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  • 7/27/2019 Olokliromata Gl

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    : 1

    x1 x2 x3 x xx0= =x -1-1x

    f()

    Cf

    0

    =1

    f(x)dx lim f( )x + + + +

    ====

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    : 2

    1.

    : () f

    F F(x) = f(x) , x .

    :

    F f .

    :

    H G(x)=F(x)+c , c ,

    f

    G f ,

    G(x)=F(x)+c , c

    y=F(x)

    y=F(x)

    y=F(x)

    y=F(x)

    +c1

    +c2

    +c3

    O

    - .304 :

    F(x)f(x) . G

    f c R :G(x)=F(x)+c x R .

    , F

    F f (, ) ,=() , F .

    f () ,

    .304 . ...-

    ( ) ( ) ( ) ( ) ,f x g x f x g x c x = = + (=)

    .

    :2

    1( ) 1 , ( , 0) (0, )f x x

    x= + = ,

    1( ) ,F x x xx

    = + , ( ) ( ) ,F x f x x = .

    1

    , 0

    ( )1

    2010 , 0

    x xx

    G x

    x xx

    + >=

    + + =

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    .

    21

    , , , , , , ,...ln

    xx xe x x

    e x x xx x x x

    .. .

    f.

    f F

    ( ), ( ) ( )F x f x x = f , .

    (. 305) .

    /

    1 ( ) 0f x = ( ) ,G x c c= R ,

    2 ( ) 1f x = ( ) ,G x x c c= + R

    31

    ( )f xx

    =

    ( ) ln ,G x x c c= + R

    4 ( ) f x x=

    1

    ( ) ,1

    x

    G x c c

    +

    = + +

    5 ( )f x x= ( ) ,G x x c c= +

    6 ( )f x x= ( ) ,G x x c c= +

    72

    1( )f x

    x=

    ( ) ,G x x c c= + R

    8 21

    ( )f x x

    =

    ( ) ,G x x c c= + R

    9 ( ) xf x e= ( ) ,xG x e c c= + R

    10 ( ) xf x = ( ) ,ln

    xG x c c

    = +

    306 307. 2, 4, 5 7 .

    21 11

    2 , 0, 0( ) ( )

    0 , 0 0 , 0

    x xx xxF x f xx

    x x

    = =

    = =

    f 0, ( ) ( )F x f x x R =

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    : 4

    2.

    :

    ) f [,].

    ) [,]

    x =-

    , -1

    [xk-1 , xk], [xk-1 , xk]

    : f(1)x + f(2) x + f(3) x + + f() x + ..+ f() x =

    = 1

    f ( ) x

    =x0 x1 x2 x3 =xxx x xx -1-2-1 +1

    1 2 3

    )

    lim++++

    (

    = 1

    f ( ) x ) .

    f .

    f(x)dx

    f ( x ) d x = lim++++ (

    = 1

    f ( ) x )

    x1 x2 x3 x xx0= =x -1-1x

    f()

    Cf

    0

    f f(x) 0 f x[,] :

    = 1

    f ( ) x

    f().

    Cf , x x x= x= .

    ( + ) .

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    : 5

    :

    f(x) 0 , f [,] :

    f(x)dx

    Cf ,

    x x x= x=.

    Cf

    :

    f(x)dx

    ()

    1 = +-

    , 2 = + 2

    -

    , .... , = +

    -

    , ...., = +

    -

    =

    .

    1 ==== = 1+ 2+ 3+ .+

    =1 =1 =1

    ( + )= + ,

    =1 =1

    = , ( )

    1+2+3+.+ =

    =1

    (+1) =

    2 12+22+32+.+2 =

    2

    =1

    (+1)(2+1) =

    6

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    I

    f ( x ) d x

    1. f(x) 0

    f(x)dx 0

    2. f(x) 0 [,] f [,]

    f(x)dx >0

    3.

    f(x)dx=0

    4.

    f(x)dx =

    - f(x)dx

    5.

    f(x)dx= f(x)dx

    6.

    [f(x)+g(x)]dx= f(x)dx+ g(x)dx

    7.

    [f(x)+g(x)]dx= f(x)dx+ g(x)dx

    8.

    f(x)dx= f(x)dx+ f(x)dx ,

    , ,

    ( = 1 + 2)

    ( )f x dx

    f

    : ( ) ( )= f x dx f t dt

    dx x.

    . ( ) ( ) ( ) ( )g t f x dx g t f x dx

    = ,

    . ( 1) ( 1)t x dx t x dx

    =

    ( )f x dx

    Cf

    21

  • 7/27/2019 Olokliromata Gl

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    : 7

    .

    2

    2 2 2 2

    1

    1

    2 [ ] 2 1 3xdx x= = = 2

    2 2 2 2

    1

    1

    2 [ 2013] 2 2013 (1 2013) 3= + = + + = xdx x

    f(x)dx

    [,] x = -

    = + -

    f ( x ) d x = lim++++ (

    = 1

    f ( ) x

    c dx = c(-) , c ( ..331 )

    .

    1

    0

    x dx = 12

    3.

    ().

    :

    1.

    2. .

    f ],[ . G

    f ],[ , =

    )()()( GGdttf

    (f (x) g (x))dx [f (x) g(x)]

    + = ++ = ++ = ++ = +

    (f (x)g(x) f(x)g (x))dx [f(x).g(x)]

    + =+ =+ =+ =

    2

    f (x)g(x) f (x)g (x) f (x)dx [ ]

    g (x) g(x)

    ====

    f (g(x))g (x)dx [f (g(x)).]

    ====

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    1dx [x]====

    f (x)dx [f(x)]=

    dx [ x]= = = =

    f (x)dx [ f (x)]=

    xdx

    2x[ ]

    2

    ====

    f (x)f (x)dx 2

    1[ f (x)]2

    =

    x dx

    +1x=[ ]

    +1

    f (x)f (x)dx +1

    1=[ f (x)]

    +1

    1dx

    x[2 x]====

    f (x)

    f(x)

    dx [2 f (x)]=

    xdx [-x]====

    f (x) f (x)dx=[-f(x)]

    xdx =[ x]

    f(x) f (x)dx=[ f(x)]

    2

    1

    xdx ====

    2(1+ x)dx=[x]

    2

    1

    f(x)f (x)dx =

    ( )21+ f(x) f (x) dx [f(x)]=

    2

    1

    xdx ====

    2(1+ x)dx=[-x]

    2

    1

    f(x)f (x)dx =

    ( )21+ f(x) f (x) dx [-f(x)]=

    xe xdx [e ]====

    f ( x )e f ( x )f (x)dx [e ] =

    1dx

    x[ln x ]====

    f (x)

    f(x)

    dx [ln | f (x) |]=

    1dx

    x

    -+1x[ ]-+1

    ====

    1

    f (x)

    -+11f (x)dx=[ f (x)]-+1

    x dx ====

    x

    1

    xdx [ ]

    +1

    ++++

    ====

    f (x) f (x)dx=

    f (x)

    1

    f (x)f (x)dx=[ ]

    +1

    +

    x

    xdx [ ]

    ln

    ====

    f(x)

    f(x)f (x)dx=[ ]

    ln

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    : 9

    4.() f(g(x)),

    g=[, ] , g f

    g().

    )

    f(g(x))g(x)dx

    ) u=g(x) , du=g(x)dx u1=g() , u2=g()

    ) T

    f(g(x))g (x)dx =

    2

    1

    u

    u

    f(u) du

    (. .313-314 337-338)

    5.( )

    :

    f(x)g(x)dx= [f(x)g(x) ] -

    f (x)g(x)dx

    :

    1. e

    x+

    2. (x+) (x+)

    3. P(x) 1 a

    ax

    x

    =

    4. ln(f(x)

    f(x) ,g(x) ,

    , g(x), g(x) = G(x) :

    f(x)g (x)dx = [f(x)g(x)] - f '(x)g(x)dx

    ,

    . (. .310-311)

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    : 10

    .9

    1

    xe dx

    9 9 9 9 (1)

    9 9 3 3 3

    1 1

    1 1 1 1

    9 (*)2

    1

    3 3 3 3(*) 32 2 2 3

    11 1 1 1

    1( ) [ ] ( ) [ ] 9 5 4

    2

    1, , 1 1, 9 3

    2

    ( ) [ ] 2 9 2 ( ) 9

    = = = = + =

    = = = = = = =

    = = = = =

    x x x x x x

    x

    u u u u u

    e dx x e dx xe x e dx xe xe dx e e e e ex

    xe dx u x u x x u x ux

    u e du u e du u e ue du e e u e du

    3

    3 3

    11

    3 3 3 3 33 3(1)

    1 1

    2[ ] 2

    9 2[ ] 2[ ] 9 6 2 2( ) 5

    +

    + =

    =

    = + + =

    u u

    u u

    e e ue e du

    e e ue e e e e e e e e e

    6.

    (x)(x+)dx

    (x)(x+)dx

    (x) .(x) =

    ,

    E (=.(x)) .

    :

    2

    1

    2x22xdx (.1iv .316)

    (x)ex+ dx

    (x) .(x) =

    ,

    E (=.(x)) .

    :

    2

    1

    x2e-xdx (.1i .316)

    ex+(g(x))dx ,

    ex+(g(x))dx

    g(x) ,

    . 2

    , .

    :

    2

    1 e

    xxdx (.1ix .316)

    (x) lnx dx(x)

    .(x) =

    .

    :

    2

    1

    x3lnxdx (.1iii .316)

    :

    )

    )

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    : 11

    7.

    (x)

    dxQ(x)

    ,

    .

    f (x)dx [ln | f (x) |]f(x)

    ==== ,

    .2 2 2

    2 2

    12 2

    1 1

    2x (x 1) 5dx dx [ln(x 1)] ln5 ln 2 ln

    x 1 x 1 2

    += = + = =

    + +

    f (x)

    f(x)

    , :

    ) <

    (314-315)

    .1

    2

    0

    2x+1dx

    x 5x 6 + , ()

    2x+1

    (x-2)(x-3)=

    A

    x-2+

    B

    x-3, xR-{2,3} 2x+1=A(x-3)+B(x-2)(1), xR-

    {2,3} (1) (A+B-2)x=3A+2B+1, A+B-2=0 3A+2B+1=0, = -5 =7

    1 1

    1 1

    0 0

    0 0

    2x+1 -5 7dx ( )dx 5[ln x 2 ] 7[ln x 3 ]

    (x-2)(x-3) x-2 x-3= + = +

    ) (x) Q(x)

    (x) : Q(x) , (x)= (x)=,

    (x) = Q(x).(x)+(x).(x)

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    : 12

    8.: * !!! *

    )2

    x2

    2xx

    1+2

    = ,

    2

    2

    x1-

    2xx

    1+2

    = ,2

    x2

    2xx

    1-2

    = , . x2

    ) 21-2x

    x2

    = , 21+2x

    x2

    = , 21-2x

    x1+2x

    = .

    xxdx dx ...

    x

    = =

    ,

    u=x, du= - xdx

    1du [ln u ]

    u

    = =

    2 2

    1 x xdx= dx= dx=

    x x 1- x

    ,

    u=x, du= - xdx2

    1du

    1 u

    = =

    2 2

    1 x xdx= dx= dx=

    x x 1- x

    ,

    u=x, du= xdx2

    1du

    1 u

    = =

    3xdx=

    2x xdx=

    =

    (1-2x)xdx=

    ,

    u=x, du= xdx

    x x

    2+1x xdx

    =

    (1- u2)du=

    2+1x xdx

    * 21-2x

    xdx= dx=2

    1-2x 1 1dx= [x] [2x]

    2 2 4

    =

    * 4 21+2x

    xdx= ( ) dx=2

    .

    x x

    21= ( 2x 2x)dx=...

    4

    + +

    2x 2xdx

    *

    R (x , x) dx

    R=x,x

    u=x

    2,

    2

    2

    1 xdu ( ) dx

    x 2

    2

    x2du (1 )dx

    2

    =

    = +

    22dudx1 u

    ====++++

    .

    x=2

    x2

    2x

    12

    + ,x=

    2

    2

    x1

    2x

    12

    +

    x=2

    x2

    2x

    1 2

    ,2

    2

    x 11

    x2

    2

    + =

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    : 13

    9.

    .

    R(ex , ex , ... , ex) dx (R= )

    : u=ex

    .

    2x x

    2x x

    e edx

    e 2e 1

    ++ +

    =(x xu e du e dx= = )

    e e 22 e

    2 2 e

    e e

    u 1 1 (u 2u 1) 1du du [ln(u 2u 1)]

    u 2u 1 2 u 2u 1 2

    + + += = + +

    + + + +

    . -

    f(x, x+, x+, ...)dx

    u= x+

    = ... (,,)

    x+dx

    , 1 dx

    x+

    u=x+ u=

    x+

    x+ x+f ( x , , , ...)dx

    x+ x+

    u=

    x+

    x+

    = ... (, , ...)

    2

    1

    k

    2 2 2

    k

    f ( x , - x )dx x=

    u

    2

    1

    k

    2 2 2

    k

    f ( x , x - )dx x= 1

    u

    2

    1

    k

    2 2

    k

    f ( x , x + )dx x= u

    .1 24 4

    2

    2 3

    0 0 02

    u+1 x=u 1x +1dx=( ) du= du

    1 u udx du

    u

    =

    =

    10.

    (-1, -2,)

    :

    =

    f(x)dx

    M ()

    , -1 , -2 ,() .

    0 , 1 , 2 ...

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    : 14

    =

    f(x)dx ,

    , .

    .

    ) =

    (lnx) dx=

    (x)(lnx)dx=[x(lnx)]-

    x((lnx))dx==[x(lnx)]-

    (lnx)-1dx=

    )

    -2 2 -2 2 -2 2 -2

    2 -2

    -2

    xdx= x. xdx= x.(1+ x-1)dx= x.(1+ x)dx- xdx=

    =....(u=x, du=(1+ x)dx)......= u du-

    =

    ) =

    xe

    x dx x>0 2 , =

    x

    -1-1

    e 1- +

    -1(-1)x.

    11.

    . ( )f x dx

    , 12

    ( ) ,( )

    ( ) ,

    f x xf x

    f x x

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    : 15

    12.

    ( ) ( ) [ , ]F x f x x = , ( ) [ ( )]f x dx F x

    = ( ) ( ) [ ( )]f x dx F x dx F x

    = = ( ) [ ( )]f x dx f x

    =

    ( )f x dx

    f. ( ) ( ) ( )f x dx f t dt f u du

    = =

    ( )f x dx

    = , ( ( ) ) 0f x dx

    =

    f [,] ( )f x dx

    ( ) ( )

    [ , ]

    f x g x

    x

    =

    , f, g , ( ) ( )f x dx g x dx

    = .

    . ( ) ( )f x dx g x dx

    = ( ) ( )f x g x=

    ( ) 0

    [ , ]

    f x

    x

    f , ( ) 0f x dx

    . .

    ( ) 0f x dx

    ( ) 0, [ , ]f x x

    f [,] ,( ) 0

    [ , ]

    f x

    x

    , ( ) 0f x dx

    > .

    ( ) ( )

    [ , ]

    f x g x

    x

    f, g , ( ) ( )f x dx g x dx

    : ( ) ( ) ( )h x f x g x= , ( ) 0 [ , ]h x x .

    . ( ) ( )f x dx g x dx

    ( ) ( ), [ , ]f x g x x

    ( ) ( )f x dx f x dx

    , : ( ) ( ) ( ) , [ , ]f x f x f x x ,

    : ( ) ( ) ( ) ( ) ( )f x f x f x f x f x

    f [,] , f ( [,])=[m,M] ( ) , [ , ]m f x M x

    ( ) ( ) ( )m f x M

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    13. F(x )= ( )x

    t dt

    A f

    F(x) = ( )

    x

    t dt

    , x ,

    f, ( ) ( ( ) ) ( ), = = x

    F x f t dt f x x

    ()

    x = (),

    = , t = () .

    F

    F f , ( ) ( )F x f x x =

    =() , F F.

    1., t = () ,

    . ( ) ( ) ( ) ( )

    x x

    a a

    g x f t dt g x f t dt = 2..

    (,) , [,) , (,] , [,] , [,+), ( +), (-,], (-,), (-,+).

    3. F f .

    F .

    4. F F(x0) =0x

    f(t)dt .

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    5. F - -

    .

    F f. f,

    c :

    1 2 1 2 , , : F1(x)=1

    x

    f(t)dt , F

    2(x)=2

    x

    f(t)dt

    : F1( x ) - F2 (x) =1

    x

    f(t)dt -

    2

    x

    f(t)dt =

    1

    x

    f(t)dt +

    2

    x

    f(t)dt

    =2

    1

    f(t)dt

    = c , x

    6. ( ) ( ), ( ) 0x x

    a a

    d df t dt f x f t dt

    dx dt

    = =

    7. :( )

    ( ( ) ) ( ( )) ( ) =g x

    f t dt f g x g x

    .

    8. 0 0( )Q Q t= ( )Q t ( )Q t ( ) ,

    00

    ( ) ( )

    t

    Q t Q Q y dy= + 9. F , F .

    10. F , ox , lim ( ) ( )=

    oo

    x xF x F x lim ( ) ( )

    o

    o

    x x

    x xf t dt f t dt

    =

    11.

    x

    f (t)dt dx F(x)dx

    = ( x

    F(x) f (t)dt= & F()=0 )

    F(x)dx (x) F(x)dx [xF(x)] xF (x)dx F( ) F( ) xf (x)dx F( ) xf (x)dx

    = = = =

    12. F(x) =x

    f(t)dt , F()=0 CF ( ,0) ( )

    F, F(x)= ( )x

    f t dt

    , ,

    f

    , f

    , , x , f

    F(x) = f(x) , x . , x ().

    ( )

    ( )( ) ( )

    g x

    h xF x f t dt = (f, g, h )

    x F

    g h

    x ( ), ( )g x h x f.

    : ( ) ( )x

    F x f t dt = ,

    ( )

    ( ) ( )g x

    F x f t dt =

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    .F(x)=0

    1

    xt

    dtt , 1

    t

    t ( ,1) (1, ) +

    0 ( ,1) , ( ,1)

    . F(x)=2

    01

    x

    t dt , 2

    1 t =[-1,1]

    0 , F [-1,1].

    . F(x)=2

    3

    1

    x

    t dt , 2 1t

    =(- ,-1] [1,+ ) 3 [1, ) + [1, )x + , F [1, )+

    . F(x)=ln 2

    3

    1x

    t dt , 2 1t

    = (- ,-1] [1,+ ) 3 [1, ) + ln [1, )x + x>0.

    :0

    ln 1

    x

    x

    >

    0

    ln ln

    xx e

    x e x e

    >

    F [ , )e +

    . F(x)=

    ln

    2

    54

    x

    xt dt

    ,

    2

    4t

    =(- ,-2] [2,+ ) ( 0x > 5x > g(x), h(x) )

    :2

    05

    59

    ln 29

    5 2

    xx

    xx e x

    xx

    x

    > > >

    0

    5

    ln 2

    5 2

    x

    x

    x

    x

    > >

    F [9, + )

    . f :[0, )+ f R g(x)=6

    1

    (3 )f x t dt,

    H f(3x-t) , 3 0 t R x t

    3 ( ,3 ] t x t x g f(3x-t) [1, 6], [1,6] ( ,3 ] x , 3 6 2 x x

    g [2, )+

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    H F F(x)= ( )

    x

    f t dt

    , , x

    f, F F(x)=f(x) , x .

    1: f g

    : ( )g x x ,

    F(x)=

    ( )

    ( )

    g x

    f t dt

    .

    (x)= ( )x

    f t dt

    (x)= ( ), f x x

    F(x)=(g(x))=

    ( )

    ( )

    g x

    f t dt ,

    F(x)=((g(x)))= (g(x))g(x)=f(g(x)) g(x).

    2

    : F(x) =

    ( )

    ( )

    ( )

    g x

    h x

    f t dt , f g h :

    ( )g x ( )h x , x . :

    g, h : I R () x

    f : R () g(x) h(x) , F . F:

    . x :

    F(x)=

    ( ) ( ) ( )

    ( )

    ( ) ( ) ( ) ( )

    g x g x h x

    h x

    f t dt f t dt f t dt f t dt

    + =

    F(x)=

    ( ) ( )

    ( ( ) ( ) ) ( ( )) ( ) ( ( )) ( )

    g x h x

    f t dt f t dt f g x g x f h x h x

    =

    . G f ( ) ( ),G x f x x =

    F(x) =

    ( )

    ( )

    ( )

    g x

    h x

    f t dt = G(g(x))-G(h(x)), F

    F(x) = G(g(x))g(x) - G(h(x))h(x) = f(g(x)) g(x)- f(h(x)) h(x)

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    : 20

    f

    f , () F

    F(x) =x

    f (t)dt , x

    (.2iii, 323 ) f(x), f (x) f (x) 2x + = ,

    ( )x x x x xf (x) f (x) 2x e f (x) e f (x) 2xe e f (x) 2xe + = + = = (1)() xxe , xxe R,:

    ( )x x x

    t t t x t t x t x x x

    0 0 0

    0 0 0

    F(x) te dt t e dt [te ] e dt [te ] [e ] xe e 1= = = = = + x x1F (x) xe e=

    : (1) ( ) ( ) ( ) ( )x x x x x

    e f (x) 2 xe e e f (x) 2e (x 1) = = , x xe f (x) 2e (x 1) c= + , xf (x) 2(x 1) ce= + .

    f(x)=lnx , x>0f, f

    x x x

    x x

    1 1

    1 1 1

    1F(x) ln tdt (t) ln tdt [t ln t] t. dt x ln x [t] x ln x x 1

    t= = = = = + G(x) x ln x x=

    . 1, 1( )ln , 1x xf x

    x x = >

    f ( ,1) (1, ) + 1, 1

    1 1

    lim ( ) 0

    lim ( ) lim (ln ) 0

    x

    x x

    f x

    f x x

    + +

    =

    = =

    f R.

    2

    1

    2

    , 1( ) 2

    ln , 1

    + =

    + >

    xx c x

    F x

    x x x c x

    f

    1, 1 21 1

    1lim ( ) lim ( ) (1) 12 + = = + = +x x

    F x F x F c c

    c1 = 0 , c2 =

    . f : f(1)=2 xf(x)-f(x)=3. N f.

    ( ) ( )( ) ( ) ( )x 0

    2 2

    xf x f x f x3 3xf x f x 3

    x x x x

    = = =

    1

    2

    3 , 0

    ( ) , 03 , 0

    +

    c x x

    f x c xc x x f 1 ,

    2 21

    lim ( ) (1) 3 2 5

    = + = =x

    f x f c c

    15=c

    3= c

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    14.

    . F : F(x)= ( )x

    f t dt

    .

    . 2

    2010( ) 1

    x

    f x t dt= ., .

    ( [1, )fD = + , ...2

    ( ) 1 0f x x = , f. , f(1))

    .

    0

    ( ) ,

    x

    txf x e dt x R= .

    f f(x)

    u=tx, du=xdt t=0, t=1 u=0 u = x22 2 2

    0 0 0

    1 10, ( ) ( ) ( )

    x x x

    u u ux f x e du f x e du xf x e du

    x x = = =

    2 2 2 2

    2 22 ( ) 2 ( ) 2 2x x x xx e f x xe f x xe x e+ = = , 0x .,

    . f . R,

    3

    1

    ( ) 2012= f t dt 5

    3

    ( ) 2013= f t dt .

    2

    ( ) ( )

    x

    x

    g x f t dt

    +

    = [1,3]

    ( ( ) ( 2) ( ) 0g x f x f x = + > , g . , g([1,3])=[2012, 2013]

    . f R , R .

    ( ) ( )F x f x t dt

    = .

    u=x-t du= -dt ( ) ( )x

    x

    F x f u du

    = . G f, G(u)=f(u) , u R , F(x)=G(x-)-G(x-) F(x) = f(x-)-f(x-)

    . .

    ( )

    x

    f t dt

    f,

    f f

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    : 22

    . f *

    :f R R 1

    ( ) 12 ( )

    xt

    f x dtf t

    = +

    ): f(x)>0 , ) f.)f f(x) 0, R, f(1)=1>0 f(x)>0

    )2 2

    2 2( ) ( ( )) ( ) ( )

    2 ( ) 2 2

    = = = +x x x

    f x f x f x c

    f x

    2 1

    ( )

    2

    +=

    xf x .

    . : (0, )f R+

    1

    1( ) 2 ( ) , 0

    x

    f x f t dt xx

    = + > (1)

    1

    2( ) 2 ( ) ( ) ( ) 2 ( ) ( ) 2 ( ) ( ) 2ln

    x

    xf x x f t dt f x xf x f x xf x f x f x x cx

    = + + = + = = = +

    (1) f(1)=2 , c=2 ( ) 2ln 2f x x= +

    1. : x ( ) ( )( )

    x

    a

    f x f t dtg x

    = +

    / g(x), ( ) ( ) ( ) ( )x

    a

    g x f x g x f t dt = +

    .

    2. : ( ) ( )x

    a

    g x f t dt t ,

    g(x) :

    ( ) ( ) ( ) ( )= x x

    a a

    g x f t dt g x f t dt , .

    3. : ( , )

    x

    a

    f t x dt t ,

    . (, ).

    . :f

    0

    ( ) ( ) ,

    x

    tf x e f x t d t x=

    (1)

    u=x-t . ..

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    : 23

    .

    . f(x)+f(+-x)=c (1) : =

    + -f(x)dx = (-)f( )= (f()+f())

    2 2

    f(x)dx= (c-f(+-x)dx=

    ( u=+-x, du= -dx)=c(-)+ f(u)du

    = c(-)- f(u)du

    2= c(-)c

    ( )2

    = , (1) x=2

    + 2f(

    2

    + ) = c

    .

    ) ... ( ) 0f x dx

    , ... ( ) 0, [ , ]f x x

    ) ... ( ) ( )f x dx g x dx

    , ( ) ( ), [ , ]f x g x x

    ( ) ( ) ( ), [ , ]F x f x g x x = .. ( ) 0F x

    .

    ) f [,], ([ , ]) [ , ]f m M =

    . ( ) ( ) ( )m f x dx M

    ) ( ) ( )f x dx f x dx

    . :2009 2009 2

    0 0

    ln ( 1) ( )2

    xx dx x dx+

    .2

    ( ) ln( 1)2

    xF x x x= + + , [ 0 , 2 0 0 9 ]x

    21

    ( ) 1 0 , [ 0 , 2 0 0 9 ]

    1 1

    xF x x x

    x x

    = + =

    + +

    F . [0,2009] , F(x) F(0)=0

    . .

    . ()

    Bolzano , Rolle , , .Fermat ...

    . , R 0

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    : 24

    i) N x0(,) gC .

    ii) N g(x0)=2+f(x0)

    i) g()= g()=0 g (0,+ ) [,]

    . Rolle . x0

    (,) : g(x0)=0 , gC (x0 , g(x0)) .

    ii)

    x x

    1g(x) 2 f (t)dt xg(x) 2x f (t)dt

    x

    = + = + .

    x

    (xg(x)) (2x f (t)dt) g(x) xg (x) 2 f (x)

    = + + = +

    x=xo ,(i )

    o o o o o og(x ) x g (x ) 2 f (x ) g(x ) 2 f (x )+ = + = +

    .f R

    1

    ( ) ( ) ,

    x

    F x xf t dt x R=

    i) (0,1) : ( ) 0F = , ii)1

    ( ) ( )f t dt f

    =

    .A :f R R :

    2( ) , ,

    x h

    x

    f t dt h x h R

    +

    . ( ) 0,f x x R=

    . F(x) f, ( ) ( ) ( )

    x h

    x

    f t dt F x h F x

    +

    = +

    02 ( ) ( ) ( ) ( )( ) ( )

    h F x h F x F x h F xF x h F x h h h h

    h h

    + + +

    :0

    ( ) ( )( ) lim 0

    h

    F x h F xF x

    h

    + = = ( ) ( ) 0F x f x = =

    . .

    ) f, F(x)=

    x

    f ( t )dt ,

    oo o

    x xlim F(x) F(x ), x

    = xlimF(x) F( ) 0

    = =

    x

    xlim f (t)dt f (t)dt 0

    = =

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    . i)2

    22

    lim( 1 )

    x

    xt dt

    , ii) 21

    2

    lim

    x

    xt d

    i) F(x)=2

    2

    1

    x

    t dt F(x) [1, + )

    x 2limF(x) F(2) 0

    = = , ii)

    3 32

    21 1 1

    2

    8 7lim lim[ ] lim( )

    3 3 3 3

    x

    x

    x x x

    t xt dt

    = = =

    ) ( )

    . o o

    1

    0lim( ln )

    x

    xx

    tdt+

    +

    1

    lnx

    x

    tdt+

    =1 1

    1 1 11( ) ln [ ln ] [ ln ] [ ] ( 1)ln( 1) ln 1x x

    x x x

    x x x

    x x

    t tdt t t t dt t t t x x x xt

    + ++ + + = = = + +

    1 *

    0 0lim( ln ) lim(( 1)ln( 1) ln 1) ln1 1 1

    x

    x xx

    tdt x x x x+ +

    +

    = + + = =

    * ( 1)ln( 1)x x+ + 0.( )

    0 0 0 0

    2

    1ln (ln )

    lim( ln ) lim lim lim 01 1 1

    ( )

    DLH

    x x x x

    x x xx x

    x x x

    + + + +

    = = = =

    D L Hospital 0/0 /

    .:

    2

    2

    02

    1lim( 1 )

    +

    +

    x

    xt dt

    x

    (x)=2

    2

    1x

    t dt+ F(x)=(2+x) , x 0lim F(x) F(0) (2) 0 = = = 2

    2

    2

    2 22

    0 0 02

    ( 1 )1

    lim( 1 ) lim lim[( 1 (2 ) .(2 ) ] 5( )

    x

    x

    x x x

    t dt

    t dt x xx x

    +

    +

    ++ = = + + + =

    )

    g(x)

    xh(x)

    lim f (t)dt+ :

    m(-) ( )f x dx

    (-)

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    .

    2

    2

    1lim

    3

    +

    + +

    x

    xx

    dtt

    .

    02 2 2 2 2 2 2 2 22 ( 2) 3 3 4 7 3 3 4 7

    >

    + + + + + + + + + +x

    x t x x t x x t x x x t x x

    2 2 2

    2 2 2

    1 1 1

    3 3 4 7

    + + +

    + + + +

    x x x

    x x x

    dt dt dt x t x x

    :

    2

    2 2

    2 2 2

    1 1 1[ ] [ ]

    3 3 4 7

    ++ +

    + + + +

    x

    x x

    x x

    x

    t dt t x t x x

    2 2

    2 2

    1 1lim [ ] lim [ ] 0

    3 4 7

    + +

    + += =

    + + +x x

    x xx x

    t tx x x

    .

    2

    2

    1lim 0

    3

    +

    +=

    +

    x

    xx

    dtt

    .

    2

    2 2 2 2 2

    1 ( 3)( ) , ( )

    3 ( 3) ( 3) 3

    += = = + + + +

    x xf x f x

    x x x x,

    >0 ( )f x < 0, f [x, x+2]

    f([x, x+2])=[f(x+2), f(x)] , m=f(x+2) M=f(x)

    2 2 2 2

    ( ) ( ) ( 2 ) ( ) ( 2 )

    x x x x

    x x x x

    m f x M m dt f t dt M dt m x x f t dt M x x

    + + + ++ + + ++ + + ++ + + +

    + + + + + + + +

    2 2

    2 2 2

    1 1 12 ( 2) ( ) 2 ( ) 2 2

    3 ( 2) 3 3

    x x

    x x

    f x f t dt f x dtx t x

    + ++ ++ ++ +

    + + + + + + + ++ + + ++ + + ++ + + +

    :

    2

    2

    1lim 0

    1

    x

    xx

    dtt

    +

    +=

    +

    .

    F 2

    1( )

    3= +f x

    x , F(x+2) - F(x) =

    2

    2

    1

    3

    +

    +

    x

    x

    dtt

    F [, +2] , (x)

    x

    x + , ( )x + (< () < +2)

    2

    2 2

    ( 2) ( ) 1 1( ) ( 2) ( ) 2 ( ) 2

    2 3 ( ) 3

    ++ = + = =

    + +

    x

    x

    F x F xf F x F x f dt

    t x

    2

    2 2( )

    1 2lim lim 03 3 ( )

    +

    + += =

    + +

    x

    x xx

    dtt x

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    Z.

    1f (x )dx

    - 1( ) ( )x f u f x u= = , ( )dx f u du=

    -

    1

    ( ) ( )f f

    = = 1

    ( ) ( )f f

    = = .

    1f (x)dx uf (u)du

    = = xf (x)dx

    [,] f 1-1.

    .

    1.

    x

    ( f ( t )dt )dx

    x

    F(x) f (t)dt

    = , F()=0 .

    x

    ( f (t)dt)dx F(x)dx (x) F(x)dx [xF(x)] xf (x)dx

    = = =

    x

    F(x ) f (t)dt=

    F ( x ) d x

    2.

    x

    ( f (t )dt )du

    . f ( t )d t

    ,

    x x

    ( f (t)dt)du f (t)dt 1du (x ) f (t)dt

    = =

    3.

    x u

    ( f ( t )dt )du

    u

    F(u ) f ( t )dt

    = , F()=0

    ,x u x x x x

    x( f (t)dt)du F(u)du (u) F(u)du [uF(u)] uf (u)du xF(x) uf (u)du

    = = = =

    4. ( f (t)dt)dx f (t)dt 1dx ( ) f (t)dt

    = = 5 . 352.

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    : 28

    .f(x)=x3+x-1

    ) f f-1

    ) =1

    1

    1

    f (x)dx

    . ) 2( ) 3 1 0 ,f x x x R = + > , f. . 1-1

    f-1 f R f-1 .

    ) =

    1

    1

    1

    f (x)dx

    , 1( ) ( )x f u f x u= = ,

    ( )dx f u du= x= - 1 x=1 -1 -1 -1([- 1,1]) [ (-1), (1)]f f f=

    3 2

    3

    1 ( ) 1 1 ( 1) 0 0

    1 ( ) 1 1 1

    f u u u u u u

    f u u u u

    = = + + = =

    = = + =

    1 1 4 22 1

    0

    0 0

    3u u 3 1 5I uf (u )du u (3u 1)du [ ]

    4 2 4 2 4= = + = + = + =

    15.

    f , ,

    (). .

    1. Cf , , x= x=

    :

    () = f ( x ) d x

    f [ , ] :

    1. f(x) 0 , [ ]x ,

    () =

    f ( x )d x

    Cfx= x=

    1 . f(x) 0 [ ]x ,

    () =

    - f ( x )d x = - f ( x )d x

    Cfx=

    x=

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    : 29

    1. f [ , ]

    f

    .

    ()=(1)+(2)+(3)=

    f(x)dx+ -f(x)dx+ f(x)dx

    , f [ ,]

    x=x=

    1

    2

    3

    Cf

    1 f ,

    :

    ) f(x)= x , xx, x=1 , x=4 ) f(x)=x , xx, x=2

    , x=

    3

    2

    ) f(x)=lnx , xx, x=

    1

    e, x=e

    2. Cf Cg x= x= .

    :

    () = f (x ) g ( x ) d x

    12 3

    Cg

    Cf

    x=x=

    f(x)-g(x) [,] . :

    () = (1)+(2)+(3) = [ ] [ ] [ ] + + f(x) g(x) dx g(x) f(x) dx f(x) g(x) dx

    , f(x)-g(x) [ ,]

    2 f , g

    : f(x)=x , g(x)=x , x=0 , x=2

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    3. f g

    x=x=

    CfCg

    f(x) = g(x) .

    x= x=

    () =

    f ( x ) - g ( x ) d x

    :

    f xx (f(x)=0)

    3 f , g :

    f(x) = x3

    , g(x) = 2x-x2

    4.

    ( , , ,

    )

    12

    3

    Cf

    CgCh

    C

    : ()=

    [ f (x ) -g (x )] d x+ [f (x ) -h (x ) ]d x + [ ( x ) - h ( x ) ] d x

    4 ,

    ) f(x)=3x

    , g(x)=x y=3 x=0

    ) Cf , f(x)= x , (1,1)

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    5. f-1

    , , x= x=

    () = 1f (x ) dx

    1f (x) 0

    - 1( ) ( )x f u f x u= = , ( )dx f u du=

    1( ) ( )f f = =

    1( ) ( )f f

    = = .

    1f (x)dx uf (u)du

    = = xf (x)dx

    1fC , x=

    Cf , yy y=

    ( 1fC , Cf y=x)

    :

    ()= 1

    0

    f (x)dx

    =0

    [-f(x)]dx

    , ( f()=)

    Cf-1

    Cfy=x

    f

    f

    -1

    ()

    ()

    =

    0

    5. f(x)= ex+x-1

    ) .

    ) f f-1

    ) C f-1

    , x=e

    1.>0 f [-,]. :

    ) f,

    0

    f (x)dx 2 f (x)dx

    = , ) f, f (x)dx 0

    =

    2. f(x)=2x+4

    x, x>0.

    ) f ,

    x= , x=+1 , >0 , ()=2+1+4ln(1+1

    )

    ) ()

    3. i) f [, ].: f (x)dx f ( x)dx

    = +

    ii)

    1 x

    x 1 x

    0

    dx , 0 >

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    4.) h , g [,] h(x)>g(x) , x[,] , h(x)dx g(x)dx

    > ) f f(x)-e

    -f(x)= x-1 , x f(0) = 0

    i) N f f ii) x

    7. f f: (0, )+ R , f(1)=2 ( ) ( ) 2 , 0xf x f x x x + = > . f.

    8. f f [1, 5] ( ) 0>f x [1,5]x .

    , (1,5) :3 4

    2 f (x)dx 3 f (x)dx 0

    + =

    9. f: R R x

    2

    3

    f (t)dt x 9, x , f(3).

    10. f, f: R R , :1

    ( ) ,x

    xf x x R

    e

    =

    fC

    + y=2.11. f f(x)=xe

    x+

    x

    t x

    0

    e f (t)dt , x R

    ) f ) xex

    = 1- ex

    ) Cf,

    12. f (0, )+ 4

    3

    f (t)dt 1= 4

    5

    f (t)dt 3= .

    x 2

    x 1F(x) f (t)dt , x 0

    +

    += > . ) F .

    ) (1,5) : f(+2) - f(+1) = - 4

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    4(2001) f, R, o:

    i) f(x) 0, xR

    ii) f(x) = dt(xt)ftx2-11

    0

    22

    , xR. g

    x-f(x)

    1g(x) 2= , xR.

    . (x)2xf-)x(f 2= 10. g . 4

    . f:

    x1

    1f(x)

    2+= . 4

    . limx + (x f(x) 2x). 7

    4(2001)

    f, (0,+) :

    . f (0,+). 3

    . f:

    0x,x

    xln1)x(f >+= 7

    . f. 6. f. 4.

    f, xx x=1, x=e. 5

    2(2002)

    =+

    x,e

    1e)x(f

    1x

    xIR.

    . f f1.

    10

    . f1 (x) = 0 .

    5

    .

    dx(x)12

    1

    2

    1f

    10

    0xdt

    x

    )t(t f

    x

    1)x(f

    x

    12

    >+=

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    4(2002)

    f, R. , :

    f(x)f(x) + (f (x ))2 = f(x)f(x) , xR. f(0) = 2f (0) = 1.. f. 12. g [0,1],

    1dt(t)f1

    x2x

    02

    )t(g=

    +

    [0,1]. 13

    3(2003) f(x) = x

    5+x

    3+x .

    . f f. 6

    . f(ex)f(1+x) xIR. 6

    . f (0,0) f f1 6. f1, x x=3 8

    3(2003 )

    x1xf(x) 2 += .. 0f(x)lim

    x=

    +

    . 5

    . f,

    x . 6. 0f(x)1x(x)f 2 =++ . 6. ( )12lndx

    1x

    1

    1

    0 2 +=+ 8

    4 (2004) f: IR IR f(1)=1. xR ,

    +=3x

    10)1(13)(z)( x

    zzdttfxg z=+iC, , IR*, :

    . g IR g. 5

    . Nz

    zz1

    += 8

    . Re(z2) =

    2

    1 6

    . A f(2)=>0, f(3)=>, x0 (2,3) f(x0)=0. 6

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    4(2004 )

    f [0, +) R ,

    += 21

    0

    2

    2xf(2xt)dt2

    xf(x) .

    . f (0, +) . 7 . f(x) = ex (x + 1). 7

    . f(x) [0, + ). 5

    . f(x)limx +

    f(x)limx

    . 6

    4

    (2005)

    f R ,

    2f (x) = ex f (x )

    x R f(0) = 0.

    . : f(x) =

    +2

    1ln

    xe

    . 6

    . :

    dtt)f(xlim

    x

    0

    0x

    . 6

    . : h(x) = x

    xdtf(t)t2005 g(x) =

    2007

    2007x

    .

    h(x) = g(x) x R . 7

    . x

    xdtf(t)t 2005 =

    20081

    (0 , 1). 6

    4

    (2005.)

    f: IR IR , 2005x

    xf(x)lim

    20x=

    .

    . :

    i. f(0)=0 4

    ii. f(0)=1. 4

    . , :( )

    ( )

    22

    22x 0

    x f(x)lim 3

    2x f(x)

    +=

    +. 7

    . f/ f(x)>f(x) x ,

    i. xf(x)>0 x0. 6

    ii. 0. g [0, 1] g(x) > 0 x [0, 1].:

    F(x) =

    x

    0

    f(t)g(t)dt

    , x [0, 1],

    G(x) =

    x

    0g(t)dt , x [0, 1].

    . F(x) > 0 x (0, 1].

    . : f(x)G(x) > F(x) x (0, 1].

    . :F ( x ) F ( 1 )

    G ( x ) G ( 1 ) x (0, 1].

    . : ( )( )

    2x x

    2

    0 0

    x0 5

    0

    f ( t )g ( t )d t t d t

    limg (t)d t x

    x+

    .

    3 (2007 .)

    ( )x

    f x e elnx= , x 0> ) ( )f x ( )1, + . 10

    ) ( )f x e x 0> .7

    ) ( ) ( ) ( )

    2 2

    2 2

    x 2 x 2 4

    x 1 x 3 2

    f t dt f t dt f t dt

    + +

    + +

    = +

    ( )0, + .8

    4

    (2008)

    f

    +=2

    0

    3 45310 dt)t(f)xx()x(f

    . : f(x)=20x3+6x45 .8. g IR .

    h)hx(g)x(glim)x(g

    h=

    0 .4

    . f () g ()

    452

    20

    +=++

    )x(f

    h

    )hx(g)x(g)hx(glimh

    g(0)=g(0)=1,

    i. g(x)=x5+x3+x+1 .10

    ii. g 11 .3

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    : 38

    4 (2008 .) f [0, +) f(x) > 0 x 0. :

    =x

    0f(t)dt)x(F , x [0, +),

    =x

    dt)t(tf

    )x(F)x(h

    0

    , x (0, + ).

    . =+1

    0

    1 1

    t )(Fdt)]t(F)t(f[e 6

    . h (0, +). 8

    . h(1)=2, :

    i . g(x) x [ , ] ,

    dx)x(gdx)x(h > . 2

    . R f, :

    ,1xe)x(f )x(f = x R f(0) = 0 .

    ) f f. 5

    ) ,f(x)xf(x)2

    x 0. 12

    ) f , x = 0, x = 1 x x,

    )1(f2

    1E

    4

    1

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    : 40

    (2011)

    f,g : , x:i) f(x)>0 g(x)>0

    ii)

    =+

    x 2t

    2x0

    1 f (x )d t

    g(x t )

    e

    e

    iii)

    =+

    x 2t2x

    0

    1 g(x ) d tf ( x t )

    ee

    1 f g f(x)=g(x) x.

    9

    2.: f(x) = ex, x. 4

    3.:

    0

    x

    l n f ( x )l i m

    1f

    x

    5

    4.

    ( )= 2x

    1

    F(x) f t dt xx yy x=1 7

    (2011)

    f : , 3 , :

    iv)

    +x 0

    =f (x)

    l i m 1 f(0)x

    v) f(0) < f(1) - f(0)

    vi) f(x) 0 x

    1. f x0=0. 3

    2. f. 5 g(x )= f(x ) - x, x:

    3. g :0x

    xli m

    xg(x) 6

    4. ( )2

    0

    f x dx>2 5

    5. g,

    xx x=0 x=1 ()= e -5

    2

    ( )1

    0f x dx (1, 2) , ( )

    0f t dt =2 6

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    : 41

    (2012) f:(0,+), x>0 :

    f(x) 0

    2 2x -x+1

    1

    x xf(t)dt

    e

    + x

    1nt tnx x= dt e f(t)f(t)

    1. f. 10

    f(x)=ex(nxx), x>0, :

    2.: ( ) +

    2

    x 0

    1lim f(x) f(x)

    f(x) 5

    3. lnx x-1, x>0,

    = x

    F(x) f(t)dt , x>0

    >0, ( 2). :

    F(x) + F(3x) > 2F(2x), x>0 ( 4). 64. >0. 0(,2) : F() + F(3) = 2F() 4

    (2012) f:A A=(0,+) :

    f() = (-,0] f (0,+ ),

    + + + +xf(x) f(t)

    11 12f(x) x e = e f (t) t dt 2x t

    = x

    1F(x) f(t)dt , x>0

    1.

    =+

    2

    2xf(x) n

    x 1, x>0 8

    2. F (x0,F(x0)), x0>0, . (x0,) >x0, F M(,F())

    : F()x(1)y+2012(1)=0 6

    3. >1,( )+( ( + + =

    35F() 1 )f() x 1) x 10

    x 1 x 3

    , x, (1,3) 5

    4. ( )

    2x x

    x 1

    tf dt t f t dt

    x, x>0 6

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    : 42

    -

    1. F (x) = xlnx - x f (x) =

    lnx.

    2. ,

    .

    3. F1, F2 f,

    c.

    4. H f (x) =1x

    1lnx2 +

    + [1, + ).

    5. f, g ,

    f (x) g (x) dx [f (x) g (x)] - f (x) g (x) dx

    =

    6. f : f (x) dx [f (x)]

    = .

    7.

    F1, F2, F3

    f,

    ,

    x0.y

    CF1 CF2

    x0

    y

    0x x

    8. F (x) = ex

    + c,

    x0.

    9. : f (x) dx g (x) dx = (f (x) g (x)) dx

    10. f (t) = t

    2 dx2x-xx , t

    22 dx2x-xx = xf (t).

    11. +

    3

    2

    dx1x

    4x-x=

    +

    3

    2

    dx1)(x

    dx4x)-(x

    .

    12. : =

    0()fdx(x)fx -

    0dx(x)f .

    13. : +

    dx(x)f 0dx(x)f

    = .

    14. : =

    0dx(x)f .

    15. :

    x

    dt(t)f = f (x).

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    : 43

    16. :

    (x)g

    dt(t)f = f (g (x)) g (x).

    17. :

    xdt(t)f = - f (x).

    18. :

    (x)h

    (x)g dt(t)f = f (h (x)) h (x) + f (g (x)) g (x).

    19.

    x

    1dt

    t

    1.

    y

    0x

    y

    x1

    y=lnx

    20. =8

    6

    4

    2 cdxcdx , c .

    21.

    +

    cdx(x)f , c 0.

    y

    0x

    y

    x

    Cf

    22. f R f (10) = 100, :

    100 = f (0) + 10

    0dx(x)f .

    23. : =1

    01-1xdx .

    24. e 2,7, =1

    0

    x1,7dxe .

    25. = 2

    0dx(x)f , :

    ) 2

    0d()f =

    ) 0

    2dt(t)f = -

    ) =2

    08-3Adz4)-(z)f(3

    26. f (x) > 0, >ln2

    10dx(x)f .

    27.

    0dx(x)f f (x) 0 x [, ].

    28. f (x) g (x) x [, ],

    dx(x)gdx(x)f .

    29. < ,

    dx(x)f

    dx(x)f .

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    : 44

    30. f [1, 3],

    +

    2

    30dxx)4-(1 .

    36. : =2x

    1 2lnxdtt

    1

    , x > 0.

    37. =

    dx(x)gdx(x)f , f (x) = g (x) x [, ].

    38.

    +=

    dx(x)fdx(x)fdx(x)f , < <

    .

    39.

    x dt(t)f = -

    x

    dt(t)f .

    40. : =ln

    ln

    x -dxe , , > 0.

    41.

    :

    = 1

    0

    23dx)x-(x .

    (

    f (x) = x2

    g (x) = x3).

    x

    y

    0

    y

    x 1

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    42.

    ,

    : = - 2

    2-dx(x)f .

    x

    y

    y

    0x

    -2 2

    43. f [0, 1] f (0) = f (1),

    1

    0dx(x)f = 0.

    44. 5

    0dx(x)f = 10, f [0, 5]

    3.

    45.

    =

    dx(x)f . x

    y

    0x

    y

    C f

    46. 1

    2.

    x

    y

    0

    y=1

    x

    y

    1

    1 2-1

    -1

    1

    x

    y

    0

    y=1+ x2

    x

    y

    1

    1 2-1

    -1

    y= x2

    2

    1. . . 2. . . 3. . . 4. 5. 6. 7. 8. , 9 2001 2012