Α16.ΟΛΟΚΛΗΡΩΜΑΤΑ ΑΣΚΗΣΕΙΣ

! " ! # $

# "% #!"% # & "

! # ' ( #

) * +,,-

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.

www.nsmavrogiannis.gr

http://lyk-evsch-n-smyrn.att.sch.gr

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 .



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. :



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. :



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



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



!

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)



"

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



!! 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



$

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 (;) () () . .



!! 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. .



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



!! 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 .



" #

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. . , , -



% &'(

. - 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



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



) * !

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 .



"

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).



) * #

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

.



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.



) *

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



) * !

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.



"

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.



) * #

() 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 () ( )

, - . " $ "



$

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]



) *

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



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 .



+ ,

) *

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



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



+ ,

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



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



+ , !

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



"

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



+ , #

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



$

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



+ ,

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



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 : +



+ ,

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)



319 : $

322 $

323



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