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0 $ f (x) : a b ], ! ) c1 c2 c3 : a b ] ZcZcZcf (x) dx = f (x) dx + f (x) dx:(7:10)3c123c1c2570 c1 < c2 < c3, (7.10) V.( , , c1 < c3 < c2 ( ) ). > V .Zc2c1Zc3Zc2Zc3Zc2Zc2c1c3c1c1c3f (x) dx = f (x) dx + f (x) dx () f (x) dx = f (x) dx ; f (x) dx:, , IIZc2c3Zc3f (x) dx = ; f (x) dxc2 # (7.10).#4$ # VI. f (x) b : a b ]Z f (x) 0 ( x 2 : a b ], f (x) dx 0.a.
> f (x) 0, (7.5) $ f (x) : a b ] :nXf (k ) T xk 0:k=1' , Zbaf (x) dx = lim0!nXk=1f (k ) T xk lim0 0 = 0:!9 VI .#4$ # VII. f (x) g (x) : a b ] f (x) g (x) ( x 2 : a b ], ZbaZbf (x) dx g (x) dx:aP $ ' (x) = g (x) ; f (x). > ' (x) ' (x) 0 : a b ], , VI IV .Zba' (x) dx 0 ()Zb ag (x) ; f (x) dx 0 ()58ZbaZbZbZbaaag (x) dx ; f (x) dx 0 () g (x) dx f (x) dx:9 VII ..
0 $ f (x) : a b ], $ j f (x) j ) < Zb f (x) dx aZb j f (x) j dx:a2 ;jf (x)j f (x) jf (x)j ! x 2 :a b] * .#4$ # VIII. f (x) : a b ] m f (x) M ( x 2 : a b ], Zbm (b ; a) f (x) dx M (b ; a):a(7:11)> , m f (x) M 8 x 2 : a b ], VII, .ZbaZbZbaam dx f (x) dx M dx:9 * IV ) III :ZbZbZbZbaaaam dx f (x) dx M dx () m (b ; a) f (x) dx M (b ; a):9 , $ (7.11) .#4$ # IX. f (x) : a b ] m f (x) M ( x 2 : a b ], " , &" m M , Zbaf (x) dx = (b ; a):59(7:12)> m f (x) M , # (7.11), ,.Zbf (x) dxm a b ; a M:(7:13)Zb1( = b ; a f (x) dx, $ (7.12) a, , (7.13), m M .
9 IX .#4$ # X ( $ 89). f (x) : a b ], * , Zbaf (x) dx = f ( ) (b ; a) 2 : a b ] :(7:14)> $ f (x) : a b ], < (. 7.2). E, # f (x) : a b ] # m ! # M , .. *x1 x2 2 : a b ] , f (x1 ) = m, f (x2 ) = M m f (x) M 8 x 2 : a b ]:(<, IX, * , .Zbaf (x) dx = (b ; a)(m M ):(7:15)( ) ) $ f (x) :a b], ) $ * 2 :a b], f (x) , .. f ( ) = . ( = f ( ) (7.15), ! # (7.14). 9IX .602 .( : a b ] $ f (x) < , .. f (x) 0, > $ (7.14) S { * , $$ y = f (x) x = a, x = b, y = 0 (. 7.3), $ (7.14) * S ABCD ABCD b ; a f ( ).
9 , $(7.14) , S = SABCD .;. 7.3> !, $ f (x) " b ; a , & f (x) 2 :a b], " " . !" #$"1. % "! ?2. * ! : ?3. * !" F = F (x) : ! Ox a b (! ! !" ! ! )?4. & !" f (x) Aa b] (c. 7.1).C! & !:) A a b ] !? ) " k ?5. * "! & A a b ]? % "!" & f (x) A a b ]? (!. 7.1).ZbC! f (x) dx:a) A a b ] !? ) " k ?616. ) ! &! (c.
7.1).D! (f (x) = 12=xx =0 <0 x 1& A 0 1 ]?7. D! & ! ! " ! &!? (c. 7.1).8. ! " ! &! . (c. 7.2).9. . + ! & ! !"! & &? (& !+ +!+ !! III & &.10. ( ! & & (c. !! IV).11. 5! f (x) g (x) &" A a b ]. G )! f (x) ; g (x) & 4 ? (c. !! IV).12. ! f (x)+ g (x) ) &") A a b ] f (x) g (x) " & 4 ?H. =!!(: f (x) { ()x ; !g(x) = 01 !! x ; !:Z313. ." ! f (x) dx, !0(0 x 1f (x) = 12 1 < x 3(c.
!! V, IV, III).14. ! (c. !! X).15. 5 & ! !"! " ! .16. o !! X "") ?H. =!! + f (x) & ! 13 A 0 3 ].6. < !. 7. < !.11. =! &") & .Z3Z1Z3Z3Z114. f (x)dx = f (x)dx + f (x)dx = dx + 2dx = 1 + 2 2 = 5:0010621 8' .)%* (:-<.=% % '&%.$ Zb( , f (x) dx,a* (. M5) , * , ! ! . (<, x ! ! (,t), , ..ZbaZbf (x) dx = f (t) dt:a( $ f (x) :a b], x { : a b ].
> f (x) , Zx : a x ], f (t) dt $ a x. !< $ K (x), ..ZxK (x) = f (t) dt:(8:1)9 * 8.1. a ZxK (x) = f (t) dtaf (x) : a b ], & -: a b ], 01ZxdK (x) = dx @ f (t) dtA = f (x)0a8 x 2 : a b ]:(8:2)3 (8:2) , , ( , & ( .63( T K = lim K (x + T x) ; K (x) :K (x) = limx 0 T x x 0Tx.(8:3)0!!' ! (8.1), II V, T K = K (x + T x) ; K (x) ==x+Z xZxaaf (t) dt ; f (t) dt =x+Z xZaf (t) dt + f (t) dt =ax+Z xxxf (t) dt:( * , )* ) x x + T x , TK =x+Z xxf (t) dt = f ( ) (x + T x ; x) = f ( ) T x:(8:4)> ) ) x x + T x, ! x T x ! 0.(< T x ! 0 : f ( ) ! f (x) ( $ f (x)).
Q <, (8.3), (8.4) :T K = lim f ( )T x = lim f ( ) = lim f ( ) = f (x)K (x) = limx 0 T x x 0 T xx 0 x.. $ (8.2) ., $ (8.2) K (a) K (b) :K = lim K(a + 4x) ; K(a) ( )K+ (a) = xlim0+0 44x x 0+04xK = lim K(b + 4x) ; K(b) ( ):K (b) = xlim0 0 44x x 0 04x0!!!0!004!4!0;44! ;$ # 8.1.! ; f (x) & x 01 d B@ Zb f (t) dtCA = ; f (x)dx x64: a b ],Zbxf (t) dt8 x 2 : a b ]:: a b ], -' V, ) -., ZbaZxZbZbaxxf (t) dt = f (t) dt + f (t) dt ()ZbZxf (t) dt = A ; f (t) dta A = f (t) dt. > A { , A = 0,a< * $ (8.2) ,0 b101Zd B@ f (t) dtCA = A ; d @ Zx f (t) dtA = ;f (x):dx xdx a. 0 f (x) { , ' (x) (x) { $$ $ , 010 (x)1bZZd B@ f (t) dtCA = f : (x) ] (x)- d BB f (t) dtCC = ;f : ' (x) ] ' (x):Adx adx @' (x)00002 ) $ , ) 8.1 8.1. 8.1. d) dx0x1ZB@ e t2 dtCA = e x2 ;0d!) dx;0110xZBBt2 dtCx px =Ce=eA@p;;00epx 2 x;Z0 sin t CdB) dx @ t dtA = ; sinx x xd) dx0 014 44Zsinxsinx4sinxB@ sin t dtC43A = ; 4 x = ; 4 4x = ;:x4t$ # 8.2.f (x)0xxx: a b ],4&, * &.
5 (ZxK (x) = f (t) dt.a. >! ! $ (8.2).65 3 / --42) & . 8.2. f (x) : a b ]F (x) { - f (x) * , 6&-4Zbaf (x) dx = F (b) ; F (a):(8:5)3 (8:5) : f (x) :a b] ( a b .2 !F (b) ; F (a) = F (x) ba ! $ (8.5) bf (x) dx = F (x)aaZb:', ! ! $ f (x) : a b ] . ( F (x) { ! f (x) : a b ] f (x) : a b ], < 8.2 $ ZxK (x) = f (t) dt ) ! f (x) : a b ]. 9a ,.K (x) = F (x) + C ()Zxaf (t) dt = F (x) + C 8 x 2 : a b ]: (8:6)( (8.6) x = a I , Zaaf (t) dt = F (a) + C () 0 = F (a) + C () C = ;F (a):( C = ;F (a) (8.6), 66Zxaf (t) dt = F (x) ; F (a) 8 x 2 : a b ] x = b $Zbaf (t) dt = F (b) ; F (a)* $ (8.5). > 8.2 .
8.2. 4 644x623) x dx = 4 = 4 ; 4 = 324 ; 4 = 32022=!=2Z2!) sin x dx = ; cos x =3= ; cos 2 ; ; cos 3 = cos 3 = 21 =35Z5 1) x dx = ln j x j e = ln 5;ln e = ln 5;1:eZ6 $ 8.3.$ f (x) : a b ], x = ' (t) & & ' (t) : ] , ) x = ' (t), t 2 : ], : a b ]0!) ' () = a ' ( ) = b.' &" :ZbaZ f (x) dx = f ' (t) ' (t) dt:0(8:7)( F (x) { -! ! $ f (x) : a b ].
> $ 1 -=! (. 8.2) .Zbaf (x) dx = F (b) ; F (a):67(8:8)3 F ' (t) ! $ $ (8.7), d F ' (t) = dF (x) dx = f (x) ' (t) = f ' (t) ' (t):dtdx dt9 $ (8.7)$ 1 -=! ( <, ' () = a,' ( ) = b), 0Z 0f ' (t) ' (t) dt = F ' ( ) ; F ' () = F (b) ; F (a):0(8:9)' (8.8) (8.9) $ (8.7). > 8.3 . 8.3. Z13px x + 3 dx:;()p(8:10)t = x + 3:> x = t2 ; 3 dx = 2 t dt. 1 $ (8.10): x = ;3 t = 0, x = 1 t = 2. 9 ,Z13px x + 3 dx =Z2 t ; 3 t 2 t dt = 22Z2 t4 ; 3 t2 dt =0010105!235t24t3AA@@:= 2 5 ; 3 3 = 2 5 ; 2 = 16 5 ; 1 = ; 1650; 8.4. Z3=2p0pdx1 + 1 ; x268:()x = sin t:(8:11)ppp> dx = cos t dt, 1 ; x2 = 1 ; sin2 t = cos2 t = cos t,t = arcsin x:(8:12)1 $(8.12):p x = p0 t = arcsin 0 = 0, x = 3=2 t = arcsin( 3=2) = =3.
9 ,==Z 3 (cos t + 1) ; 1Z3=2Z 3 cos t dtdxp 2 = 1 + cos t =dt =1+cost1+1;x000p=0p1=Z 30! =3 1tA dt = t ; tg = ; tg = ; 3 :@1 ;2 cos2(t=2)2 0363 8.4.$ u (x) v (x) & : a b ]. ' bu (x) v (x) dx = u (x) v (x) aa b u (x) v (x) = u (b) v (b) ; u (a) v (a).aZb0Zb; v (x) u (x) dx0a(8:13)3 (8.13) $ , , v (x)dx = dv, u (x)dx = du, ! 0Zba bu d v = u v a0Zb; v d u:(8:14)a, $ u (x) v (x) ! $ (u (x) v (x) ) = u (x) v (x)+ u(x) v (x), < $ : a b ]. (<, $ 1 -=! , .0 bu (x) v (x) + u(x) v (x) dx = u (x) v (x) a :Zba000690> : a b ] $ u (x) v (x), u(x) v (x) ), < $ * 0ZbaZbu (x) v (x) dx +0a0 bu(x) v (x) dx = u (x) v (x) a 0 ) (8.13). > 8.4 .
8.5. Z0(2 x + 1) sin 3 x dx:() u = 2x+1, dv = sin 3xdx. > du = 2dx, v = ;(cos 3x)=3.' $ (8.14), Z !Zcos3x(2x+1)cos3x; 3 2 dx =(2 x + 1) sin 3 x dx = ; ;3000= ; (2 + 1)3 cos 3 = 2 3+ 1 + 13 +Zcos02+ 3 + 3 cos 3 x dx =02 sin 3 x = 2 ( + 1) :3 3 03 8.6. Ze12x2 ln x dx:() u = ln x, d v = x dx. >d u = (ln x ) dx = x1 dx0' $ (8.14), Ze1v = x3 :3 e3Ze x3 dx e3 ln e ln 1 1 Ze 2x2x ln x dx = 3 ln x ; 3 x = 3 ; 3 ; 3 x dx =1117033ex=3 ; 9 e3e = 310 3@e1312eA; 9 ; 9 = 9+ 1 : !" #$"1. & & ) (c. 8.1).2. * " !! & & ?(c. !! 8.1).3. < "0 Zx10 Z01pdtd4) dx @ 1 + t3 dtA 1) dx @ 2 ; sin t dtA :x14.
< Zx( arctg t )2 dtp:x2 + 1H. .!! >.5. 5! f (x) { ", ' (x) (x) { " . (, 0 (x)1Zd B f (t) dtC = f A (x) ] (x) ; f A ' (x) ] ' (x):Adx @limx!+1000' (x)6. < +0 x31ZdBCA :t2edt@dxp3x7. (, + " A a b ] f (x) 4 + (c. !! 8.2).8. " ! & & ( <+>) (c. 8.2).9. ." !Z1 dxZZ=3 dx) ex :) (2 x + cos x) dx1)cos2 x 110=2;7110. " & (c. 8.3).11. ." !Zln 2pex ; 1 dx:p0. t = ex ; 1.Z3 p312.
& x2 1 ; x3 dx ! x = sin t?013. (, " A ;a a ] f (x) !"!8:ZaZa) f (x) dx = 2 f (x) dx, ! f (x) 1a0;)Zaf (x) dx = 0, ! f (x) .a;14. & ! & & (c. 8.4).15. ." !Z3Z2Z2 2x) x arctg x dx1) x e dx1) ex sin x dx:p010p3. a) 4 1 + x31 ) sin xx ; 2 :4. 0 252.36. 3x2ex6 ; p31 2 e x2 :3 xp9. ) 0 752 ; 11 ) 31 ) e ; e 1:11.
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