Α16.ΟΛΟΚΛΗΡΩΜΑΤΑ ΑΣΚΗΣΕΙΣ
DESCRIPTION
Μαθηματικα Γ ΛυκείουTRANSCRIPT
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! " ! # $
# "% #!"% # & "
! # ' ( #
) * +,,-
LATEX
-
1.1
1. :
1. ex+1
2. e2x
3. 2ex
4. e2x+1
2. :
1. 2x2 x + 12. 1x + e
x
3. x23 +
x
4. x + x
3. :
f Fg G
f + gfG + gFfGFg
G2
3fF 2 + 4gG3
4. F f (x) = x + 1 F (0) = 1.
5. f F (x) = xex + (x + 1)2.
G f G (ln 2) = 1
6. F (x) = ex+1 + x, G (x) = ex+1 x
7. - f
f (x) dx.
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1. 4.
2. A(0,2).8. f 2009.
1.2
9. . 4N 3m/sec. 6m/sec;
10. (t) t - . t = 3 t = 2 1.
11. R :
Cf M (x0, f (x0)) x20.
12. : x2 + k dx =
12
[x
x2 + k + k ln(x +
x2 + k
)]+ c
13. f : R R
f(x) ={
2x 1x 1x x = 00 x = 0
0 F : R R
F (x) ={
x2 1x x = 00 x = 0
g : R R g(x) =
{x x < 0
x + 1 x 0 0 .
14. R f(0) = 0 f (x) = 2e2xf(x) x.
15. f . F
f (x) dx R
R.
16. f . F f .
17. f R Cf .
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18. :
1.
(x + 1) dx
2. (
x + 2x)dx
3.
x+1x2 dx
4.
x3+1x6 dx
19. :
1.
x + 1dx
2.
(2x) dx
3.
xex2
dx
4.
ex1dx
20. :
1.
13
x18dx
2.
11+ 1x
dx
3.
(x + 1) (x + 2) dx
4.
x+1x+2dx
21. :
1. (
x2 + y)dx
2. (
x2 + y)dy
3. (
1x+1 + x
)dx
4.
3x + 2dx
22. :
1.
x
xdx
2.
x+x+ dx
3.
x+yxydx
4.
x+yxydy
23. :
1.
(x + 1) (x + 2) (x + 3) dx
2.
3x2+2x+1x3+x2+x+1
dx
3.
(xm + xn) dx
4.
(x + 3
)dx
24. :
1.
xex2dx
2. (
x + 11 x) dx3.
xxdx
4. (
x2 + y2)dx
25. :
1.
ex+2 (x 1)dx2.
xex+1dx
3.
x2ex+1dx
4. |x| dx
26. :
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1.
exxdx
2. (
ex2 + 1x2)dx
3.
ttdt
4.
ex
e+1dx
27. :
1.
x3xdx
2. (
pp+q
)tdt
3. (
pp+q
)tdq
4.
1x23x+2dx
28. :
1.
x+1x23x+2dx
2.
6x3+x2x23x+2dx
3.
x1xdx
4.
(ex + xe) dx
29. :
1.
x (
x + 1)(x2 + 1
)dx
2.
1213xdx
3.
5x2+3x+1x1 dx
4.
5x2+3x+1(x1)(x2)dx
30. :
1.
xe3x1dx
2. (
x3 + e3 + x)dx
3.
(2x) (3x) dx
4.
(3x + ) dx
31. :
1.
(3x + ) (2x ) dx
2.
ex+1ex1dx
3.
(2x + 3x) dx
4.
(1 + 4x)35 dx
32. :
1.
x3ex4dx
2.
x3exdx
3.
(x + 1)2
4. (
(x 2)2 + (x 3)2)dx
33. :
1. (
1(x2)2 +
1(x3)2
)dx
2.
(x 2)2 (x 3)2 dx
3.
ex+xdx
4.
2x32x53xdx
34. :
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1.
u5
13u6+18du
2.
x ln
x
3.
(x + ) dx
4.
2(47
)tdt
35. :
1.
f (x) dx f (x) ={
x x < 0x2 x 0
2.
x1+|x|dx
36. :
1.
1(x)(x)dx
2.
e2x
ex1dx
3.
2x 3dx
4.
1x1dx
37. :
1. (ln(x))5
x dx
2. (
x3 1) 13 x2dx3. 31+x
xdx
4.
xe4x2+5dx
2.1
38. x1, x2, x3 x37x2+6x. , , x x1, x2, x3
1x3 7x2 + 6x =
x x1 +
x x2 +
x x3 :
1x3 7x2 + 6xdx
39. f (x) = 1(x1)2(x2)3 .
1. A,B,,, E f (x) = Ax1 +B
(x1)2 +
x2 +
(x2)2 +E
(x2)3 .
2.
f (x) dx
40.
2xdx
41. x+
x+ dx.
42.
1(x1)(x2)(x3)dx
43. f (1, + ) :1. f (x)
x 1 = 1
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2. f (2, f(2)) 3y 3x + 5 = 0
44.
ln2 xx dx
45.
x41+x5
dx
46. Euler, t = x2 ,
1
xdx
47.
x1x22x+3dx
48. u = x+ 1x -:
ex+1x
(1
(1x
)2)dx
49.
I =
xdx
J =
xdx
:
1.
I = 1x 1x +
1
I2
2.
J =1x1x +
1
J2
50.
x2
(x3+) dx - 1.
51. f (x) = 2x + . , f (1) = 2
30f (x) dx = 7.
52. 10
ln(1 + x2
)dx.
53.
1x+xdx.
54. :
1.
x31+x4
dx
2.
3x+13x2+2x+1
dx
3.
(2x 3)x2 3x + 2dx55. :
1.
(x + x)
x xdx2.
x(1+x)2
dx
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!
3.
xx(xx+x)dx
56.
1x2
(1x
)
(1x
)dx.
57. (x)
x2(x)dx.
2.2
58. In =
eaxnxdx. In In2.
59. 1
lnxdx
- -.
ex lnxdx
.
60. P (t) P (t) = kP (t) ( ). L . LP (t). P (t) = kP (t) :
P (t) = kP (t) (L P (t))1.
P (t)P (t) (L P (t))dt =
kdt
2. u = P (t) P (t)
P (t) (L P (t))dt =
du
u (L u)
3.
duu(Lu)
P (t) =LeL(kt+c)
1 + eL(kt+c)
c .
4.
P (t) =LP0
P0 + (L P0) eLkt P0 = P (0)
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"
3.1
61. 10f (x) dx = 14 ,
40f (x) dx = 64
31f (x) dx = 20 4
3 f (x) dx.
62. 0
(2f (x) + 3g (x)) = 5
4 0
f (x) dx + 5 0
g (x) dx = 7
0
f (x) dx
0
g (x) dx
63. f R . : 43
f (x) dx 23
f (x) dx = 52
f (x) dx + 45
f (x) dx
64. f
f (x) dx = 4 :
f (x) dx
f (x) dx
3f (x) dx
65. f (x) = ex.
1. t0f (x) dx = 3
2. 1+t0 f (x) dx = 3
3. 10 f (t + x) dx = 3
4. 10tf (x) dx = 3
66. b > 1 b1
(b 4x) dx 6 5b
67. , 1994.
1. F (x) = x
2(t + 1) (2t) dt.
2. .
68. 32
(x2 + x +
)dx = 2
10
(x2 + x +
)dx + 2
21
(x2 + x +
)dx
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!! F (x) =
x
f (t) dt #
3.2
69. 20
(t log2 ) dt = 2 log2(
2
)
70. f : [, ] R
f (x) dx = 0
f [, ].
71. < f
|f (x)| dx = 0
f(x) = 0 x [, ].72.
2e
g (x) dx
f (x) dx <
g (x) dx. (C)f , (C)g .
75. f : R R f (x) > 0 x.
ex1x f (t) dt = 0.
3.3
76. Berkeley, 1981. f : [0, 1] R .
limt+
10
xtf (x) dx
! " # F (x) = x
f (t) dt
4.1
77. :
1. f (x) = x1 e
tdt
2. g (x) = 1x
tdt
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$
3. h (x) = x2+x+11
t 2dt
4. s (x) = 4x2x+1 e
tdt
5. w (x) = x1
(axet
2+ b
)dt
78. (x) = x0 tdt.
(4
),
(2
),
(4
),
(2
).
79. F (x) = x2
(2t 5) dt. Cf 2 3.80. :
x
dt
t= x1
dt
t
81. f R f (x) =xx f(0) = 0.
82. F (x) = x2 e
tdt, G (x) = x3
(t2 + 1
)dt. -
:
1. (F (x) + G (x)) = ex x2 1
2. (F (x) G (x)) = ex x3 (t2 + 1) dt x2 x2 et dt x2 et dt3. (F G) (x) = (x2 + 1) e x3 (t2+1) dt
83. xx
et+t2
et+1 dt. %&'
84. limx0
x0 (t2)dtx(1x)
85. f R
x = f(x)0
11 + 4t2
dt
f f (x) = 4f (x).
4.2
86. f : [0,+) (0,+). x > 0
x
x0
f(t)dt >
x0
tf(t)dt
87. I : [1, 1] R I (x) = x0 2t+1t2+5t+6dt (;) () () . .
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!! F (x) =
x
f (t) dt
88. f : [0,+) R x = x
0
f (t) (t + 1)dt = f2 (x)
f .
89.
( (x)(x)
f (t)dt
)= (x) f ( (x)) (x) f ( (x))
90.
f (x) = ex2
ln tt
dt
91. 1995, IV. G (x) = x1
f (t) dt f (t) =
3t1
euudu x > 0, t > 0 :
1. G (1)
2. limx0+
xG(x)3
x+11
92.
g (x) = 1x2
ln tdt
93. f :
1. f (x) = x1
tt dt
2. f (x) = x1
txtx dt
94. 1995, I. - , 0 < < f : (0,+) R
f (t) dt = 0
g (x) = 2 +1x
x
f (t) dt, x (0,+)
x0 (, ) :1. g (x0, g (x0))
xx.
2. g (x0) = 2 + f (x0)
95. f (x) = x1x+1 ln tdt.
1. .
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2. -.
96.
f (x) = x1
t 1et
dt
1. f
2. f
97. f : R R .
g (x) = (1 x) x0
f (t) dt
x0
f (t) dt = (1 x) f (x) .98. f : R R x : x
0
etf (x t) dt = x
99. F (x) = x1
(t4 t3 t + 2) dt
.
100. (0,+) : 1
dx
1 + x2= 1
1
dx
1 + x2
101. x3x a
3dt.
102. x x+3x
t (5 t) dt ; ;
103. f (x) = x0
(et t 1) dt . (& ' ' ex x + 1
104. R , x, x
0
f (t) dt = xex + d
105. I (x) = x0
(1 + t) dt.
(1 + x) I (x) I (x) = 0106. x0 :
(x) = x0
(t 1) (t 2)3 dt
107. f x0
f (t) dt = f (x) ex
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!! F (x) =
x
f (t) dt
108. f : R R f (x + y) =f (x)+f (y) x, y. > 0
f (x) dx =
0.
109. f [0, 2
]
2 x
f (t) dt = 2x 1
[0, 2 ]. .110. f F (x) = x+x f (t) dt. F
(x) = f (x + ) f (x ).111. f : R R x
x20
f (t) dt = x (x)
g (x) = f(x2).
112. Harvard-MIT, 2006. - f g
xg (f (x)) f (g (x)) g (x) = f (g (x)) g (f (x)) f (x)
x. , f g . 0
f (g (x)) dx = 1 e2
2
g(f(0)) = 1 g(f(4)).
113. f : [0,) [0,) .
g (x) =
x0 f (t) dt x0
tf (t) dt
1. limx0+
g (x).
2. g .
114. R . f : R R
x
f(
t
)dt =
x1
f (t) dt x. f ; '&' %
x
f (t) dt
115. g : R R g (x) > 0 x. f : R R
f (x) = x0
g (f (t)) dt
1. f .
2. f .
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" #
116. f R :
f (x) = x0
f (t) dt) + 1
117. Harvard-MIT, 2003. > 1 a2
a
1x
lnx 132
dx
;
118. 1993, . - f : R R : x
etf (t) dt = ex e exf (x)
x, R.119.
f (x) =
{ x2
0
xdx
x3 , x = 0 , x = 0
.
Charles Hermite")#$
Jacques Salomon Hadamard" )#
120. 1999, I. h : [1,+) R
h (x) = 1999 (x 1) + x1
h (t)t
dt
x 1. 1. h(x) = 1999x lnx, x 1.2. h [1,+) .
121. f : R R F (x) =
x0
f (t) dt .
4.3
122. Hermite- Hadamard. f [, ] :
f
( +
2
) I =
t
x2 2dq.
() u = 2(x + 1x
)
() u = 2 (ex + ex)
() u = x
I =
2
(t
t2 2 2 ln(t +
(t ) (t + )
)+ 2 ln
)
244. . , , -
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% &'(
. - y = f (x) f : R f . [, ] . L A = (, f ()) B = (, f ()).1 ', %&2 /' [, ] ' ,/' , 3 +' (
x0 = , x1 = +
, ... , x =
4( '&
X0 = A = (x0, f (x0)) , X1 = (x1, f (x1)) , ... X = B = (x , f (x))
35 6 ,' // + ('+ X0X1...X1X& ' L //
lim+
X0X1...X1X = L
.
X0X1...X1X = (X0X1) + (X1X2) + ... + (X1X) =
(x0 x1)2 + (f (x0) f (x1))2++
(x1 x2)2 + (f (x1) f (x2))2 + ... +
(x1 x)2 + (f (x1) f (x))2. 7' (- +' f ' ( + , /'
[x0, x1] , [x1, x2] , ... [x1, x ]
4&'
f (x0) f (x1) = f (1) (x0 x1)
f (x1) f (x2) = f (2) (x1 x2)...
f (x1) f (x) = f () (x1 x)3.
X0X1...X1X = |x0 x1|
1 + (f (1))2 + ... + |x1 x |
1 + (f ())2 =
=
k=1
1 + (f (k))2
,+% L = lim+
k=1
1 + (f (k))2 =
1 + (f (x))2dx
L =
1 + (f (x))2dx
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-
L -:
1 %
10
1 + x2dx (+' x = t
' (
245. :
1.
f (x) dx =
2.(
f (x) dx) =
3.( x
a f (t) dt)
=
4.(
f (x) dx)
=
5.
f (x) dx =
6. x f
(t) dt =
246. f : [0, 1] R 10
xf (x) dx 13 10
f2 (x) dx
1. 10 (f (x) x)2 dx 0
2. f (x) = x x
247. u = 1 + lnx :
(1 + lnx)2
xdx
248. f
F (x) = x+x
f (t) dt
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) * !
F (x) = f (x + ) f (x ).249. 1996, . f - [, ]
f (x) + f ( + x) = c c . :
f (x) dx = ( ) f(
+ 2
)=
2
(f () + f ())
250.
621136
ex2dx =
20
(3x) dx
251.
limx+
x+1x
ln tdt
limx+
x+1x
et ln tdt
252.
limx+
x+1x
t
tdt
253. f : (0,+) R : f (x) > 0 x 3
0f(13xu
)du = f (x)
f (1) = e2
254. 2001. f , R :i) f (x) = 0, x R.ii) f (x) = 1 2x2 1
0tf2 (xt) dt, x R.
g
g (x) =1
f (x) x2
x R.1.
f (x) = 2xf2 (x)2. g .
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-
"
3. f
f (x) =1
1 + x2
4. limx+ (xf (x) 2x)
255. g : R R R.1. f : R R
(f g) (x) x (g f) (x)
x f = g1.
2. f : R R
(sinh (x)) x sinh ( (x))
x,
sinh (x) =ex ex
2 .
() .
() 10 (x) dx.
256.
f (x) = x3 3x2 + x
Cf .1. f
, , .
2. Cf Cf P (2, 2).
257. 1997, IV. f R f (x) 2 x R.
g (x) = x2 5x + 1 x25x0
f (t) dt, x R
1. g (3) g (0) < 0
2.
g (x) = 0
(3, 0).
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-
) * #
258. :
x+1x
ln tdt = ln(x + 1)x+1
xx 1
259. 2002.
1. h, g [, ]. h (x) > g (x) x [, ],
h (x) dx >
g (x) dx.
2. R f , :
f (x) ef(x) = x 1, x R f (0) = 0
() f f .
() x2 < f (x) < xf (x) x > 0.
() E f , x = 0, x = 1 xx
14
< E 0 x R.
F (x) = f (x t) dt, x R
x0 R F (x0) = 0 F (x) = 0 x R.285. 2003. f(x) = x5+x3+x.
1. f f .
2. f(ex) f(1 + x) x R.3. f
(0, 0) f f1.
4. f1, x x = 3.
286. x 0 f (x) > 0 F (x) = x0
f (t) dt x (0,+) 1xF (x) < F (x).
287. f : R R 2f (x)+3f (x) = exexex+ex .1. f .
2.
f (x) dx.
288. f R - g (x) =
10f (tx) dt .
289. f [0, 1] 10f (1 x) dx = 1
0f (x) dx.
290.
limx1
x1 e
t2dtx 1
.
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-
291. 10x (1 x) dx = 1
0x (1 x) dx.
292. 1998, . f : (0,+) R
f (x) > 0, x > 0
f (x) + 2xf (x) = 0, x > 0
A (1, 1).
1. f (0,+) f .
2.
x 12x2
f (x) < x1
f (t)2t2
dt 1
3.
F (x) = x1
(1 +
12t2
)f (t) dt, x > 1
4.
2e x1
et2dt < 1
x .
293. f : [0,+) R f (x) > x x > 0 f (0) = 0
> 0 0 xf
3 (x) dx ( 0 xf (x) dx)2294. f - [1, 4], f (x) 3, f (1) = 1 f (4) = 7.295. 1. f : [0, ] R
f ( x) = f () f (x) x. 0
f (x) dx =f()
2 .
2.
40
ln (1 + x) dx.
Pafnuty Lvovich Chebyshev" ) "#
296. Chebyshev. f, g < :(
f (x) dx
)(
g (x) dx
) ( )
f (x) g (x) dx
297. 2004. g (x) = exf (x) f R f (0) = f
(32
)= 0.
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-
) *
1. (0, 32) f () =f ().2. f (x) = 2x2 3x
I () = 0
g (x) dx, R
3. lim I ().
298. g (0,+). > 0
1
f (x) + 2x1 + x2
dx + 1
1
f(1x
) 2x1 + x2
dx = 2 ln
/ % 3 + (+' u = 1x
299. f : [, ] R . f [f (x) , f (x)].
1.
f (x) ( ) <
f (x) dx < f (x) ( )
2.
f (x) 0, f (3) = > x0 (2, 3) f (x0) = 0.
301. 2005. f
f (x) = ex, > 0
1. f .
2. f , y = ex M .
3. E () , f , M yy,
E () =e 22
4.
lim+
2E ()2 +
302. 2005. f : R R
limx0
f (x) xx2
= 2005
1.
() f(0) = 0
() f (0) = 1
2. R
limx0
x2 + (f (x))2
2x2 + (f (x))2= 3
3. f R f (x) >f(x) x R, :() xf (x) > 0 x = 0()
10f (x) dx < f (1)
303. 1. 10x (x + t) dx = 0
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-
) * !
2. f : R R 10f (x) g (x) dx =
0 g : R R f (x) = 0 x.3. f (x) = x2 + x + 1
0
f (x) g (x) dx = 0
g 1, x, x2. f (x) = 0 x.
304.
f (x) =[ x
0
et2dt
]2
g (x) = 10
ex2(1+t2)
1 + t2dt
:
1. f (x) + g (x) = 0, x
2. f (x) + g (x) = 4
3. limx+
x0
et2dt =
2
305. 2005. f R ,
2f (x) = exf(x)
x R f (0) = 0.1. :
f (x) = ln(
1 + ex
2
)
2. :
limx0
x0 f (x t) dt
x
3. h (x) = xx t
2005 f (t) dt g (x) = x20072007 . h (x) = g (x) x R
4. xx
t2005 f (t) dt = 12008
(0, 1).
306. 1998, IV.
h (x) = 212(e4x ex) , x 0
4.
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-
"
1. limx+h (x) = h (0) = 0.
2. h(x).
3. x1 x2 - h(x) x1, x2.
4. M = 33475 ln 20 h (x) dx = 8.
307. f : [, ] R , f , f(x) 0 = x = x = . M f .
1. f
(x) dx 0.2. x0 (, ) f (
Rolle). 1 (, x0), 2 (x0, )
f (1) f (2) 4f (x0)
3. p, q < p < q <
f (p) f (q) 4M
4.
|f (x)|M + f (x)
dx 2
308. Baccalaureat, 1986. - :
I =
4
0
dx
2x, J =
4
0
dx
4x
1. () ;
() I.
2. () f :[0, 4
] R f (x) = x3x . f
[0, 4
] x
f (x) =3
4x 2
2x
() I, J J .
309. Baccalaureat, 2006.
1. f (x) = x2e1x, R. C 2cm.
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-
) * #
() f +. ;
() f . f .
() f .
2. . I = 10xe1xdx.
() I+1, I .
() I1 I2.
() I2 - 1)
3. () x [0, 1] x xe1x ex .
() I +.
310. Putnam, 2005. 10
ln (x + 1)x2 + 1
dx
x = 1u1+u
311. f : R R f (1) = 2007 f (x + y) = f (x) + f (y)
x, y
1. f (0) = 0.
2. m R : x+mx
f (t) dt = mf (x) + m0
f (t) dt
3. f .
4. f (x) = f (0).
5. f .
312. 1997, IV. f R f (x) > 0 x R. , R < . :
1. f (x) f () f (x) (x ) x [, ].2. 2
f (x) dx f () ( )2 + 2f () ( )
, - . " $ "
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-
$
313. f 312
f (x) dx 1.
1. () |f (y)| 2.() y, y0 |f (y) f (y0)| 2 |y y0|.
2. < .
F (y) =
ex2y2dx
.
317.
fk (x) = 249k2x2 +
1 k2k
x , k [0, 1]
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-
) *
1. fk (x).
2. fk (x) xx ;
318. Berkeley, 1980.
F (x) = xx
et2+xtdt
F (0).
319. 2007. f [0, 1] f(0) > 0. g [0, 1] g(x) > 0 x [0, 1]. :
F (x) = x0
f (t) g (t) dt, x [0, 1]
G (x) = x0
g (t) dt, x [0, 1]
1. F (x) > 0 x (0, 1].
2. :f(x) G(x) > F (x)
x (0, 1].
3. :F (x)G (x)
F (1)G (1)
x (0, 1].
4. :
limx0+
( x0
f (t) g (t) dt) ( x2
02tdt
)( x
0 g (t) dt) x5
320. 2008. f R
f (x) =(10x3 + 3x
) 20
f (t) dt 45
1. f (x) = 20x3 + 6x 45
2. g R. -
g (x) = limh0
g (x) g (x h)h
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-
3. f (1) g - (2)
limh0
g (x + h) 2g (x) + g (x h)h2
= f (x) + 45
g(0) = g(0) = 1,
() g(x) = x5 + x3 + x + 1() g 1-1
321. f [0, 2] 20
(t 2) f (t) dt = 0
H (x) = x0
tf (t) dt, x [0, 2]
G (x) =
{H(x)
x x0 f (t) dt, x (0, 2]
6 limt0
11t2t2 , x = 0
1. G [0, 2].
2. G (0, 2)
G (x) = H (x)x2
, 0 < x < 2
3. (0, 2) H() = 0.
4. (0, )
0
tf (t) dt = 2 0
f (t) dt
322. , -
, 1999. limx+
x0
4+t2dt
x3 .
323. , -, 2001. f, f f [0, ln 2], f , f (0) = 0, f (0) = 3, f (ln 2) = 6, f (ln 2) = 4
ln 20 e
2xf (x) dx =3.
ln 20
e2xf (x) dx.
324. f : R R x : x+x
f (t) dt = b
, . f .
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-
+ ,
) *
1 1. ex+1
2. 12e2x
3. 2ex
4. 12e2x+1
2 1. 23x3 12x2 + x
2. ln |x|+ ex
3. 353
x5 + 23
x3
4. x + x
3 F + G, F G, FG , F 3 + G4
4 F (x) = 12x2 + x + 1
5 G (x) = xex + (x + 1)2 + c , c = 52 ln 2 ln2 2
6 35
7 9 :; 9:
9 16N
10 e2
11 f (x) = 13x3 + c
14 f(x) = 2x
17 f(x) = ax
18 12x2 + x + c
23
x3 + 2 ln x + c
lnx 1x + c
12x2
15x5
+ c
19 23
x + 13+ c
122x + c
12ex2
+ c
ex1 + c
20 1331 13
x31 + c
x ln |x + 1| + c
13x
3 + 32x2 + 2x + c
x ln |x + 2| + c
21 13x3 + yx + c
x2y + 12y2 + c
ln |x + 1| x + c
29
3x + 2
3+ c
22 25
x5 + c
x + |ln (x + )| |ln (x + )| + c
x + 2y ln |y x| + c
y 2x ln |x y|+ c
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-
23 14x4 + 2x3 + 112 x2 + 6x + c
2
x3 + x2 + x + 1 + c
xm+1m+1 +
xn+1n+1 + c
(x+13
) + c
24 12 ex2+ c
23
x + 13 + 23
1 x3 + c
x xx + c
13x
3 + y2x + c
25 2ex+2 + ex+2x + c
2ex+1 + ex+1 (x + 1) + c
5ex+14ex+1 (x + 1)+ex+1 (x + 1)2+ c
12x |x|+ c
26 12 exx + 12 exx + c
ex2 1x + c
12
2t + c
ex
e+1 + c
27 xln 33x 1ln2 33x + c
1ln p
p+q
(p
p+q
)t+ c
11t p
(p
p+q
)t+ 11t q
(p
p+q
)t+ c
ln |x 2| ln |x 1| + c
28 3 ln |x 2| 2 ln |x 1| + c
18x + 3x2 5 ln |x 1| + 48 ln |x 2| + c
2x + 23
x3 + c
ex + xe+1e+1 + c
29 12x2 + 14x4 + 25
x5 + 29
x9 + c
113 13x + c
52x
2 + 8x + 9 ln |x 1|+ c
5x 9 ln |x 1| + 27 ln |x 2| + c
30 19 e3x1 (3x 1) + c
14x
4 + e3x x + c
12x 1105x + c
133x + c
31 1105x + 12x + c
2 ln |ex 1| x + c
1ln 22
x + 1ln 33x + c
532
51 + 4x8 + c
32 14 ex4+ c
x3ex 3x2ex + 6exx 6ex + c
32x 2x 12xx + c
13 (x 2)3 + 13 (x 3)3 + c
33 1x2 1x3 + c
15x
5 52x4 + 373 x3 30x2 + 36x + c
1+ e
x+x + c
1ln 2+2 ln 3+3 ln 52
x32x53x + c
34 178 ln(13u6 + 18
)+ c
14x
2 ln x 18x2 + c
(x+)+1
(+1) + c
2ln 47
( 47
)t + c
35 f (x) dx ={
x22 + c x < 0
x33 + c x 0
|x| ln (1 + |x|) + c
36 1 ln |x | + 1 ln |x |+ c
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-
+ ,
ex + ln |ex 1| + c
13
2x 33 + c
2 (x 1)
1x1 + c
37 16 ln6 x + c
143
(x3 1)4 + c
323(
1 +
x)4 + c
18 e
4x2+5 + c
38 16 ln |x| 15 ln |x 1| + 130 ln |x 6| + c
39 A = 3, B = 1, = 3, = 2, E = 1
1x1 3 ln |x 1| 12(x2)2 +
2x2 + 3 ln |x 2| + c
40 x x + c
41 x+|ln(x+)||ln(x+)|2
+ c
42 12 ln |x 1| ln |x 2| + 12 ln |x 3| + c
43 f (x) = 43
x 13 x + 1
44 13 ln3 x + c
45 251 + x5 + c
46 ln |1 + x| ln |x| + c
47 x2 2x + 3 + c
48 ex+ 1x + c
50 13(1)(x3+)1
51 = 1ln 2 , = 73 ln2 21ln2 2
52 ln 2 2 + 2
53 23()(
x 3 x 3)
+c
54 1. 12
1 + x4 + c
2.
3x2 + 2x + 1 + c
3. 23
x2 3x + 23 + c
55 23 (x x)32 + c
11+x + c
ln (xx + x) + c
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-
56 122 1x + c
57 2
x+ c
58 In = exn1xx+nx2+n22 +n(n1)22+n22
In2
61 1754
62 0 f (x) dx = 211 ; 0 g (x) dx = 1711 64
65 t = ln 4
t = 1 + ln 4
t = ln 31+e
t = 31+e
66 b = 2
67 xx + 122x + xxx
69 35 (& (&
75 x = 0
77 f (x) = ex
g (x) = x
h (x) = (2x + 1)
(x2 + x 1)
s (x) = 4e4x2 ex+1
w (x) = a
x1 e
t2 dt + axex2+ b
78 ( 4 ) = 22 ; ( 2 ) = 1; ( 4 ) = 22 ; ( 2 ) = 079 y = x + 2 y = x 3
81 f (x) = x xx
83 x33 + x
84 23
87 ' I ( 12 ) = 120 2t+1t2+5t+6 dt = ln 2 + 5 ln 5 8 ln 3 +' max (I (1) , I (1)) =max (5 ln 3 + 8 ln 2, 13 ln 2 8 ln 3) = 13 ln 2 8 ln 3
88 f (x) = x24 + x2
90 & Df = R f (x) = x ,+% f &
'&% 7(&' ' (, 0]
'&%6' ' [0,+)
91 9: e33 9: 6
3
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-
+ , !
92 & Df = (, 0)(0,+) g (x) = 4x ln |x| g
'&% 6' ' (,1], (0, 1]
'&% 7(&' ' [1, 0), [1,+)
93 xx
2x2 x21 tt dtx+ x1 tt dtx2
95 9:(1,+) 9: & ' ,/& '
96 < f &
'&% 7(&' ' /' (, 1]
'&% 6' ' [1,+)
' $ x = 1
98 f (x) = x x
101 3x2a3 a3
102 1 x = 1 332
104 f (x) = ex + xex
106 : =, +' x = 1; , ' x = 2 : >, x = 54
107 %&2 ( 4& f (x) f (x) = ex >, ,,'' ex 84& f (x) = xex + ex
109 f (x) = x; = 6
111 f (x) ={
(x)2x +
(x)2 , x = 0
, x = 0
112 e16
113 +
114 . = 1 f ,& & ,/, ' '' . = 1 ( & f = 0
116 f (x) = ex
117 a = 3
118 f (x) = x
119 = 23
123
ln 3
23
$
124 18 46
(e8 1 + e6 + e2) e4 4 + ln 3
1
125 23
3 23
3
14
22 lnln1
13p
6 13p3
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"
126 ln 2 ln 3
263
ln 7 ln 2 ln 5 3
1
127 t + 13 t3 s 13 s3
0
14
1 + ln 2 ln (e + 1)
128 1
52
x4 2x2 + x
2
129 132
172
172
132
130 6581(ln 2ln 3)
32 s
6 ln s 14 s6 12 s2 ln s + 14 s2
43
223 + 2
131 1 2e
14
2 2
12 ln 3 12 ln 2 + 14 19
3
316 e
4 + 116
132 13
72 + ln 2
ln 2
1 1
133 2 9
13mn
3 + 12n3 13m4 12nm2
x412x2
134 20 f (x) dx = 13 (42 1)
135 983
136 ln 2 316
137
138 = 3
139 = 7, = 6, = 3
140 x = 1; x = 2
141 ##
143
144 13
145
146 9 ln 11 + ln 2
147 12
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-
+ , #
149 1 +
k=2
k1ln k
150 : 1 e e : 6 e3 32e 6e 6e
152 12 ln 3 + 16 ln 3 13 ln 2 + 112 ln 112
153 12 ln2 2
155 2
156 1
157 x2
159 1148 + 564
161 : 0 f(x)f(x)+f(x) dx = 2 3: 12162 I1 = 5; I2 = 1312 12 ln 2
163 : I = 12x
(x2 a2) 12a2 ln(x +
(x2 a2)
)+ c
: F (1) = 12 ln(2 +
3)3
164 122
165 = k + 4
167 : 24
168 ln 2524
169 32 1e
170 * 1 = 2x + 2x 8 4& 433
172 4865
173 = 1, = 1, = 0
ln x 12 ln(x2 + 1
)+ c
174 9: 6 ln 2 9: ii. 6 ln 2
177 '' u = 3 + x ln x ( 4& ln (3 + 2e2) ln (3 + e)182
I = ex1x
x + x
2 + 2+
( 1)2 + 2
I2
I4 = ex
44x + 424x + 1222x + 433x + 16 sin x3x + 24xx + 24
(4 + 202 + 64)
183 I5 = 12 ln 2 14
185 : 13x3 ln x 19x3 + c : 19
186 4
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$
188 21 f (x) dx = 52189 : 4 , 4
190 = 1
192 9 & f (x) = 2(x)x22 ,+% f &
'&% 6' ' [, 0]
'&%
7(&' ' [0, ]
197 $
199 1e(
bb
aa
) 1ba
202 9 = 0 9 12 ln 2
205 9 2x+xf(x)2(+1) f (x)
206 12
209 S = 112
210 S = 1355
12
211 S = 132
212 E = e1 (ln x ln2 x) dx = 3 e213 E = 10 (x (2x 1)) dx = 23214 +' f(2) = 2 3' f(0) = f(1) = 0 = 2 4/ & E = 20 (2 f (x)) dx =103
215 23
216 = 36
217 = 3
218 94
219 736
220 10
221 9 1 x y = 3x x 0 x = 0 9 E () = 12 9 12
222 1+1
223 4e 6e1
225 12
227 3ln 2 43
228 p = ln 1+e2
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-
+ ,
229 = 1
231 1
232 9 93: A(2, 2); B (3, 3) 943: 13
233 y = 2x + 122 + 18
2
E =28
(1 + 4
)2 12234 = 336
235 9 t (x) = 2x2 4x; 9 E = 20 |t (x)| dx = 83236 f (x) = x+x2 + 1
237 13 + 12
240 >( '' 7 f (x) = cx3, c > 0, x 0
244 2; 132713 827 ; 2 ; 12
2 12 ln
(2 1
)
245 f (x) + c
f (x)
f (x)
0
f () f ()
f (x) f ()
247 13 + ln x + ln2 x + 13 ln3 x + c
250 = 1, 2, 3
251 +; +
252 $
253 f (x) = e2x2
254 9 $
255 9 (x) = ln(x +
y2 + 1
)9 ln
(2 + 1
)2 + 1
256 >& x 1; x 1
< 7,+ + 6&'%' y = x 2 4/ & 274
259 9 i.? f (x) = ef(x)1+ef(x)
260 :
261 13
263 12 e+1
e1
264 x = 1 x + y = x 4
265 f (x) = ln (x + 1)
F (x) = x ln (x + 1) + ln (x + 1) x
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-
A (0, 0) , B (e 1, 1)
266 13
267 x = 13 + 137; x = 13 13
7
268 +
270 : 1 1e3
+ 2 ln 2
272 ' ( x0 = 1
9@: & 2 f (x) 2 x94@: A&'
x+1x
f(t)dt =
2 ln(
x+1x
)x < 2
2x2 + x + 23 1 x 023 23x3 + 2 ln (x + 1) 0 < x < 1
2 ln(
x+1x
)x 1
< 6&'%' + / ' x = 12 ; x = 1e112 1
273 9; 9; 9
275 9 y = 2x + 5
277 f (x) = x +
279 1. f (x) = ln (x + 1)
2. x0
f (t) dt = ln (x + 1)x + ln (x + 1) x281 1'&% 7(&'
285 9 2512
287 f (x) = exexex+ex
ln(ex + ex
)+ c
290 & ,'/' 7 00 ,+% & limx1ex21 =
1e
292 : f (x) = e1x2 : F (x) = 12 12 e1x2
x , x > 1
293 '(& '' (x) = x0 tf3 (t) dt ( x0 tf (t) dt)2 , x 0295 : 8 ln 2
297 : I () = 7 (22 7 + 7) e : !299
300 g (x) = 3x2 |z| f (x3) 3 z + 1z 301 : +
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-
+ ,
302 : "
303 : t = 23
305 : 0
306 : < h + +' h(
ln 44
)= 210 4(
4
) 4
: x2 = 2x1 : 2004
308 : ii. 1 : (b) 2 = 3J 2I; 43
309 : +; $ 1 : b f (x) = xe1x (x 2)
:
: I+1 = ( + 1) I 1 :4 e 2; 2e 5
: 4 $
310 ln 28
311 f (x) = 2007x
313 4' '?
314 = ,/& ' ,&2 , x > 4 0 < x < 1
317 498(1 k2) x = 494 k1 k2
1 /(+ k 4/ &
E (k) =
492
(1k2)k
0fk (x) dx =
2401
24
(1 k2
) (1 k2)k
& E (k) = 2401241 k2 (1 2k) (2k + 1) ,+% E (k) & +' k =
12 ,+% +' 4/ x
x '&2 f 12(x)
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-
319 : $
322 $
323
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