1. Интегралы ФНП Диф_ур (853736), страница 7
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11.1).17. !+ !):! & " !):!? (!. 11.5).18. ! " !+ !)! !!&& ! ! " & (!. 11.6).10619. 5, !!" &"++Z sin 3 xZ cos x + cos 2 xpdx1dx210xx+1xlnx1211+! !+ !):!.20. 5 !!& & ! ! " &, !):&! ! (!. 11.8).p4. 0.6. =!)! 11 !)! > 1:2. ln 4 3:15. a) c)!1 ) !)!1 ) !)!.107 12< '/ )*( $ f (x) ) : a b ), : a b ; " ] ! " > 0, ) : a b ), .. x = b f (x) (. 12.1, ).
6 f (x)( x = b) : a b ) bZ " f (x) dx " ! +0.;a! !:ZbaZbaf (x) dx. ',f (x) dx = "lim+0!bZ ";af (x) dx:(12:1)0 * (12.1),Zb ! f (x) dx (". Ea (12.1) * ( !Zb), ! f (x) dx (a".;)!). 12.1108)4 f (x)( x = a) ( a b ] :Zbaf (x) dx = "lim+0!Zba+"f (x) dx(, $ f (x) ) ( a b ], : a + " b ] ! " > 0, ) ( a b ] - . 12.1, !).0 ) $ ) ( $ f (x) ( ! ;1 +1, c !), ) # ), ) ! $ f (x) (. 12.1, ). < (", ( + , (", ( +( (.> !, $ , $ !) . 12.1, , :+1ZZbaaZcZd+1bcdf (x) dx = f (x) dx + f (x) dx + f (x) dx +Zf (x) dx:Z1 12.1.
( , ! ln x dx 0, < .2 $ ln x ! x = 0, <Z10Z1ln x dx = "lim+0 ln x dx:!"(12:2)2 (12.2) $ . 2< ) u = ln x, dv = dx- du = dx, v = x. 9x ,Z10 1 Z1 ln x dx = "lim+0 x ln x " ; dx = "lim+0 (;" ln " ; 1 + ") =!"!109" ; 1 = ; lim (ln ") ; 1 = ; lim 1=" ; 1 = lim " ; 1 = ;1= ; "lim+0 ln" +0 (1=")" +0 ;1="2" +01="( "lim+0 " ln " =).0!0!!!!Z1', , ln x dx = ;1.0Z1 12.2.
( , ! 1 dx;x0.2 $ 1 ;1 x ! x = 1, <Z10dx = lim 1Z " dx = ; lim ln (1 ; x) 1 " = ; lim ln " = +1:0" +0" +01 ; x " +0 0 1 ; x;;!!!9 , .) * * ) * . 12.1. $ ba < b > 0 { .Zdx ( & 0 < < 16 a (x ; a) ( & 1.1. 2 $ (x ; a) ! x = a, <Zbadx = lim Zb dx =(x ; a) " +0 a+" (x ; a)!8+1 b>(x;a)>lim 6= 1>>< " +0 ; + 1 a+"==>>b>>limln(x;a): a+" = 1" +00111(b;a)"A 6= 1lim @;;!!8>><=>">:;!+0;1;1;=lim : ln (b ; a) ; ln " ] = 1" +0!11081>< (b ; a)= > 1;:+1; < 1 1 ! ). 12.1.
( a < b > 0 { .Zb dx2 , ! (b ; x) !a0 < < 1 ! 1.2 ( f (x) { ) : a b )$ , x = b. $"& S , $ $ y = f (x) 0 x = a, x = b y = 0, * ABCD (. . 12.1, ) " ! +0 ( *).', :S = "lim+0 SABCD :(12:3)!> bZ "SABCD = f (x) dx, (12.3) , ;aS = "lim+0!bZ ";aZbf (x) dx = f (x) dx:a> !, (" - " . $ , * x = b ( ).111#4$ # 12.1.( $ f (x) ): a b) x = b ! f (x). >, a < a1 < b,ZbZb ! f (x) dx f (x) dx ! aa! .1#4$# 12.2 (4$!).bbZZ) f (x) dx = f (x) dx { aaZbZbZb!) f1(x) + f2(x) dx = f1(x) dx + f2(x) dxaaa( ), ) ! , * ).#4$ # 12.3 (6 BC" !D -41E). ( F (x) { -! ! ) : a b)$ f (x), x = b ! f (x).
> $Zba bf (x) dx = F (b) ; F (a) F (x) a(12:4)F (b) = xlimF (x). P (12.4) : ! b 0 (12.4) , ! ! (12.4) .9 ! $ * * .! ; 12.2 (#F $#F).$ f (x) ' (x) : a b), x = b f (x) ' (x). ', ( x 2 : a b) 0 f (x) ' (x), &":) Zbaf (x) dxZba' (x) dx (,112(, -) Zba' (x) dxZbaf (x) dx(, - (. 12.3 (# F $#F).$ f (x) ' (x) : a b), x = b f (x) ' (x). ', " f (x) = Alimx b 0 ' (x)(0 < A < +1)! ; ( Zbaf (x) dx( (.Zba' (x) dx2 12.2, 12.3 ! ! (.
11.3,11.4).. >, ! 12.2 12.3, $ ! ! .' 12.2 * ).$ # 12.1. 0 * M > 0 , ) 0 f (x) (b ;Mx) x 2 : a b), 0 < < 1, Zb! f (x) dx aM!) f (x) (b ; x) x 2 : a b), 1, !Zb f (x) dx .a 12.3. ' ! +ZZ 5 + cos xZ sin x) ( ; x)2 dx) dx!) x px dx > 0:x00011132 $ ) ! x = . > ;1 cos x 1, x 2 :0 ) x 4 12.1(.
!)) 5(+;cosx)2 ( ; x)2 ) .2 $ f (x) = xsinpxx !) ! x = 0. () ' (x) = p1x # $ f (x) ' (x) x ! +0 :f (x) = lim sinpx : p1 = lim sin x = 1:limx +0 ' (x) x +0 x xx x +0 x ( 12.3 ) , Z 1px dx. ( !) ! ), 0Z 1 px dx (. 12.1), !) )0 *.( $ f (x) = x1 ) ) : 0 +1) ! : x = 0 x = +1. (<+Z dx Z1 dx +Z dx= x + x :x001( 0 < < 1 1, 0 < 1 > 1. , ! > 0 ) . 12.1. 1! !!!11Zbaf (x) dx(12:5)) & (", ! Zbaj f (x) j dx114(12:6) $ !) (", (12:5) , (12:6) .4 ) ! $ ! .
12.4. 8& (" - (.2 9.5. 12.4. ' ! Z2 sin x + sin 3 xpx ; dx:(12:7)sinx+sin3x> 0 px ; px2; x 2 ( 2 ] Z2 2! px ; dx (. 12.1), Z2 sin x + sin 3 x px ; dx .9 , ! (12.7) ! *. ( 12.4 ! * , *.33333 !" #$"Zb1. !!& & f (x) dx, !:a) x = b ! ! f (x)1) x = a ! ! f (x)1!) x = c (a < c < b) ! ! f (x).Z2 dx2. 5 (!) ), & 4 ; x2 !)!.0Z2 dx3. 5, & p!)! " ! 4 &.24;x04. K!! !)! !!" &!Zb 11(x ; a) + (b ; x) dx ( 2 R ) :a1155. . + ! & ! !"! !!& & & ?6.
5 ! !! !!") & & ") .7. + + " ! !!") & & ") (c. " 12.2, 12.3).8. K!! !)! !+: !!" &":+Z1 x arcsin pxZ1 ex ; eZ dx)dx1 ) (1 ; x)2 dx1 )2 :2tgxxlnx0011H. .!! ! !!") &.9. * !!" & & "!:) !+ !):!? ) ! !):!? (!. 12.1).Z sin 5 x10. 5, !!" & p3 5 4 dx ! !+x +x0!):!.O3. 0 5: 4.
)! < 1 < 11 !)! &) ) :8. ) c)!1 ) !)!1 ) !)!.116 13)* (/ %/ $$ $ $ . $ , < .. 3 , , ) .> ) (), ) $ ) $ . >, D = fxg ) x ) x 2 D y, , )D y = f (x).] $ . 1, f (x) = x2, $ ! y = x2(. . 13:1).;.
13.15 $ x !) $ y $, ! x.0 ) D ) f(x y)g ) (x y) 2 D z , , , ) D $ z = f (x y), ( (.] $ f (x y)) z = f (x y), <117;$ . 1, f (x y) = x2 + y2, $ ! * z = x2 + y2 (. . 13.2).. 13.25 $ (x y) !) z $, ! , , x y.(! ) ! ) $ . 0 D = f(x y z )g ) ) (x y z ) 2 D u, , ) D ( (u = f (x y z ). $ ! < ) . $ ( f (x y z ) = const ) $ u = const.1, f (x y z ) = x2 + y2 + z 2, $ x2 + y2 + z 2 = const.2 $ ! # ! n > 3.
13.1. %) n (x1 x2 : : : xn) n{ ( ) Rn.& < ) M (x1 x2 : : : xn), x1 x2 : : : xn { . 13.2. 0 ) M (x1 x2 : : : xn) n) D R z , , ) D 118 n (z = f (x1 x2 : : : xn) z = u(M ):0 " Rn13.3. ( M (x1 x2 : : : xn) N (y1 y2n : : :nyn) 2 R . / ) < R q(M N ) = (x1 ; y1)2 + (x2 ; y2)2 + : : : + (xn ; yn)2:, R1 R2 R3 < $ ! < .
( n > 3 < $ ) ! #. 13.4. {& M0 2 Rn ) M 2 Rn, (M M0) < :!: (M0)., (M0) :{ R1 M0 2-;. 13.3{ R2 M0 -;. 13.4119{ R3;# M0 .. 13.5 !* (M0) Rn n{ + M0 . 13.5. ( ) D Rn (.. D ) ) Rn ).> M ) D, Rn* { M, (M) D:> M D, ! { Rn , ! !, M1 2 D M22D.;.
13.6 13.6. %) D, * . 13.1. R2 D = f(x y)jx2 + y2 < R2 g.. (0 0) R ! * ), ). 13.7. %) D, )* , ).120 13.2. R2 D = f(x y)jx2 + y2 R2 g.. (0 0) R * ) , ).5, < ), 13:1 ). 13.8. %) D , ! ) , ) ) D. 13.3. R2 D = f(x y)j1 < x2 + y2 < 4g ).;;. 13.7 13.4. R2 )D = f(x y)j(x ; 2)2 + y2 < 1g f(x y)j(x + 2)2 + y2 < 1g ).. 13.8121& ) . ' M1 , M2 , , ! ) D..
!* Rn x1 = x1(t) x2 = x2(t) : : : xn = xn(t) x1(t) x2(t) : : : xn(t) { $ . 13.9. ) &. ! &. 13:1 13:3 ) D !. 13:2 ) D ! . 13:4) D ! , .13.10. %) D nnR , R * { {! M0, D (M0), , Rn * {!#, )* ) D. # ) D . ( ) R2 f(x y)jy 0g R2 (.. ). n13.11. 5& M0 2 Rn ! ! R , )* < .. > ), $ !$$ $ . < $ ).! _(M0 ) { { M0, ..(M0) ! M0.
13.12. ( $ u(M ) n M0 2 R , , ) ! , M0.122G a u(M ) M0, 8" > 0 9_ (M0) : 8M 2 _(M0 ) ju(M ) ; aj < ":!: a = MlimM u(M ):Q ju(M ) ; aj < " < ! " > 0. 1 ", ju(M ) ; aj < " ! $ u(M ) a. Q ! 8M 2 _ (M0). G # ", # _(M0), < . (< a = MlimM u(M ), $ u(M ) ! a M , ! M0 (, , M0).0 M0 ! D $ u(M ), ! _(M0) , $ u(M ) (.
. 13:9) ju(M ) ; aj < " ) 8M 2 _(M0). (< < .!0!0;. 13.9 13.13. 0 M D $ u(M ), 0{ ! -a = MlimM u(M ) 8" > 0 9_ (M0) : 8M 2 _(M0)\D ju(M );aj < ":!0 < !, ! ju(M ) ; aj < " M 2 _(M0), # , ) ! D $ u(M ).0 13:12 13:13 $ a = xlimx f (x), ) , , $ 123!0 ! f (x) a _ (x0), $ * n{ _ (M0), ! n{ _(M0) \ D. (E, $ u(M ) ! ) $ f (x).)9 ! , $ ( , , , $ ) $ .
13.14. 3 u(M ), M0, , ) ! , M0, M ! M0, lim u(M ) = 0:M M0! ! $ ) ! $ . R2 $ u(M ) ) $ M (x y) M0(x0 y0), u = f (x y), ! a = xlim!x f (x y ):y!y002 < ! ( ), ! (x0 y0) Ox (. . 13:10).;. 13.10124,x ; x0 = cos y ; y0 = sin lim f (x y) = lim0 f (x0 + cos y0 + sin ):x!x0y!y0! 13.5. 12 + y2xpx2 + y2 + 1 ; 1 :limx!y!00x2 + y22pplim=limx!x2 + y2 + 1 ; 1 0 2 + 1 ; 1 =0y !0q 2= + 1 ; 1 13.6.
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