I. , , , , , , . , : , f a b R 1 ], x - = nx b,x b (.1.)y

( )1f ( )2f ( )nf 0=x a1x2x 1 nx= nx b x

(.1.) [, a b] 0 1 1, , , ,n nx x x xK, 10 1 2 1 n nx a x x x x b < < < < K n- 1 1 2 1, , , , ,na x x x x b 111 ] ] ]K 1 1 2 1 2 1, , , .n n nx a x x x x x x x K 1,i ix x 1 ]1,2, 3, ,i n K i, ( )i if x ( )i=1=ni iSn f x .( )i=1=ni iSn f x f [, a b] i. , ( )if ix , 1,2, 3, ,i n K. , . S Sn , i>0 >0, max ix . 0 1 2 1 n nx a x x x x b < < < < K , a b 1 ] 0 1 1 2 1, , , , ,n nx x x x x x 111 ] ] ]K 1 1 0 2 1 2 1, , , .n n nx x x x x x x x x K 1,i ix x 1 ], =1,2, , i n K, i ( )i if x . ( )i=1 ni if x ( ) f , a b 1 ]. i. S 0imax x , >0 >0 S < imax x < i. ( ) , , f , a b 1 ] ( ) ba f x dx .,( )0 =1( )=bni imax x iaif x dx f xlim a , b . , a b 1 ] . f , a b 1 ] . f , a b 1 ] , a b 1 ].4 f , a b 1 ], 1,i ix x 1 ] i ( )i if x , . , . x x , a b 1 ],a b < , i ( )=1=0ni iif x , i ( )=1=ni iif x b a , . .1. : , f a b R 1 ] , a b 1 ].2. : , f a b R 1 ] [ ], a b.3. : , f a b R 1 ] , a b 1 ].: 120x dx:51, ( )=0,f x'( )2= fx x 0,1 1 ], 120x dx. 0,1 1 ] n 0 1 2 1=0, = , =2, = ,nx x x x x x n x K 1= xn. i .( ) ( ) ( )( )2 2 21 22 2 23 2 2 2== 2 = 1 2nx x x x x xx x x x n x xx n + + + + + + + + +KKK

. 1201=3x dxIII. 1.6001 1 1= 1 261= 261=3nnlim limn n _ _

, ,+ +( ) ( )331 2 11=61 1 1 = 1 26n n nnn n _ _

, ,+ ++ +( ) ( )2 2 21 2 11 2 =6n n nn+ ++ + + K ( )y fx , a b 1 ] ( )y k fx , k , ( ) ( ) .b ba ak f x dx k f x dx 2. ( )y fx ( )y g x , a b 1 ], ( ) ( )y fx g x + ( ) ( ) ( ) ( ) b b ba a afx g x dx fx dx g x dx 1 ]+ + . 3. ( )y fx , a b 1 ], , a c1 ] , c b 1 ], a c b < < , ( ) ( ) ( )=a c bb a cfx dx fx dx fx dx + . 4. ( )y fx ( )y g x , a b 1 ] ( )( ) ( ), x a b fx g x 1 ] ( ) ( ) b ba afx dx g x dx 5. ( )y fx , a b 1 ] ( )( ) ( )( ), 0 0x a b fx fx 1 ] 7( )( )02=0aaafx dxfx dx'( ) ( )0 0b ba afx dx fx dx _ , ( )y fx ( )=0 ba fx dx ( )=0 fx [ ], a b. ( ) ( )0 0b ba afx dx fx dx _ , 6. ( )y fx , a b 1 ]

, f , f 7. = b a, f=0, ( )=0 ba fx dx 8. a b > f , a b 1 ], ( ) ( ) b a defa bfx dx fx dx IV. -

8 , , - , .: : , f a b R 1 ] , a b 1 ] F f. :( ) ( )( )ba fx dx F a F b (1): ( ) ( ) xax fx dx . (2) ( ) ( ),x fx . , f, ( ) ( ),x F x C C R + .(3) (2) x a , ( )0 a . , (3) ( ) ( )a F a C +, ( )C F a . x b , (3) ( ) ( ) ( )( )b F b C F b F a + . . ( )( ) ( ) b ba ab f tdt fx dx ( ) ( )( )ba fx dx F a F b - 9( ) ( )( )ba fx dx F a F b x b , . x, ( )( ) ( ), xax a b f t dt F a 1 ] + , ( ) ( ) ( ) xaF x f t dt F a + . ( )F x. (1643 -1727) (1646 1716) , 17. . . - , ( ) ( )( )ba fx dx F a F b , - . . , f , a b 1 ], , , F f , a b 1 ]. , . , . :101. . - !2.2200cos sin sin sin0 12x dx x I3.2 33 22112 1 72 2 3 3xx dx I4.( )sincos cos sinbbaadx x b a + I1112 22 2 4241 1233 -321133 -3 3 -321

3 3 32 22 2 211 2 3 3 3 3 3x x dx x x x dxxx x x _ _ , , 1 1 1 1 1 1 ] 1 1 1 1 1 1 ]+ + + + + 323138 1 2 2 1 1 2=3 3 8 2 3 3 38 1 4 2 2 =3 24 3 310 1 4 2 =3 24 3 1 1 1 1 1 1 ] 1 1 ] + 80-1-32 2=2479-32 2 =24V. M - . , , - .1. : ( )y fx , a b 1 ]. :1o ( )x g t , 1 ] ( )g t a ( )g t b , ( )g a ( )g b

2o( )( ), , , t g t a b 11 ] ] ( ) ( ) ( ) ( )' 1b ba afx dx f g t g t dx1 ] ( )1 .: ( )y F x ( )y fx , a b 1 ], . ( ) ( )'Fx fx . - ( )2 ( ) ( )( )ba fx dx F a F b , 1 ]