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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
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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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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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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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\ p j J Vy F(x) = ln (x + J x 2 + ) l
f(x) = 1 (pcm)
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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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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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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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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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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* 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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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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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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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
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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 | -. (
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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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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
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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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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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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
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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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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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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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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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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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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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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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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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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|
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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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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 '
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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
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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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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
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- 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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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)
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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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* 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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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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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
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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
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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
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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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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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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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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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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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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
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Dng 6p 1 p 1
I = f- dx v J = ; f . dx- cosx 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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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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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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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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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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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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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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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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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-
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- -
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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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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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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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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' ' ^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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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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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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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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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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ng gp PDF bi GV. Nguyn Thanh T