Download - 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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: 3
.
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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:
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
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.
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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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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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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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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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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: 17
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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: 18
.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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: 19
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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: 21
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
-
7/27/2019 Olokliromata Gl
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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 . ..
-
7/27/2019 Olokliromata Gl
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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
-
7/27/2019 Olokliromata Gl
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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
= =
-
7/27/2019 Olokliromata Gl
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: 25
. 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
(-)
-
7/27/2019 Olokliromata Gl
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: 26
.
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
-
7/27/2019 Olokliromata Gl
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: 27
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.
-
7/27/2019 Olokliromata Gl
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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=
-
7/27/2019 Olokliromata Gl
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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
-
7/27/2019 Olokliromata Gl
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: 30
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)
-
7/27/2019 Olokliromata Gl
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: 31
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 >
-
7/27/2019 Olokliromata Gl
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: 32
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
-
7/27/2019 Olokliromata Gl
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: 33
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
>+=
-
7/27/2019 Olokliromata Gl
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: 34
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
-
7/27/2019 Olokliromata Gl
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: 35
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
-
7/27/2019 Olokliromata Gl
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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
-
7/27/2019 Olokliromata Gl
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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
-
7/27/2019 Olokliromata Gl
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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