a = 0 v p - th ta c th gii ging v d 12

m ^racosx bsinx , , , , - V _ .2) Tnh A = ----------------- dx (a, b l cc hng s)J cosx + sinxa ,

p p < .

* Phn tch A - af---------------dx bf----sm x----- dx hay A = al bJJ cosx + sinx Jcosx + sinxa a

Tnh I v J (nh ni v d 3)

- .-2 3 2

Bi 10. a/ I = f-----COS xrdx v J = f0 cos X + sin X Q

GII 7C

71- BC 1 Ta c I + J = Jdx =

0 2Jt2

- Bc 2 Xt I = j0

sin3xcos3x + sin3x

-dx (Xem v d

c o s X d x

cos3x + sin3x



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65/146

t t = - X 2

dt -d x

sn t = cosX, cost = sinx

71X 0 2

t 7t 0

71

\ r ~ T _ f sin3t(.-dt) 2j*Vy I = 5--^ r = I^ s i n t + c o s t Q2

sin t

COS t + sin t-


66/146

- -

c/ A = ------------------------------ dx v B = f-- d0 co sx + sinx 0J cosx + sinx

71

Bc1 Ta c A + B = fl.dx = J 40

71

4

- Bc 2 A - B = -Jc0 L

t t = sinx + cosx

c o s x - s i n x ,---------------------- d xCO'SX + .sinx

d t = ( co s X - s i n x ) d x

u0 .

' V2 ,Vy A - B = J-dt = lntj

1 1

. 1 , = ^-ln22

Tm li, ta c

A + B = -4

A - B = ln 22

A = ( - + ln 2)4 2

B = ( - ln 2)4 2

TC4

Suy ra K = J0

71 71

Ta C K = 3 4[---- C- -X- dx - 2 [ co s X+ sin X '

3cosx - 2snxcosx + sinx

dx

---- dx = 3A - 2B = ( - + 51n2)co s X+ sin X 4 2

T a71

2

10a) I = J0

s i n X

cos X + sin Xdx

cossx + sin5xdx

Hng dnN %' K

- Bc 1 Chng minh I = J bng cch t X = - t . (t = - x)

Bc 2 Tnh I + J. p s

66



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67/146

A - 4f cosx J 2sinx + c

71

10b)

* Hng dn

- Bc

dx v B2sinx + cosx

-dx

L4

1 Tnh 2A - B = J-0*

:dx (t t = 2 sinx + cosx)

n4

- Bc 2 Tnh 2B + A = J--""-dx -0 ..........

, 2A - B = ....... ;Gii h dt = ^ dx = (1 + tg2x)dx.t t = tgx => dt = ^ dx =COS X

7 1'

1 4

Vy I = J(t4 - t2+ l)dt - jl.dx.0 - 0 :

V d 2

Bin i (tng t v d 1). cotg5x = (cotg X - cotgx)(l + cotg x).+ cotg

t t = cotgx => dt = - (1 + cotg2x)dx /

Vy J = f(t3 t) d t + K vi K = fcotgxdx - [? x dx (Xem dng 0 * * sinx

4 4

7t n4 1 3

J hz F T ---- i T dx b / J = hsin X. cos X ^ s:6 6

Bi 11. a/ I = I---- --------- dx b/ J = ^os^xsin x.cos X

7t

4 71 1Cc/ K = jtg 5xdx d/ B = Jtg 4 dx

0 0 4GII

- -4 1 4 A ' ~

a/ I = f------------ r-dx = f -dx = -2co tg2x |4 =1(1 i sin 2 x

sin 2x I

2 V33

71

Cch khc I =[ 1 dx =(tgx - cotgx)|4 =q U os X sin x j -T

6

Tt K

- . T 3r cos2xdx 3fCos2x - sin2x ,b / J = 1 X 2 1 = I " T o i f - d x

K sin X. co s X JJ sin X. cos X6 ' 6

if ' -= J -772--------V = [G0t gx +tgx] I3 = 0'

^ s i n X c o s X J V

6 6

2J33

68



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69/146

Cch khc J = 3[4cos2xdx, sin 2 x6

71

t t = sin2x

.

Vy J= 2 . ] ^ = 0kt

dt = 2 COS 2Xdx => COS 2x dx = 4- dt2

7t 7t

6 3

V32

V32

n4

c/ K = Jtg5xdx0

Ta c tg5x = (tg5x + tg3x) - (tg3x + tgx) + tgx

= tg3x ( 1 + tg 2x) - tgx(l + tg2x) + tgx

= (tg3x - tgx)(l + tg2x) + tgx = (tg3x - tgx).

71 71

4 -1 4

Vy K = f(tg3x - tgx) ~dx + tgxdx0 cos x 0

71

4 -1

Tnh Ti = f(tg3x - tgx)-TdxQ , COS X

COS2 X+ tgx

- t t = tgx

t2

COS Xdx

Vy T1= f(t3 - t)d t = - 4_4_ 0 4 2

1

7 Tt4 4

Tnh T2 = tgxdx = fsmx dx o c o s x

Ta c T2 = ln2 (t t = cosx). Sau cng ta dc K = - + ln2

69



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70/146


71/146

7c'2

4

b) T = Jcotg6xdx

Hng d n

,cotg6x = (cotg6x + cotg4x) - (cotg4x + cotg2x) + (cotg2X + 1) - 1

= (cotg4x - cotg2x + 1 ) ( 1 + cotg2x) - 1 = ( ).---5------ 1sin X

' ._ 0 2 .t t = cotgx, ta c T = J(t4- t 2+ 1)(-dt) - Jl.dx

1 K4

, - _ 13 71p s T = 15 4

7C8

e ) A = Jtg32xdx0

Hng dn

tg 2x = (tg 2x + tg2x) - tg2x = tg2 x(l + ....) - tg2x

= tg2 x. ----- tg 2 xCOS 2x

. 1

Vy A = jtg2 x. J dx - jtg2xdx = I - J.

Q COS Q

Tnh bng cch t t = tg2x.

Tnh J (bng cch vit tg2x = Sinj -X v t t = cos2x)COS2x

p s' A = (1 - ln2)4

T c gh

I81 . Tnh I Jsinx.sin3xdx

0

4 4

C) I = v 2 -1 D) I =

71



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72/146

' ' ^Bi 2. Tnh a Jasin4x(2c os 22 x - ljdx = 1

.0

A) a = 16 B) a = 8 C) a =

Jt ' 4

Bi 3. Tnh I - Jsin32 xdx0

o

71

C) a = 4 D) = 1

3i 4. Tnh J = j | l - 2 s in 2- j dx

; A) J = - B) J = C) 15 8

n4Bi 5. Tnh A = J^4cos4x -4 co s2x + ljdx

0

A) A = - B) A = - C) A = D) A = 1 2 16

A) A = u ; A = 4 8

rt

Bi 6 . Tnh A = Jsin2x^2cos2- l j dx

" , A)A = B) A = . C) A = D) A = 16 8 4 2

T12

Bi 7. Tnh I = J tgxdx0

r 1 , A) I = l n 2 B) I = ln2 C ) I = - l n 2 D ) I = - l n 23 . . . 4. 6

n

Bi 8 . Tnh J = 8f 1 ~ y 22xdx t g2 x

24

A) J = ln 3 B) J = ln 3 . C ) J = l n 2 D) J = -] n 2\ J o = - m o4

n4 -

Bi 9. Tnh I = f----- -J 1 + C0

dxl + cos 2 x

A )I = 1 B) 1,-1 C )I = i D) I = I

72



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73/146

6 -

Bi 10. Tnh J = f---- =r (J 1 - sin X

-dx

A )J = V3-1 B) J = \/3 + 1 C )J = y - l D ) J = + 1

TC

2Bi 11. Tnh K = J

0

dx

2 + V3 cos X + sin X

Bc 1 2 + V3cosx + sinx = 2 + 2V3 1 .- r - C OS X + x sin X 2 2

1 + CO S X

7t

1 2Bc 2 K = =- f----4 J0 COS

= 2 =4 cs

k

2

0

,2 1 X_ n _

2 12

Bc 3 K = 2

. 71

3 * 1 2\ y

Bi gii trn ng hay sai, nu sai th sai u ?

A) ng B) Sai t bc 1 C) Sai t bc 2 D) Sai bc 3n6 2

Bi 12. Bng cch t t = cos3x, tch phn 1= I- dx c bin i sin 3x12

thnh tch phn no sau y ?1

" V t ) *

C) " [ 'J L + _l_'d6 0J U - t l + t j

Tt

Bi 13. Xt tch phn A = f----0 3 si

B> * I T ) d

D> ( * h )

dx

dt

. Bng cch t t = tgx, tch3 sin12 X - 2 cos2X - 2

phn A c bin i thnh tch phn no sau y ?

, } - * _ B) J - A - C ) . j- 5 - D) j * f - 0J t 2 - 4 0 t + 4 0* 1-2 0

73



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74/146

Bi 14. Tnh A = sin 2xVcos2X + 2s in 2X + 3 dx bng cch t0

t = VCOS2 X + 2'sin2X + 3

A ) A = ~ (V 5 + 2) B ) A = - ( n / 5 - 2 )

C) A = ~(5 +8) " D) A = -( V 5 -8 )

Ko i - H I- m - , r ^S;sn:Bi 15. Tnh I = j - ~ =

i'SHNpsi

8s in Xcos X dx

; ' ;: -u :.7y:'OiAX A *r X

: A) I = 2 (s/2 - a ) B) I = 2 ( ^ + 1 ): C) I = 4(72 - 1 ) D) I = 4 ( 7 2 + 1 )

JT

Bi 16. Tnh I = dx >;sin X

A) I = B) I = - C) I = D) I = 13 3 3

Bi 17. t t = tg-^ thi I = J c bin i thnh 2|f(t)dt.0 C OS6 ^ 0

2

Hy xc nh f(t).

A) f(t) = 1 - 2t2+ t 4 lB ) f(t) = 1 + 2t2 + t 4 .

C)f(t) = l + t2 D) f(t) = 1 - t2

Bi 18. Bit rng nu "Hm s f(x)-lin tc trn

n T

2 ' 2|f(sinx)dx = f(cosx)dx0 0

TC n

Tnh I = [----S-------dx v J = "s inx + eosx J si

0 ; -2

th

COS X ,----- ----- dx

sm X+ cos X

A) I = J = - . B) i = J = - C) I = J = D) Kt qu khc2 4 8

74



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75/146

n J

Bi 19. Tnh A = v B =) -Q s i n X + cos X 0s i

sin xdxsin 3X + c o s 3X

0' . 3 ,

Bc t t = - X ta c A = ----5 ^ 2 -----5 dt) = B2 cog t + sin t

l

2Bc 2 A + B = fldx =

2r , . T , nBc 3 Vy A = B

4

Bi gii trn ng hay sai ? Nu sai th sai u ?

A) ng B) Si t bc 1 C) Sai. t bc 2 D) Sai bc 3

Bi 20. Tnh 1 = ]t g 2- d x

A) l =-S3 - n B) = y/ + n C) I = 3 V3 - n D) I = 3\3 +n

Hg d

sin 2 x - s in4 x2 4

X1 8 -

Bi 1. I = j(cos2x -cos 4x )dx =2 0J 2

t K16 a aK

Bi 2. A = asin4 x.cos4xdx = - 7 s in 8 xdx = - cos8 x0 00

= - I cos- cosO = = 1. Vy a = 16.1 6 1 , 2 J 16

7 T4 4

Bi 3 . I .= s in 22x.sin2xdx = J(l - COS2 2x).sn2xdx0

y/2 - 1

t t = COS 2x :* dt = - 2 sin 2 xdx

X = 0 => t = 1 ; X = = > t = 0l 4

75



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76/146

B 4.

Bi 5.

Bi 6.

Bi 7.

Bi 8.

Bi 9. I

J = f c o s5 d x = f c o s4 . c o s d x = f i l - s i n 2 .COS d x 2 l 2 2 J{ 2 j 2

^ , X d t = c o s x t t = sin => j 2 2

^ [ x = 0:=>t = 0 ;x = 7r=> t = l

= Vy J =J(l - 1 2 )2 ,2dt = 2 J(-l - 2t2 + 1 4).dt = 2^t - - t 3+ t 5'

n t - K .

4 2 4 l 4 1%- j^2 cos2X- 1 J dx = jcos22xdx = ~ j( l + COS 4x) dx

0 0 2 0

= fx + s in 4 x V = 4 J0 8

n 7t * 7t

2 ^ 1 1 ^ ^. = Jsin2x.cos2xdx = j sin 2 2 xdx = J(l-c os 4x )dx

0 04 8 0

i r 1 .. ] 2 7T= X ---- s i n 4 x

. sL 4 J 0 16

xc

I = [ sm ^x dx (t t = cos4x) ' cos4 x

1

. = l n l - In 1 = l n 24 t J 4 t 4 1 4 V 2) 4

2 . . 2

TC Jt 7

J = 2 8f --T dx = 2 8f cotg4xdx = 2 8f .gggig.dxi 2 tg 2 x 2 ^Sin4x

24 * 2 4 24

t t = s inx , ta c J = J- = ln|t| 1

= ln 2 22

4 1

76



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phÂn loẠi vÀ phƯƠng phÁp giẢi toÁn tÍch phÂn vÀ cÁc bÀi toÁn Ứng dỤng - lÊ...

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