1. Интегралы ФНП Диф_ур (853736), страница 6
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1 s = 2 (1 ; cos ') (. .9.10).' , = 2 (1 ; cos ') = 2 sin ' 2 + 2 = 4 (1 ; cos ')2 + 4 sin2 ' = 4 (2 ; 2 cos ') = 16 sin2 '2 :Q, , $(10.8) Z qs = 2 2 + 2 d' =0ZZ s''2 '= 2 16 sin 2 d' = 2 4 sin 2 d' = ;16 cos 2 0 = 16:00( ;AB y = f (x), a x b, f (x) : a b ]. P;AM A M x f (x) (. 10.4). 2 < !$ x , (10.2), ) $00000s = s (x) =Zx qa1 + :f (z ) ]2 dz:0> $ : a b ], 01ds = d @Zx q1 + :f (z ) ]2 dz A = q1 + :f (x) ]2:dx dx a5 $ s = s(x), $$ :qqq222d s = s (x) dx = 1 + :f (x) ] dx = (dx) + :f (x) dx] = (dx)2 + (dy)2 :', $$ d s ) $qd s = (dx)2 + (dy)2 (10:9) dx = T x { * , dy = f (x) dx { $$ $ .870000003 (10.9) .2 < ;AB M x f (x) (.
. 10.4). ' $$ $ y = f (x) , ! dx = MD , $$ $ , .. dy = CD. 9 , MCD ($ pq(10:10)MC = MD2 + CD2 = (dx)2 + (dy)2 :9 (10.9) (10.10), , d s = MC , . . M x f (x) x + dx.. 4 , ;AB x = ' (t), y = (t) = f (') ,p 2 2 $$p 2 ) $ d s = x + y dt d s = + 2 d',.0005 6( T ) ) x = a, x = b (. 10.5) * S = S (x)! , Ox ! x.%) , S = S (x) : a b ], 9 V T ZbV = S (x) dx:a(10:11) 10.4.
1 !O <x2 + y2 + z 2 = 1:a2 b2 c2(o < { a, b, c. 9 < , Ox ! x,88;a < x < a, < (. 10.6), 8 222>< y + z = 1; x 2c2a2>: xb = const()8>y2z2<+= 12 (1 ; x2 =a2)2 (1 ; x2 =a2)bc>: x = const ;a < x < a:(10:12). 10.5. 10.6' (. 9.1), { v<,10u2utb2 @1 ; x2 A * S = . ( = uav01u2utc2 @1 ; x2 A { < (10.12), =ua102x* S = bc @1 ; a2 A, ;a < x < a.5 * <, $ (10.11) !O:010123 aZaZaxxV = S (x) dx = b c @1 ; a2 A dx = b c @x ; 3 a2 A =aaa;;;! 4aa= b c a ; 3 + a ; 3 = 3 a b c:', 9 * a, b, c V = 43 a b c.
, a = b = c = R, !O # S = 43 R3 R.( T ! * Ox , $ $ y = f (x) x = a, x = b, y = 0 (. 10.7). > T , Ox ! x,a x b, y = f (x). 9 , * S = y2 = f 2 (x), a x b, $ (10.11) !O *89ZbZbaaZbVx = S (x) dx = f (x) dx = f 2 (x) dx:2a> !, 9 T , " Ox , y = f (x) x = a, x = b,y = 0, ZbVx = f 2 (x) dx:(10:13)a. %) , 9 T , " Oy , y = f (x) x = a,x = b, y = 0, $ZbVy = 2 x f (x) dx:(10:14)a.
10.7. 10.8 10.5. 1 !O , ! * , y = e x, x = 0, x = ln 2,y = 0 ) Ox (. 10.8)- !) Oy (. 10.9).) ' $ (10.13), ;Vx = Zln 20f 2 (x) dx = ln 2= e 2 x dx = ; 2 e 2 x = ; 2 e 2 ln 2+ 2 = ; 2 14 + 2 = 38 :00!) 2 ) !O , ! * Oy (. . 10.9), $ (10.14):Zln 2;;Vy = 2 Zln 20;Zln 2x f (x) dx = 2 x e x dx:090;. 10.92 $ (c. 8.4).() u = x, d v = e x dx- d u = d x, v = ;e x dx.9 ,;Zln 2Vy = 2 x e ln 20= 2 ; 2 ; e;0x dx = 2 B@;x e;;1 ln 2 lnZ 2x + e x dx CA=0;0 ln 2 1! ln 2 !x 0 = 2 ; 2 ; 2 + 1 = (1 ; ln 2 ) :;91 !" #$"1. " & .
* "!!? (c. 10.1).2. " & ! J ! 0 !n ! + 1?3. * ! ! + f (x) A a b ], ", &") ) y = f (x), a x b, "!? * 4 ? (!. 10.1).4. < & y = a !h xa , " + !!!"x = 0 x = a.5. !, ") , ! x = ' (t), y = (t), t0 t T , !. * 4 ? (!. 10.2).6. C8 ! x2 + y2 = R2 ! . K! 10.2, " ! 4 !.7. !, ") , ") ) = f ('), ' , !. * 4 ? (!. 10.3).8. < ") ) ! L) = a '.9.
* "! " & &") )? & ! !"! " & .10. * M T , !& " !!x = a, x = b, : S = S (x) +& & & ! !!+, ! Ox ! !!! x? (!. (10.11)).11. ." ! M 4! x2 + 36 y2 + 100 z2 = 3600 (!. 10.4).12. * M T , & : & ! Ox , & & " y = f (x) " x = a, x = b, y = 0? (!. (10.13)).13. < M T , & : & ! Ox &",& & y = 2 x ; x2 y = 0.14. * M T , & : & ! Oy , & & " y = f (x) " x = a, x = b, y = 0? (!. (10.14)).15. < M T , & : & ! Oy , & & y = sin x (0 x ) y = 0.4: a sh 1:pqp8: 2a 1 + 16 + a ln 4 + 1 + 162:29213: 1615 :15: 22: 11< Zb1 * f (x)dx a : a b ] b ; a < +1, ! $ f (x) : a b ].B < * ! :1) ) !2) $ ./ ( $ f (x) ! ) :a +1) :a b] ! b > a.6 ( Zb f (x) dx b ! +1.!:Za+1af (x) dx.
',Z+1af (x) dx = b lim+!1Zbaf (x) dx:(11:1)0 * (11.1),+Z ! f (x) dx (". Ea (11.1) * ( !+Z), ! f (x) dx (a".11934 :Zbf (x) dx = a lim!;1;1Zbf (x) dxa(, $ f (x) ! ) ( ;1 b ] : a b ] ! a < b).( $ f (x) (;1 +1) ! : a b ]. > ! +Z f (x) dx * !:1;1Z+1Zbf (x) dx = alim!;1 f (x) dx:b!+1;1Za+1( < f (x) dx (" * , < ) , ! a b ;1 +1.+Z 11.1. ( , x e x dx , 0 < .' ! ,:+ZZbZb1xxx d(;x2 ) =x e dx = b limxedx=;lime+2b + 000b 111xbe 0= ; 2 b lime ; 1 = 2:= ; 2 b lim+++Z', , x e x dx = 05.;11;1;2;!!2;1!;22;1!21211;Z20 11.2.
( , cos x dx .' ! ) ,:ZZcos x dx = a lim cos x dx = a lim sin x a = ; a lim sin a:;1!;1;1a!;194!;1> *, .+Z 11.3. ( , x2 + 4dxx + 20 , < .> !Zb d(x + 2)Zb1b+2a+2dx== arctg 4 ; arctg 4 22a x + 4x + 20 a (x + 2) + 16 4, ! ) , :1;1Z+1;1!dx1b+2a+2arctg 4 ; arctg 4 =!;1x2 + 4x + 20 = 4 ablim!1+ !! 1= 4 2 ; ;2 = 4:', 025 .) * * ) * . 11.1. $ a > 0 +Z { dx ( & > 1 . 6 a x( & 1..
' ! , :8+1 b>x+Zb dx >Z dx< b lim 6= 1+ ; + 1 a==lim=b>b+xx>aaln x a = 1: b lim+1;1!!11!8><=>:1a1 > 1;11 1 ! ).;952 ( f (x) { ) :a +1)$ .$"& S , $ $ y = f (x) 0 x = a y = 0, * aABb (. 11.1) b ! +1 ( *).', :;S = b limSaAB b:+!1(11:2). 11.1> , Zb, SaAB b = f (x) dx, (11.2) , aS = b lim+!1Zbaf (x) dx =Z+1af (x) dx:> !,(" ( " . 4 - ! ! ) .96 ! ( ).#4$#11.1.0 a1 > a, ! ++ZZf (x) dx f (x) dx ! ! .1a1a1#4$ # 11.2 (4$ !).ZZ+1+1) f (x) dx = f (x) dx { a- a+++Z ZZ!) f1(x) + f2(x) dx = f1(x) dx + f2(x) dxaaa( ), ) ! , * , ).#4$ # 11.3 (6 BC" !D -41E).
0F (x) { -! ! ! ) : a +1) $ f (x), $11Z1 +f (x) dx = F (+1) ; F (a) F (x) a+1a1(11:3) F (+1) = x limF (x). P (11.3) : !+ (11.3) , !! (11.3) . 11.1{11.3 * ! ! . 2), , 11.3. ' ! $ 1 -=! , !Z1+1af (x) dx = b lim+!1Zbaf (x) dx = b limF (b) ; F (a) =+!1= b limF (b) ; F (a) = F (+1) ; F (a):+9 , $ (11.3) .!197 11.2 ( 6# 9$ F # $1$ # 6). $ u = u (x) v = v (x) (( : a +1). " Z+1axlimu (x)v (x)+!1v du (, :Z+1a +u dv = u (x) v (x) a1;ZZ+1a+1au dv( -v du:2 ! $ ./ . 9 ! $ * *.
11.3 (#F $#F).f (x) ' (x) : a +1)$ ( x a0 f (x) ' (x):'+Z) 1 Z+1aaf (x) dx Za(, - (-!) +1' (x) dx(11:4)' (x) dxZ+1af (x) dx(, - (.98.) ( ! Z+1Z+1a' (x) dx -. 2 f (x) dx a * * $: : a +1) K (b) b ! +1.ZbP $ K (b) = f (x) dx: ( b2 b1 a. >, f (x) 0 =)Zb2b1af (x) dx 0 Zb2Zb1Zb2Zb1aab1aK (b2) = f (x) dx = f (x) dx + f (x) dx f (x) dx = K (b1):9 , K(b2) K(b1) $ K (b) : a +1).E , # (11.4), :ZbZb+1aaaK (b) = f (x) dx ' (x) dx Z' (x) dx = const:> !, $ K (b) : a +1), , blimK (b) = b lim++!1!1Zbaf (x) dx*, .. ! !) ( ! ), ! : Z+1aZ+1aZ+1aZ+1af (x) dx .f (x) dx .
2-' (x) dx ) . 2-' (x) dx . > 99 ) < ! Z+1af (x) dx ) Z+1, . 9 , ' (x) dxa .' 11.1 11.3 * ). 11.1. ( $ f (x) x a > 0.>, * M > 0 , ) 0 f (x) Mx x a, > 1, !Z+1f (x) dx !) f (x) M x a, 1, !x+Z f (x) dx .a 11.4. ' ! ++Z 1 + cos xZdxpdx:(11:5)22+x;1x125x11> ;1 cos x 1, x 1 x 2 11.1 (. )) 0 1 +xcos2x2! (11.5) .> x 1 p1 2 , 11.1 (.
!)) p 21125 x + x ; 1 5 x! (11.5) .a111333 11.4 (# F $#F).$ f (x) ' (x) : a +1) " f (x) = Alimx + ' (x)!(0 < A < +1):1' ( Z+1af (x) dx ( (.100Z+1a(11:6)' (x) dx- 11.4 * 11.3. , (11.6) , " = A2 > 0 ) a1 > a, ; A2 < 'f ((xx)) ; A < A2 () 0 < A2 ' (x) < f (x) < 32A ' (x) (11:7) x a1:ZZ+1+10 ' (x) dx , ' (x) dx aa+Z 3A' (x) dx ) (.
11.1 11.2 !2a), ( (11.7)) 11.3 (. ))++ZZ f (x) dx, f (x) dx, ) a. a+Z4 , '(x)dx1111111aZ+1 f (x) dx.a 11.5. ' ! Zx sin x13 dx-+112Z!3x ln 1 + x4 dx:+11(11:8)Z9 (11.8) x1 dx. 11 # $ :1 : 1 = lim x2 1 : 1 = 1:2limxsinx +x3 x x +x3 x+Z 1> x dx (. 11.1), 111.4 (11.8) ) .+Z 1 (11.8) x3 dx.
9+1!!11111101 # $ :! 13 : 1 = 3:3:=limxlimxln1+x +x4 x3 x + x4 x3+Z 1> x3 dx (. 11.1), 111.4 (11.8) ) .. 4 ! ! ) , ) ! ! ) .!!1117- < ! $ , * : a +1 ) ) , . 11.1. 1! Z+1af (x) dx(11:9)) & (", ! Z+1aj f (x) j dx(11:10) $ !) (", (11.9) , (11.10) .4 ) ! . 11.5. 8& (" (.Z.( ! ! *, .. 102Z+1a+1af (x) dxj f (x) j dx .Z+1>! , ! f (x) dx )a.' , ;jf (x)j f (x) jf (x)j () 0 f (x) + jf (x)j 2jf (x)j( 8 x a ):(11:11)> (11.10) , 2 !+Z 2 j f (x) j dx ) , (11.11) a (.
11.3) +Z f (x) + j f (x) j dx.11a5 , f (x) = f (x) + j f (x) j ; j f (x) j, 2 ! Z+1af (x) dx =Z +1af (x) + j f (x) j dx ;Z+1aj f (x) j dx. > 11.5 . 11.6 ($ 9"4 8H 1$D 4 $I$). f (x) x a > 0 "& -x a > 0,M > 0 > 1, Z+1..aj f (x) j Mxf (x) dx (& (-Zdx > ! Ma x > 1 (.
11.2 ! 11.1), ( 11.3) +Zj f (x) j dx ) . 9 , ! +11aZ+1af (x) dx ! *.103 11.6. ' ! Z cos x+11cosx> x2 x12 +Z cos xx2 dx:x 1, 11.6 !-1dx ! *. ( 2x1 11.5 ! * , *. 11.7. ' ! )Z sin x+11x dx-!)Z cos 2 x+11x dx-)Z sin2 x+11x dx:) () u = x1 , dv = sin x dx.
> du = ; x12 dx, v = ; cos x.' $ ! (. 11.2), : +++Z cos xZ cos xcosxdx = ; x ;dx = cos 1 ;dx (11:12)22xxx1111cos x = 0. ( ! x lim+x # (11.12) (. 11.6), +Z sin xdx ) .x1!) 4 , ! +Z cos 2xdx *.x1) > sin2 x = 1 ; cos 2 x = 1 ; cos 2 xx2x2x2xZ sin x+1!111111 ! Z 1+112 x dx (. 11.1), 104Z cos 2x+1! 1Z sin2 x+112 x dx (. .
!)), !-x dx .Z sin x 11.8. ( , ! dxx1 *.P $ :++Z j sin x jZ sin x dx =dx:(11:13)xx11+1112jsinxjsin> j sin x j sin x, x x x 0 ! x 0.2Z sin2 x+1dx (. 9 , x1 11.7()), ! +Z sin x(11.13) . (< dx !x1 *. 9 , 11.7() ,+Z sin x dx . 9 , < x 1*.11 !" #$"1. * ! !!" &:) ! ! " ) ?) ! ! " ?) ! ! " ) ?+Zdx!)! " ! 4 &.2. 5, &2x+x;22Z0 x dx3. 5 (!) ), &x2 + 4 !)!.+Z4.
5, &x e x2 dx !)! " ! 4 &.1;11;;11055. K!! !)! !!" &(!. 11.1).Z dx, & a > 0, 2 Rxa+1Zdx , & a > 1,6. K!! !)! !!" &x ln xa 2 R.7. . + ! & ! !"! !!& & ! ! " &?8. ( ! !!& & ! ! " & (!. !! 11.2).+Z9. (, ! !!" & f1(x)dx !)!, !!a++ZZ " & f2(x)dx !)!, !!" &f1(x) + f2(x) dxa!)!. a10. " ! !!& & ! ! " & (& " <+->) (c. !!11.3).11. & ! !! &! ! " & (c. y 11.2).12. + + " ! !!") & ! ! " & (c. " 11.3, 11.4).13. (, ! f (x) 0 !) x a > 0 ! !:! "+Z x limf (x) x = A > 0 > 1, & f (x) dx !)!+a(!! 11.4).14. (, ! f (x) 0 !) x a > 0 ! !:! +Z " x limf(x)x=A>01,&f (x) dx+a!)!.15. K!! !)! !+: !!" &":+++Z 2 + arcsin(1=x)Z (px + p4 x + 2)3Z x arctg xp dxpx dx1 )) p 5 dx1 )2 3 x21+x+1(5x;4)110+11111!11!1111(!! 11.3 11.4).16. * !!" & ! ! " &"!: ) !+ !):!? ) ! !):!? (!.
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