Mô tả:
(p'(x) = -xe"
Vay nguyen ham ciia f\x) = -xe" la F(x) = (-x + De" + C.
~T\g minh F(x) = In| x + Vx^ + K | la mot nguyen ham cua
fix) =
, ^
Vx^+K
tren R.
Gidi
^
,
i a CO :
1+
.
(X + V 7 7 K ) '
r (x) =
X + Vx^ + K
Suy ra : F'(x) =
, ^
X
VX2 + K
x + Vx^ + K
VX^ + K +
Vx^ + K ( x + Vx^ + K )
= f(x) Vx e R.
Vx2+K
Do do : F(x) la mot nguyen ham cua f(x) tren R.
sl
Cho ham so fix) = xV3 - x vdi x < 3.
Tim cac so' a, b sao cho ham so' F(x) = (ax^ + bx + c) V3 - x la mot
nguyen ham cua f(x).
Dai hoc Su pham Ki thudt
TP.HCM
Gidi
Ta CO :
F'(x) = (2ax + b) V 3 - x
(ax^ + bx + c).
2V3-X
1
2V3-X
.
.
.2
[(2ax + b).2(3-x) - (ax' + bx + c)]
1 . [-5ax^ + (12a - 3b)x + 6b - c)]
2A/3-X
F(x) \k mot nguyen hkm cua f(x) k h i x < 3
-5ax^ + (12a - 3b)x + 6b - c = 2(3 - x)x
o
F'(x) = f(x)
o
Vx < 3
Vx < 3
6
2
a = 5
-5a = -2
12a - 3b = 6
b = -^
5
6b - c = 0
c - - —
el
Chvlng m i n h F(x) = — I n X - a v 6 i a > 0 l a m o t nguyen h a m ciia
x + a
2a
1
x^-a^
f(x) =
v d i Vx ^ ± a.
Gidi
r
^x-a^
T a CO :
F'(x) =
1
^x + a j
F'(x) =
1
2a' rx-a^
U
Suy r a : F'(x) =
2a
af
(X +
Vx
^x-a^
2a'
9^
± a.
vx + a.j
+ a,
2a
+a
X
2a ( x + a f
Vx ;t ± a
x-a
1
1
(x + a)(x - a)
- a^
= flx)
V a y F'(x) l a m o t nguyen h a m cua f(x) v d i Vx ^ ± a.
10 1 ChiJng m i n h
F(x) = •
neu x ^ O
X
1
neu x = 0
(x-De^+l
x^
f(x) = <
-
I
I V
l a nguyen h a m cua
neu
x^O
neu x = 0
2
Gidi
* Khix^Othi
F'(x) =
e^.x-Ce"-l).l
(x-De^+l
X
= f(x)
X
V a y F(x) l a nguyen h a m cua f(x) t r e n (-oo, 0) u (0, + « )
e" - 1
* K h i x = 0 t h i F'(0) = l i m ^^^^
x-^'O
^^^^ = l i m
X - 0
(1)
-1
x
7
IT]
Vay
F'(0) = l i m
X-.0
^
^ = lim
x^o
FXO) = l i m — = - = fTO)
x^o 2
2
2x
^
(Quy t^c L'Hopital)
(2)
TiT (1), (2), t a suy r a F(x) l a nguyen h a m cua f(x) t r e n R.
T i n h dao h a m cua F(x) = (x^ - l ) l n 1 1 + x | - x^ln | x i .
Suy r a nguyen h a m cua f(x) = x l n
1+x
A2
Gidi
Ta
CO
:
Suy r a
Taco:
F(x) = (x^ - l ) l n
11
+ x | - x^ln | x I
x^-1
F'(x) = 2 x l n I x + 1 1 +
x +
1
- 2xln I X I
-
= 2 x l n I X + 1 i - 1 - 2xln | x | v 6 i x
0, x ^ 1
2
f l + x^2
+
x
+
x
il
= xln
= 2xln
f(x) = x l n l
X
X
1
h
x
1
i - I n 1X1 ]= 2 x l n 1 1
+
X
f(x) = F'(x) + 1
Tir ( * ) , ( * * ) t a suy r a :
Suy r a
1-
(**)
ff(x)dx = [F '(x)dx + f l d x = F(x) + x + C
V a y nguyen h a m cua f(x) = x l n I
1
+
X
1
la :
F(x) = ( x 2 - D l n l l + x | - x ^ l n l x l + x + C.
12 I T i m a. b, c sao cho F(x) = e""^ (atan^x + btanx + c) l a m o t nguyen h a m
n n
cua f(x) = e'"^ .tan^x t r e n
I
2'2
Gidi
Taco :
F'(x) = 7 2 . 6 " ^ ( a t a n ^ x + b t a n x + c) + e ' ' ^ [ 2 a ( l + tan^ x ) t a n x + b ( l + tan^ x)]
v d i Vx e
7t
F(x) l a nguyen h a m cua f(x) t r e n
71
{'2'
2)
<=> F'(x) = f i x ) , Vx
7t _ 71
2' 2.
n
71
'2' 2
8
2atan^ x + (yl2a + b ) t a n 2 x + (^^b + 2 a ) t a n x + (V2c + b) = e ' ' ^ t a n ^ x
<=> e
Vx e
2' 2
1
a = 2
2a = 1
V2a + b = 0
b = -
A ^ b + 2a = 0
2
.
1
c =—
2
V2c + b = 0
13 I Chutog m i i i h F(x) = | x | - l n ( l + I x I ) la mot nguyen ham cua fix) 1+
Dai hoc Tong
hap TP.HCM
-
1993
Gidi
x - l n ( l + x)
Ta
CO
:
F(x) =
neu x > 0
0
neu
- x - l n ( l - x)
0
X =
neu x < 0
neu x > 0
Ta
CO
:
f(x) =
1 + x
0
neu x = 0
V
1-x
Do do, t a
* Khi
neu
CO :
0 thi
F'(x)
= 1 -
* Khi x < 0 thi
F'(x)
= -1 +
* vu* Ehi
X >
X =
n.u^
0 thi
1+x
1+x
1-x
= f(x)
1-x
(1)
= f(x)
(2)
l^vn+^ r
F(x) - F(0>
,.
x - l n ( l + x)
F (0 ) = l i m
= lim
X - 0
x^O"
= lim
x-»0*
F'(0*) = 1 - l i m
x^.0^
F'(O^) = 1 - 1
x^O"
X
,.
l n ( l + x)
lim
X
x-»0* X
Suy ra :
0
X <
^—
1+ X
= 0
x
(do quy t i c L ' H o p i t a l )
TV/r-.iu'
v
F(x)
-
F(0)
Mat khac : F (0 ) = l i m
,.
- X - ln(l
-
x)
= lim
x - 0
x-»o-
X
FXO-)=-l-liml^^^
x->0"
X
-1
^
F'iOl
= - 1 - l i m -i-tiL = - 1 + 1 = 0
1
x-»0-
Vay
F'(0*) = F'(0") = 0
(3)
F'(0) = 0 = flO)
Nen
TCr (1), (2), (3) suy ra F(x) la nguyen ham ciia f(x) tren R.
14 I Churngminh
F(x) =
— In x
(X
2
4
•
0
fx In X
f(x)
>0)
la nguyen ham cua
(x = 0)
(x > 0)
(x = 0)
0
Dai hoc Yduac
TP.HCM
Gidi
K h i X > 0 ta CO : F'(x) = x l n x +
2
2
X
X
(1)
= xlnx = f(x)
x^,
In
Khi
0 ta
X =
CO
X
-
: F(O^) = l i m ^^""^ ^^^^ = l i m
X - 0
x^O*
= l i m — In
X
x-yO* 4
x^
X
-0
2
x^.0*
(quy t^c L'Hopital)
2
x^o-
^
x ^ O *
- lim — = lim
2
x-^O*
= lim
= lim
= 0 = f(x)
x^o^V
(2)
2)
Tix (1), (2) ta ket luan F(x) la nguyerI hkm cua fix) tren [0, +oo).
15 i Tinh dao h^m cua ham so F(x) = In
fx'-2^
+ 1]
^x^ + 2^R + lj
x^-1
Roi suy ra ho nguyen ham cua f(x) =
xUl
10
Gidi
Ta CO :
F(x) xac d i n h vdfi m o i x.
x^ - 2^
Ta
CO
:
F'(x) =
+1
2V2(x2 - 1)
x^ + 2 A / X + 1
x^ + 2V^ + 1
(x^ + 2V^ + 1)2 ' x^ - 2 A / i + 1
x^ + 2Vx + 1
2V2(x2 - 1)
2^{x'^ - 1)
= 2V2f(x)
x^ + 1
f(x)dx =
Vay
x^-l
x^+1
16
x^ - 2 7 ^ + 1^
^ : F ( x ) = i l n
2A/2
2V2
x^ + 2A/X + 1
dx =
In
fx2-2>^ + l
x^ + 2Vx + 1
T i m ho nguyen h a m cua f(x) = max ( 1 , x^).
Gidi
Ta
CO
:
f i x ) = max ( 1 , x^) =
'x^ neu
1
X <
neu - 1
- 1 V 1
< X <
< X
1
Vay:
j x ^ d x neu x < - 1 v 1 < x
+ C neu
m a x ( l , x2)dx =
Idx
neu - 1 < x < 1
17 I T i m ho nguyen h a m ciia f i x ) = | l + x| -
X
+C
x < - l v l < x
neu - 1
< X <
1
|l-x|.
Gidi
Ta
CO :
vay
1 +x -
j ;1 + x -
l +x
dx =
-2dx
neu x < - 1
2xdx
neu - 1 < x < 1
2dx
neu x > 1
-2x + C vdi
l +x
dx = x^ + C
2x + C
X <
-1
vdi - 1
< X <
vdi
1
X >
1
11
18 I T i m ho nguyen h a m ciia f(x) = | x |
Gidi
xdx
Ta CO :
neu x > 0
+C
neu X > 0
I X I dx =
-xdx
neu x < 0
+ C neu X < 0
2
19 I T i m ho nguyen h a m ciia f(x) = x i x |.
Gidi
x^dx
Ta CO :
neu x > 0
J x I x I dx =
-x^dx
20
— +C
3
neu X > 0
x^
neu x < 0
T i m ho nguyen h a m ciia f(x) = (x +
+ C neu x < 0
\x\f.
Gidi
Ta CO :
(x+ I X I )2dx = Jlx^ + 2x I X I +x2 )dx
'
4x
2
4x dx =
+C
O.dx = C
21 I T i m ho nguyen h a m ciia f(x) =
neu x > 0
neu X <
0
cos X
Dai hoc Yduac
TP.HCM
-2001
-He
nhdn
Gidi
T a CO :
F(x)
' d(sin x)
•COS xdx
= f
• cosx
• d(sin x)
sin^ X - 1
1 - sin^ X
COS'^ X
1,
sin X - 1
+ C.
-In
2
22I
I sin X + 1
T i m ho nguyen h a m cua fix) = 2\3^\5^\
Gidi
Ta CO :
f(x)dx =
2 \ 3 2 \ 5 3 M X = f(2.32.5^)''dx =
2250"
hi(2250)
+ C.
12
23 I Tim ho nguyen ham cua :
X* +
a) f(x) =
X
2x^
+ X +
2
b) g(x) =
+x+1
x^ + x^ + 1
X^ + X +
1
Dai hoc Ngoqi thuang - 1998
Gidi
f(x)dx =
a)
x^ + 2x2
X
+ X +
+ X +
2
1
x^ + x^
1
fx''
x" + X
g(x)dx = —
dx=
J x^ + X + 1
b)
dx =
(x^ -
X +
x^
2)dx = 3
C o
x"^
X^
3
2
( x ^ - x + l)dx =
J
(hi
24 I Tim ho nguyen ham cua f(x) =
2 + 2x + C
+ X + C.
X)'
Gidi
Ta
f (x)dx =
CO :
f(lnx)4
dx = j ( h i x)"* d(ln x) = - (In x)^ + C.
25 \m ho nguyen ham cua f(x) =
——
e" - 4e "
DH Quoc gia Ha Ngi - D/1999
Gidi
Ta
f(x)dx =
CO :
f e^dx
Tim ho nguyen ham cua f(x) =
d(e'')
e" - 2
=lln
+ C,
4
6^+2
e^^
e" + 1
Gidi
Ta
CO :
f (x)dx =
'(e" + IKe^" - e" + 1)
^-^dx
+1
I 27 I Tim ho nguyen ham cua f(x) =
=
1-e
f ( e 2 ' ' - e ' ' + l ) d x = i e 2 ' ' - e ^ + x + C.
J
2
2x
Gidi
Ta
f(x) =
CO :
f eMx
1-e
2x
d(e'')
die"")
2
e" - 1 + C = - h i 6=^+1
+C
2
e" + 1
e" - 1
13
2 8 ] T i m ho nguyen h a m cua fix) = Ve" + e
- 2.
DH Y Thai Binh
-
1997
Gidi
Ta
CO
f(x) = Ve" + e"" - 2 =
-e"2
e2
X
X
f(x) =
X
e2 - e 2
.
neu
'
X
X
—>
2
e2 - e 2
—
2
X
-e2 + e 2
X
f(x) =
.
neu
'
X
X
—<
2
—
2
X
e2 - e'2
X
neu
0
X >
X
-e2 + e 2
r
neu
0
X <
xA
X
e2 + e 2
Nen
neu
0
X >
f (x)dx =
e2 + e 2
29 I T i m ho nguyen h a m ciia f(x) -
neu
0
X <
Inex
1 + xlnx
HV
Quan he Quoc te -
1997
Gidi
T a CO :
d ( l + x l n x ) = (1 + xlnx)'dx = ( l . l n x + — .x)dx
X
= (Inx + l ) d x = Inex.dx
Vay
f(x)dx =
1 + xlnx
1 + xlnx
rd(l + xlnx)
Inex.dx
f (x)dx = I n 1 + x l n x
+ C.
3o] T i m ho nguyen h a m cua f(x) = x ( l - x)-°.
DH
Quoc gia Ha Ngi
-
1998
Gidi
Ta
CO
: fix) = [(x - 1) +1](1 - x)^" = (x - 1)^' + (x - 1)^°.
14
f(x)dx=
Nen
(x-l)2Mx +
(x-l)^M(x-l) +
(x-l)2°dx
(x-l)2°d(x-l) =
(X-1)22
(x-l)21
22
21
+ C.
,2001
31 I Tim ho nguyen ham cua f(x) =
(1
+
X2)1002
DH Quoc gia Ha Ngi - 2000
Gidi
.2001
Ta
CO
:
f(x) =
,
( l + x^r^^
f
2000
( l + x ^ r o ' d + x^f
/-
.2001
J f (x)dx =
(1^^2)1002
2
a) Dat x = tana.
X
1000
NIOOO
X
•(l + x^)^
\1000
dx
2
1
^
1 + x^
1 + x'
^
1 + x^
1 + x^
^
32 1 Tinh tich phan
,
'
dx =
..2
2002 1
+ C.
+
X ^
x^dx
—
bang hai each bien ddi sau :
(x^ + if
b) Dat u = x^ + 1 So sanh hai ket qua t i m ducfc.
Dai hoc Tong hap TP.HCM
~ A/1977
Gidi
Ta
CO
a) Dat
:
X = tana
II
=
=>
dx = (tan^a + l)da,
x^^dx
rtan^ a(tan^ a + l)da
(x^ +1)^
(tan^ a +1)^
tan^ ada
rsin^ a
(tan^ a +1)^
cos a
sin^ a. cos ada =
=
thi :
da
(vi tan^a + 1 = — ^ — )
cos a
cos'' a
sin ad(sin a)
— sin* a + C = - (tan* a. cos* a) + C
4
4
tan* a
1
x^
1
+ C.
+
C
=
-.4 • (tan^ a + lf
4 ' (x^ + 1)^
15
b)
Dat
I,
u =
r
+ 1
x^dx
=>
du = 2xdx
• x^.xdx
(x^+l)^ ~ J (x^+l)^
-2
-3o..
(u"" -u-')du
=
1 r(u - D d u
'2 .
1 . 1 .
l - 2 u , ^
- l ( l + 2x^)
+ — + Ci =
— + Ci = —
+ C,
4u^
4 (x^ + 1)2
'
4u^
2u
T a xet : I i - I2 =
4(x2 + 1)2
I i - I2 =
1 (1 + 2x2)
+ C+ - C,
4 ( l + x2)2
x^ + 2 x ^ + 1
4(x2 + i f
+ C - Ci
I i - I2 - - + C - C i .
4
33 I T i m ho nguyen h a m cua fix) =
Vx^ + x"" + 2
Gidi
Ta
CO :
Vx^ +x-^
f(x)dx =
I x2
+ X-2
I