ΣΥΝΑΡΤΗΣΕΙΣ -ΟΛΟΚΛΗΡΩΜΑΤΑ
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ΣΥΝΑΡΤΗΣΕΙΣ ΠΟΥ ΟΡΙΖΟΝΤΑΙ ΑΠΟ ΟΛΟΚΛΗΡΩΜΑΤΑ. ΣΧΕΤΙΚΑ ΘΕΜΑΤΑ ΑΝΤΩΝΗΣ ΚΥΡΙΑΚΟΠΟΥΛΟΣ Μαθηµατικός – Συγγραφέας – µέλος του ∆.Σ. της Ε.Μ.Ε. – Πρόεδρος της Συντακτικής Επιτροπής του περιοδι- κού «Ευκλείδης Β΄»
Transcript of ΣΥΝΑΡΤΗΣΕΙΣ -ΟΛΟΚΛΗΡΩΜΑΤΑ
. ..... - 2 . 1. : ) f(x)dx f .x- . ) f(x)dx, , , f x, , , - . : f(x)dx f(t)dt f()d ... = = = 2. xF(x) f(t)dt = f . , : xF(x) f(t)dt =. , x t. To x - F. t (), , x, . : .Ff . , F : F (x) f (x), x = . , f (- F). 3 - , : .f G f , , : f(x)dx G() G() = . , (,,,,).-, : 2xt02F(x) e dt=, , , . , -. . 3. . -: g(x)h(x)F(x) f (t)dt =, f, g h . : xF(x) f (t)dt =g(x)F(x) f (t)dt =. F. Bf .,g- h. xF, ,x (B ) f - h(x) g(x). 4 F. f gh- :g(x) h(x) , x I . : ,F.- F : . . , x I : g(x) h(x) g(x)h(x) F(x) f (t)dt f (t)dt f (t)dt f (t)dt = + = + . : x(x) f (t)dt, x = , : (x) f (x), x = . , : ( ) ( ) F(x) h(x) g(x) = + . , : ( ) ( ) F (x) h(x) h (x) g(x) g (x) = + ( ) ( ) f h(x) h (x) f g(x) g (x) = + . . G f , : G (t) f (t) = ,t . , : ( ) ( )g(x)h(x)F(x) f (t)dt G g(x) G h(x) = = . , : ( ) ( ) F (x) G g(x) g (x) G h(x) h (x) = ( ) ( ) f g(x) g (x) f h(x) h (x) = . 4. 1. : x20tF(x) dtt 1=. 5. . : 2tf (t)t 1= :A ( , 1) ( 1,1) (1, ) = + . x F , , f -0 x. : 1 < x < 1. F :FA ( 1,1) = . . F FA: 2xF (x) f (x)x 1 = =. . : x22tF(x) dtt 1= FA ( , 1) = : x23tF(x) dtt 1= FA (1, ) = + . , : 2xF (x)x 1 =. 2. : x21F(x) 9 t tdt. = . . : 2f (t) 9 t t = - : = [3, 3]. : g(x) x = B [0, ) = + . x F , ,x f -( ) )( )(1 1 + 8 80 61g(x) x = . : x 0 x 0x 00 x 9x 93 x 3 x 3 . , F FA [0, 9] = . .g(x) x = (0, ) + . , F I (0, ) [0,9] (0, 9] = + = . G f , : 2G (t) f (t) 9 t t, t = = . , : x1F(x) f (t)dt G( x) G(1). = = , : IF (x) G ( x)( x) 9 x x2 x = = =9 x x2 x . () 0. : x 0F(x) F(0)F (0) limx 0++ = ( 00) =( )x 0F(x) F(0)lim(x)+= x 09 x x 3lim2 2 x+ = = . : 9 x x, 0 x 92 xF (x)3, x 02 < = = 3. : x 2t1x 1F(x) e ln t dt=. . . tf (t) e n t = : A ( , 0) (0, ) = +. ( ) )(+ 8 80 7g(x) x 2 = B= 1h(x)x 1= ( ,1) (1, ) = + . : ( ,1) (1, ) = + . x F , ,x ( ) f h(x) g(x). , : x 1 x 1x 1 x 1x 2 0 x 2 0x 2 x 2 (x 1 x 2)1 10 0x 1 x 1x 1 x 1 | | | >| < | < > < > | | ||> \ . | \ .. , F : FA ( ,1) (2, ) = + . .i) 1I ( ,1) = . g(x) x 2 = 1h(x)x 1= 1I 1 ( , 0) = , 1x I . , F - 1I . 1 ( , 0) = . 1x I :1 x 2 x 2 x 2x 11 1 x 1 x 1F(x) f (t)dt f (t)dt f (t)dt f (t)dt f (t)dt. = = + = + : x(x) f (t)dt, x ( , 0) = , : (x) f (x), x ( , 0) = . , 1I : 1F(x) (x 2).x 1 = + , 1I : 1 1F (x) (x 2)(x 2)x 1 x 1 = + 21 1f f (x 2)(x 1) x 1 = + 1x 2x 121 1e ln e ln x 2(x 1) x 1= + . ii) 2I (2, ). = + . 85. 1. :x t21 1F(x) 1 d dt = . ) F. ) F . . ) : t21f (t) 1 d = . : 2() 1 = -:( , 1] [1, ) + , f : [1, ) = + . f - 2f (t) t 1 = . , f , , : x1F(x) f (t)dt = [1, ) = + . ) x A , : x x21 1F (x) f (x) 1 d ()d = = = 2F (x) f (x) x 1 = = . F[1, +)(). x (1, ) + . [1, x] , 2() 1 0 = , = = 1. , :x1()d 0 > x (1, ) + . :F (x) 0 > ,x (1, ) + F- [1, ) = + . F[1, ) + x (1, ) + : 2F (x) x 1 0 = > ., F [1, ) = + . 9 2 : x2x1F(x) dt.1 t= ) F. )F , :2x 0 > . .) : 21f (t)1 t= (1, 1). x F , : 1 x 1 1 x 1x ,1 x 1 1 x 1 2 < < < < | | | < < < 0 , 2x 0 > ., F . 3 f, . : 120F(x) x t f (tx)dt = . . , x : 10F(x) (xt)f (xt) xdt. = 10: = xt ,d = xdt .t = 0 1 0 = t 1 = , 2 x = . , : x0F(x) f ()d = :F (x) xf (x) = . 4 f, . , , . : F(x) f (x t)dt. = .: x t = ,d dt = .t = :1 x = t = , : 2 x = . , : x x x x F(x) f()d f()d = = . G f , : G () f (), = . , : [ ]x x F(x) G() G(x ) G(x ).= = : F (x) G (x )(x ) G(x )(x ) = f (x ) f (x ) = . 5 : ) 21(tx)F(x) dt,x (0, )t= +. ) 2xx(xt)F(x) dt,x (0, )t= +. ) x>0. = xt, o d = xdt. t = 1 1 x = t 2 = 2 2x = . , (0, +) : 2x 2x 1 2xx x x 1 d F(x) d d dx x= = = + = 2x x1 1 d d = . 11, (0, ) + , : 2x x 2x xF (x) (2x)2x x x = = . ) x>0 = xt : 32xxF(x) d=, 3 23x 2xF (x)x = . 6 f : , x : x x20 13 f (t)dt f (t)dt 2x 2x 1. = + + (1) f. (1) , x : 3f (x) f ( x)( x) 4x 2 3f (x) f ( x) 4x 2 = + + = + (2). (2) x x x : 3f ( x) f (x) 4x 2 + = + .(3) (2) (3) ( f( x)), : 1f (x) 2x , x2= + . (4) . (4) (1) f . . (1) : 232x 2x2+ + , - , (4). (1)( ) f. 7 f :[0, ) + , :f (1) 1 =( ) ( )101f tx dt f x , x (0, )4= +.(1) .f :[0, ) + -. x > 0 : = tx, d = xdt. 12t = 0 1 = 0 t = 1 : 2 = x. (1) - x > 0: ( ) ( ) ( ) ( )1 x0 0x xxf tx dt f x f d f x4 4= = . f ( ) 0, + () :( ) ( ) ( ) ( ) ( )( ) ( )3 26x f x 3x f x4f x f x xf x xf x 3f x 0 0x = + = =( )3f x0x = ( )( ) ( )33f xc c f x c xx= = . f(1) = 1, c = 1 : 3f (x) x , x 0 = > . 3x 0 x 0f (0) lim f (x) lim x 0,+ + = = =: 3f (x) x , x [0, ) = + (2). .,(2) . 8 f : (0, ) + , x 0 >: ( )2x 2xx 1 2 tf x f dt2 t x = + .(1) . f : (0, ) + -.x>0: tx= 1d dtx= .t = x 1 1 = 2t x = 2 = x. , (1) : ( ) ( )x x 2 21 1x 1 1 x 1 f ()f x 2 f () xd f x 2 d2 x 2 = + = + . f ( ) 0, + :2 2f (x)f (x) x 2 xf (x) x 2f (x) xf (x) 2f (x) xx = + = + =3x f (x) 2xf (x) x2 = 24 2x f (x) 2xf (x) 1 f (x)(ln x)x x x = = ( )2 22f (x)ln x c c f (x) x ln x cxx= + = + . 13 (1) :f (1) 0 = , : c = 0 :2f (x) x ln x, x (0, ) = + .(2) .,(2) . 9 f : (0, ) + , x 0 >: ( )12x1 xf x 2lnx f dtt t = .(1) . f : (0, ) + -.x>0: xt= , 21d xdtt= . t = x 1 1 = t 1 = 2 = x. , (1) :( )x x1 11 1f x 2ln x f ()d f (x) 2ln x f ()dx x= + = + (2) x1xf (x) 2x ln x f ()d = +.(3) (2)f(0, ) + , (3) () : ( )2 2ln xf x xf (x) 2ln x 2 f (x) f (x)x x + = + + = +2 2ln x 1f (x) dx 2 dx 2 (ln x) ln x dxx x x = + = + = 22ln x ln x c (c ) = + + . (1) : f(1) = 0, c = 0 :( )2f x 2ln x ln x, x (0, ) = + + .(4) .,(4) . 10 >0.-f : , x : 14( )x2 t0f x 3x e f (x t)dt= + . (1) . f : . x .: x t = ,d dt = .t = 0 1 x = xt= 2 0 = . , (1) :( )0 x2 x 2 x x 0f x 3x e f ()d f (x) 3x e e f() d = = + (2) xx 2 x 0e f (x) 3x e e f()d = +.(3) (2) f . , (3) (), :x x x 2 x x 2e f (x) e f (x) 6xe 3x e e f (x) f (x) 6x 3x + = + + = +2f (x) (6x 3x )dx = +2 3f (x) 3x x c = + + . (1) x = 0 f(0) = 0, c = 0 :( )2 3f x 3x x = + .(4) .,(4) . 11 > 0 f [0,]f (0) 0 = f (x) 0 > ,x [0, ] . x [0, ] :f ( x) x10 0f (t)dt f (t)dt xf (x)+ = .(1) : f , f1 f(). . f [0, ], - 1-1 - f1 ( ) [ ] [ ] f [0, ] f (0), f () 0, f() = = . : 15f (x) x10 0F(x) f (t)dt f (t)dt xf (x).= + F [0, ] : 1F (x) f (x) f (f (x)) f (x) f (x) xf (x) = + f (x) xf (x) f (x) xf (x) 0 = + = . F(x) c = , x [0, ] (c ) F(0) = 0, c=0.F(x)=0,x [0, ] .,(1) x [0,] . 12 f : x0F(x) f (x) f (t)dt = . : f (x) 0, x = . :2x0g(x) f (t)dt = . .F.g : x x x0 0 0g (x) 2 f (t)dt f (t)dt 2f (x) f (t)dt 2F(x). = = = g . g (0) 2F(0) 0, x = = , : x 0 g (x) g (0) 0 g (x) 0 = x 0 g (x) g (0) 0 g (x) 0 = . g0-,g(0) 0 = ., x :g(x) 0 g(x) 0 , : 2x x20 0g(x) 0 f (t)dt 0 f (t)dt 0 = = = x0f (t)dt 0 f (x) 0 = = . 16 1.. : . 2.. : 3.Louis Brand: . 4.G. Thomas R. Finney: . 5.M. Spivak: . 6.. : . 7. ...: . 30/3/2006 -.-. , . - , - . . - . .. - .
