1. Интегралы ФНП Диф_ур (853736), страница 25
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% # % $% 0o'(,# $' # ' 3 ( "#% $#% # $.375 6.9. @(Z5 (p25 ; x2)3dx:4x25( < #'#Zp 2 .( R(x a ; x2)dx a = 5 # # x = 5 sin t:? # "( /#0 ' t = arcsin x5 3 x = 2 5 " ' t = 6 6 % x = 5 { t = 2 : "3 x(t) :" #6 6 2 ! C2 56 5] $ /( x(t) = 5 sin t $ ( .
:Z5 (p25 ; x2)325px4dx = jx = 5 sin t6 dx = 5 cos tdtq25 ; x = 25 ; 25 sin2 t = 5j cos tj = 5 cos t" #! =Z 2 cos4 tZ 2 (5 cos t)3 5 cos tdt == sin4 t dt:# ## t 2 6 6 2 =4 sin4 t5=6=6F ZR(sin t cos t)dt4v R(u v) = u4 " (:44v(;v)R(;u ;v) = (;u)4 = u4 = R(u v): $' 0$ . " # y = tg t y = ctg t: I "( #$ # y = ctg t:2cos4 t dt = y = ctg t6 dy = ; 1 dt = ;(1 + ctg2 t)dt )4sin2 t=6 sin t=Z2376 !p !dy) dt = ; 1 + y2 6 y 6 = ctg 6 = 36 y 2 = ctg 2 = 0 =pp1Z0 y4Z 3 y4Z 30 y 4 ; 11= ;p 1 + y2 dy = 1 + y2 dy = @ 1 + y2 + 1 + y2 A dy =003p3p3p3p3013ZZ dyy2@A= (y ; 1)dy + 1 + y2 = 3 ; y + arctg y =0000p 3 p 1p= @ ( 3) ; 3A + arctg 3 =3 6.10.
@(: >30pZ 3 p1 + x2dx:2x1Zp2 2< = R(x a + x )dx a = 1 $ x = tg t :p" #Z 3 p1 + x2x2 dx = x = tg t t 2 4 6 3 61p1 = 1 6 dx = 1 dt =1 + x2 = 1 + tg2 t = j costj cos tcos2 t ===Z3 1Z 3 dtZ 3 cos tdt11= cos t tg2 t cos2 t dt = cos t sin2 t = cos2 t sin2 t ==4=4=4q ! p2 ! p3 dsint= (1 ; sin2 t) sin2 t = y = sin t6 y 4 = 2 6 y 3 = 2 ==4p3=2Z= p (1 ;dyy2 )y2 :=Z32=2F 0'( /#0, # ', 11+ 1 :=(1 ; y2 )y2 y2 1 ; y2377p3=2p3=2p3=2pdy = dy + Z dy = ; 1 ! 3=2 +y p2=2p2=2 (1 ; y2 )y2 p2=2 y2 p2=2 1 ; y2pp 10 1 y + 1 !p3=2 p3=21+211++ 2 ln y ; 1 p = 2 ; p + 2 @ln p ; ln p2=2 A =31 ; 3=21 ; 2=22=2ZZpppp= 6p; 2 + 12 ln 2 + p3 ; 12 ln p2 + 1 = 6p; 2 +32; 32;13p 2ppp23)2+1)6;2(2+1(12++ 2 ln 22 ; 3 ; 2 ln 2 ; 1 = p + ln p 3 : >32+1 6.11.
@(Z2 px2 ; 1x dx:p1< ' /#0 R(x x2 ; a2) (a = 1:) 4 $ x = sin1 t :Z2 px2 ; 11xdx = x = sin1 t 6cos t dt6 x = 1 t = 6dx = ; sin2t2v cos t cos t up2u1tx = 2 t = 6 6 x ; 1 = sin2 t ; 1 = sin t = sin t = 1 !;1 cos t==Z 2 cos2 tZ 2 1 ; sin2 tcostdt == ; sin t sin t sin2 t dt = sin2 t dt =2sint=6=6=2=Z6=2 ! p dt= sin2 t ; dt = (; ctg t) ; 2 ; 6 = 3 ; 3 :=6=6=6=Z2=Z2 6.12. @(uZ vut 1 ; x dx :1+x x378>vuu='$ # t = t 1 ; x 1;x 2t=1+x1+xt2 6(1 + x)t2 = 1 ; x x(1 + t2) = 1 ; t2 x = 11 ;+ t2dx = (1(;+4tt2))2 dt "$ # <Z102 ;11;tA (;4t)t@Z2t1 + t2(1 + t2)2 dt = ;4 (1 ; t2)(1 + t2) dt =! 1Z 1ZZ1= ;4 1 ; t2 ; 1 + t2 2 dt = ;2 1 ;dtt2 + 2 1 +dt t2 = q(1;x)=(1+x);1t;11= 2 2 ln t + 1 + 2 arctg t + C = ln q+(1 ; x)=(1 + x) + 1 vuu x+2 arctg t 11 ;+ x + C: > 6.13. @(Z p2x + 1x2 dx:< F1 { ( ( 0 u Z B nvu + bCR @x t axcx + d A dx ( p n = 26 c = 06 d = 1). '$ # t = 2x + 1 t2 = 2x + 1 tdt = dx:2Z 0 t2 ; 1 1 ; 2ZZ p2x + 1t@A tdt = 4 (t2 ; 1)2 dt:x2 dx = t 2=:012 !!2tt2 = t !2 = @A = 1 1 + 1(t2 ; 1)2 t2 ; 1(t ; 1)(t + 1)2 t;1 t+1 =011111= 4 @ (t ; 1)2 + 2 t2 ; 1 + (t + 1)2 A : 3Z 2 1Z dtZ d(t ; 1)111454 4 (t ; 1)2 + 2 t2 ; 1 + (t + 1)2 dt = (t ; 1)2 + 2 t2 ; 1 +37911t;11t;1+ (t + 1)2 = ; t ; 1 + 2 2 ln t + 1 ; t + 1 + C = ln t + 1 ; pp2t p2x + 1 ; 1 p2 2x + 1; t2 ; 1 + C = ln 2x + 1 + 1 ; ( 2x + 1)2 ; 1 + C =ppj2x+1;1j= ln p2x + 1 + 1 ; 2xx+ 1 + C: >Z d(t + 1)$ #I IH 3$I( ":Z xp1 + xdxA1) p1;x2)Zdxp:(5 + x) 1 + xp1) 12 arcsin x ; x +2 2 1 ; x2 + C Ap2) 12 arctg x2+ 1 + C:|||||380 7 S , y = y1(x) y = y2(x), y1(x) y2 (x) x = a x = b ( .
7.1) $ ;ZbS = (y2(x) ; y1(x)) dx:a(7:1). 7.1% , y1(x) 0 y2(x) = y(x) ZbS = y(x) dx;a(7:2) , ($ . 7.2.. 7.2) , (7.1), (7.2) * , $ x = a x = b ( $* ( . 7.3).381;. 7.3 7.1. - , :y = xe;x x = 1 y = 0:< ) . /: y(0) = 00 y(1) = 1e 0 xe;x > 0 x 2 (0 1): ), 0, * , $ * 0x ( . 7.4). 2, 3-; ( , * .. 7.44 ( , 0). 5 6 (7.2), :S0AB =Z10xe;x dx = ;Z1001x de;x = ;xe;x +Z1 ;xe dx =0= ;e;1 + (e0 ; e;1) = 1 ; 2e;1: > 7.2.
- , :y = x2 + 2x y = x + 2:382< - :(y = x2 + 2xy = x + 2x2 + 2x = x + 2 x2 + x ; 2 = 0 ) x1 = ;2 x2 = 1 )y1 = 0 y2 = 3: / : (;2 0) (1 3).4 ( y = x2 + 2x y = x + 2 ( . 7.5).;. 7.52( , * * * , 0 y1(x) = x2 +2x0 y2(x) = x + 2: (7.1) :S=Z1 ;2Z1(x + 2) ; (x + 2x) dx = (2 ; x2 ; x) dx =2;201 1 1! !32 1xx8= @2x ; ; A = 2 ; ; ; ;4 + ; 2 = 4 5:32;23 23 , :p1) y = 0 y = ln x x = 2 x = 32) y = x2 y = x3) y = ex y = e;x x = 1:1) ln 274 ; 12) 13 3) e + e;1 ; 2:|||||383>/ ( ( (7.1), (7.2) x y. 9, S , x = x1(y) x = x2(y) (x1(y) x2(y)) y = c y = d ( . 7.6) $ ZdS = (x2(y) ; x1(y)) dy(7:3);;c.
7.6 7.3. - S , - : y = ln x x = 0 y = 0 y = ln 2 ( . 7.7).. 7.7< 5 , y = ln x ) x = ey $ $x1(y) 0 x2(y) = ey c = 0 d = ln 2: / (7.3) :S=Zln 20eydy0 = eln 2 ; e0 = 2 ; 1 = 1:ln 2= ey >;, ( (7.1), y1(x) :384y1 = 0 <0 1] y1 = ln x <1 2] ($ :Z1Z201S = ln 2 dx + (ln 2 ; ln x) dxZln 2 , ey dy.0 7.4. - , :y2 = 2x + 1 x ; y ; 1 = 0:< y2 = 2x + 1 { 6 (, 00 x ; y ; 1 = 0 { . - :( 2y = 2x + 1 )x ; y ; 1 = 0 y ; 1!2;1y) 2 = y + 1 ) (y + 1) 1 ; 2 = 0 ) (y + 1)(3 ; y) = 0 )) y1 = ;1 y2 = 3 ) x1 = 0 x2 = 4:9 (, A(0 ;1) (4 3), ( ( . 7.8).
% 6 $ ( (7.3), p $ $ y = ; 2x + 1 y = x ; 1,2y { $ : x = 2; 1 x = y +1, .;. 7.82y% x1(y) = 2; 1 0 x2(y) = y + 1: ) (7.3):12Z3 0y;1AS = @y + 1 ;;121Z3 0 y23dy = @; + y + A385;122 dy =10 3 27 9 ! 1 1 !2 33yyA@= ; + + (3 ; (;1)) = ; + ; + +6 = 5 1 :62;12626 2 3 >"#$ $ , %:y2 + 8x = 16 y2 ; 24x = 48:p: 32p32 :||||| , , @ , 9x = x(t) = y = y(t) x = a x = b 0x ( . 7.9), ZS = y(t) dx(t)(7:4) a = x(),b = x( ): 7.5. % , , :9x = cos3 t = :y = sin3 t < 5 . t 0 2 x 3 1 0, y 0 1:yx 0 ) 0 ;!t ) 1 ;!0 ;!1:2), * $ I (x 0 y 0).
A 386 y(x), " # t 2 0 2 !23sintcost= ;3 cos2 t sin t = ; tg t < 0 t 2 0 2 !1110000yxx = (; tg t)t tx = ; cos2 t x0 = 3 cos4 t sin t > 0 t 2 0 2 :t), y(x) (* , * * t = 0 (yx0 ! 0 t !0 + 0) * { t = 2 (yx0 ! 1 t ! 2 ; 0) ( .7.9).0yt0yx = x0t;. 7.9" #B $ t 2 2 " 3 #t 2 2 ..
- ( ( * * * ($." # M (x y) t 2 0 2 .9 cos( ; t) = ; cos tsin( ; t) = sin t#! M1, * t1 = ; t " t1 2 2 e (;x y) , , M 0y ( . 7.9).9 M (x y) { I , II (x 0 y 0) $ 387, 0y.( $ (, , cos(2 ; t) = cos tsin(2 ; t) = ; sin t * M2, * t2 = 2 ; t, M * 0./ .4, , . , I , (7.4): a = 03 b = 1 ( $ )0 = 2 0 = x() = cos 0 = 01 = x( ) = cos3 :S1 =Z0=2sin t d cos t = 333=Z20cos2 t sin4 t dt ===Z2Z23322= 4 (2 sin t cos t) sin t dt = 4 sin2 2t sin2 t dt =00=Z 213= 4 2 (1 ; cos 4t) 12 (1 ; cos 2t) dt =02=Z23= 16 (1 ; cos 4t ; cos 2t + cos 4t cos2t) dt =031! =2 1 =Z2136= 16 4 2 ; 4 sin 4t + 2 sin 2t 0 + 2 (cos 2t + cos 6t) dt75 =0"! =2# 3113= 16 2 + 4 sin 2t + 12 sin 6t 0 = 32 :/ S = 4S1 = 38 : > $ , x = t ; sin t y = 1 ; cos t #$* %#)##.388()(:;,%#$ - t : 0 t 2: .#(- (/ { #.
7.10. 1$ / 3.. 7.10||||| , , = () = = ( . 7.11) $ Z 21S = 2 () d;(7:5). 7.11 7.6.- , = cos 5:< ) ( 2 <0 ] ( cos 5 = cos 5(;), , * 2 <; 0] $ , * 2 <0 ] ) :389! 101!003!1010<03100! 410!14101! 510!0510! 710<07100! 810!18101;! 910!0910!<0C . 7.12.. 7.12D 5 6 ( ). % S1 (3 . 7.12), (7.5). = :% () = cos 5 = ; 1010==Z 10 2Z 1011S1 = 2cos 5 d = 4(1 + cos 10) d =;=10;=10" !# 1= 4 10 ; ; 10 = 20 :A S : = : >S = 5S1 = 5 204 $ , = sin 2:.
4 :|||||390 8 l { .@ y = y(x) y(x) 2 C 1<a b] a x bl=Zb qa1 + (y0 (x))2 dx:(8:1)@ x = x(t) y = y(t) x(t) y(t) 2 C 1< ] t l=Z q(x0 (t))2 + (y0 (t))2dt:(8:2)@ = () () 2 C 1< ] l=Z q(())2 + (0 ())2d:(8:3)2( , , $* , $ 3.pp 8.1. - y = ln x x1 = 3 x2 = 8:< ; . (8.1):pZ 8qpvZ 8uut !2Z 8 px2 + 1pl = p 1 + ((ln x)0 )2dx = p 1 + x1 dx = p x dx =333pxdx = tdt 0= x2 + 1 = t0 x2 = t2 ; 10 xdx = tdt0 dx=x x2 t2 ; 13911 Z3 t2dtZ3 0 t2 ; 11t( 3) = 20 t( 8) = 3 = t2 ; 1 = @ t2 ; 1 + t2 ; 1 A dt =pp223dt1t;1= dt + t2 ; 1 = (3 ; 2) + 2 ln t + 1 =222 3;1!12;1= 1 + ln; ln= 1 + 1 ln 3 :Z3Z32 3+12+12 22, 3 6 ( $ , $ 3, p* py = ln x < 30 8] (6 .
8.1). >;. 8.1 8.2. - 6 $ , x = R(cos t + t sin t)0 y = R(sin t ; t cos t)0 t 2 <0 ] (R > 0):< (8.2), x(t) = R(cos t + t sin t)0 y(t) = R(sin t ; t cos t)0 = 00 = :l==Z rh0ihi(R(cos t + t sin t))0 2 + (R(sin t ; t cos t))0 2dt =Z q0R2(; sin t + sin t + t cos t)2 + R2 (cos t ; cos t + t sin t)2dt =2 2tR= jRjjtjdt = Rtdt = R 2 = 2 : >ZZ0039202 -, , $./ 3 , x(t) = R(cos t + t sin t)0y(t) = R(sin t ; t cos t) { <0 ]: (C { 8.2.);.
8.2 8.3. - = .< C = $ , , $ * ({ * ), * * ( { )( . 8.3). D = 2: 9(, ( , = 2 <0 2]:;. 8.3 (8.3): = 00 = 20 () = )l=Z2q02+ (0)2d =393Z2p01 + 2d: * " (" ( . ).2 22Z221 + d = 1 + ; pd =21+0001Z20 2 + 1p1A d == 2 1 + 42 ; @ p;pZ2pp0p= 2 1 + 4 ;p2= 2 1 + 42 ;pZ2p01 + 2Z21 + d + p d 2 =1+0221 + 2d + ln j + 2 + 1j =0Z2ppZ2p01 + 2= 2 1 + 42 + ln(2 + 42 + 1) ;01 + 2d pp 2 121 + d = 2 2 1 + 4 + ln(2 + 4 + 1) =Z2p0p2pp= 42 + 1 + 12 ln(2 + 42 + 1): > 8.4. % , py = arcsin x + 1 ; x2:< D , ( "x": @ , ( . % 6 <;1 +1]./, $ p , y = arcsin x + 1 ; x20 ;1 x 1: (C { .8.4.) 9 y0 = p 1 2 ; p x 2 = p1 ; x 21;x1;x1;x394{ <;1 1], (8.1) .
* (. 1 2 { $ . F 3 , ** * x (;1 + 1) (1 ; 2) (A , B . 8.5):py = arcsin x + 1 ; x2 x 2 <;1 + 10 1 ; 2] 6 l : G$ (8.1). ; 1 2 *. @ 1 2 * l l l ( 3 :121 21 2l =1 2Z s1;2;1+1p1 + (arcsin x + 1 ; x2)02dx =vv1Z;2 uZ u2uu(1;x)t1 +t1 + 1 ; x dx ==dx=21;2=;1+1v1Z;2 uu1;xp1+xp1;2d(x+1)==221+x=;1+11+xq p qt 2;1+1;1+1= 2 2 2 ; 2 ; 1 :H , p p * 1 2 $ 2 2 2 = 4: /,lim! l = 4 , 4.120!01 2;; .
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