1. Интегралы ФНП Диф_ур (853736), страница 3
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*+ !, " " ! &Z sin6 xcos9 x dx?3. (, !:!+ !" A B , Z a1 sin x + b1 cos xa sin x + b cos x dx = Ax + B ln ja sin x + b cos xj + C:(! a1 b1 a b { !"1 a2 + b2 6= 0).1. 5.2, 5.1.2. <) ! sin x = t, sin6 x = sin6 x cos x = sin6 x cos x:cos9 x cos10 x(1 ; sin2 x)53. 5 " a1 sin x + b1 cos x !+: :a1 sin x + b1 cos x = A(a sin x + b cos x) + B (a cos x ; b sin x):425 4" sin x cos x, !! ") Aa ; Bb = a 1Ab + Ba = b1:+ b1b 1 B = ab1 ; ba1 , !:=8 , A = aa1a2 +b2a2 + b2Z a1 sin x + b1 cos xa sin x + b cos x dx =Zx + b cos x dx + ab1 ; ba1 Z a cos x ; b sin x dx == aa1a2 ++ bb21b aa sinsin x + b cos xa2 + b2 a sin x + b cos x+ b1b Z dx + ab1 ; ba1 Z d(a sin x + b cos x) dx == aa1a2 +b2a2 + b2a sin x + b cos x+ b1b x + ab1 ; ba1 ln ja sin x + b cos xj + C:= aa1a2 +b2a2 + b2. !:x ; b sin x =(Ax + B ln ja sin x + b cos xj + C ) = A + B aa cossin x + b cos xx + (Ab + Ba) cos x = (Aa ; Bba) sinsin x + b cos x+ b1b B = ab1 ; ba1 :Aa ; Bb = a1 Ab + Ba = b1 ) A = aa1a2 +b2a2 + b2043 6 (/,1 $ ) < $ .
< $ , * $ , , < $ . - ( R(u v) { $ Z pn u v:) ' ) R(x ax + b)dx a 6= 0 n 2 , aZb { . (pnpny = ax + b R(x ax + b)dx $ . 2 ,n;bpnyny = ax + b- y = ax + b- x = a - dx = na yn 1 dy:> !,Z yn ; b !ZZpnnR(x ax + b)dx = a R a y yn 1dy = na R1(y)dy: yn ; b !5 R1(y) R a y yn 1 { $ y:0 v1Z B uu+ b CA dx n 2 { - a b c d { !) R @x tn axcx + d , ad ; cbv6= 0:uu + b ax + b9 y = tn ax cx + d = yn: ' cx+dn;bn 1 (ad ; bc)dyny x = a ; cyn dx = (a ; cyn)2 dy:> !, ;;;;Z0 v1uZuR B@x tn ax + b CA dx =cx + d dyn ; b ! n(ad ; bc)yynRa ; cyn44(a ; cyn)21;dy =Z= R1(y)dy dyn ; b ! n(ad ; bc) R1(y) R a ; cyn y (a ; cyn)2 yn 1 { $ y: 6.1.
uZvut 6 ; x dx:x ; 1489>>6;xv2>>y=>>2Z uZ<=u6;xyx;14tx ; 14 dx = >>> dx = 16y >>> = 16 (y2 + 1)2 dy =:(y2 + 1)2 Z 2ZZ dyy ;= 16 (yy2 ++1)1 2 dy ; 16 (y2 dy=16;8+ 1)2y2 + 1 y2 + 1Z;8 y2dy+ 1 = 8 arctg y ; 8 y2 y+ 1 + C =;Zvvuuuu6;xx (x ; 14) + C:t= 8 arctg x ; 14 + t x6 ;; 14 ppm n) R x x ::: x dx n 2 ::: m 2 { .! S # !* n ::: mS = 1Efn ::: mg: 1 S = nl ::: S = mk- l k { .9 ppyS = x dx = SyS 1 dy- n x = yl ::: m x = yk :9 ,Z ZZ ppR x n x ::: m x dx = R yS yl ::: yk SyS 1 dy = R1(y)dy R1 (y) R yS yl ::: yk SyS 1 { $ y:4 $ 0 !!* .1vvuuZ B u+ b ::: mut ax + b CA dy n 2 ::: m 2 { ) R @x tn axcx + dcx + d- a b c d { , ac ; bd 6= 0:45;;; 6 $$ $$ -xm (a + bxn)pdxa b { , - m n p { ,p 6= 0: ..'1"2#.0 ( & * + ( (:1) p { , 2) m + 1 { , 3) m + 1 + p { .nnP ! < .1) p { .0 ! r # !* m n ( x = tr )Z mZpnprx (a + bx ) dx = R(x x)dx = dx = rtr 1dt =;ZR(tr t)rtnZ=dx = R1(t)dt:( p { ! .
9 xn = z :ZZZ11(1=n) 1mnpm=npx (a + bx ) dx = z (a + bz ) n zdz = n (a + bz )pz (m+1)=n] 1dz:2) m n+ 1 { .pp( p = sl : ' t = S a + bz = S a + bxn :ZZZ pxm (a + bxn)pdx = R z S a + bz dz = R1 (t)dt:1;;;3) m n+ 1 + p { .ZZ a + bz !pp(m+1)=n] 1(a+bz ) zdz =z p+(m+1)=n];;z460 v1uu1dz = R B@z tS a + bz CA dzZzvupus { p: 5 t = tS a +z bz = S ax n + b 10 vZZ B uuS a + bz CtA dz = R1(t)dt R @z;z$ t: 6.2. Z q3 1 + p4 xpx dx: < m = ; 21 n = 14 p = 31 : ( m n+ 1 == ;(11==2)4 + 1 = 2 ( 2). 1! q3t = 1 + x1=4- x1=4 = t3 ; 1- x = (t3 ; 1)4- dx = 4(t3 ; 1)33t2dt:> !, Zx1=2;ZZ(1 + x ) dx = (t ; 1) t 12t (t ; 1) dt = 12 t3(t3 ; 1)dt =4 1=3322;33r7 r1212741=4= 12 t dt;12 t dt = 7 t ;3t +C = 7 1 + x ;3 1 + x1=4 4 +C: ZZ633389m>>>Z< x = n sin t >=p 2 22=a) R(x m ; n x )dx = >> dx = m cos tdt >>:Zmn!mZ= R n sin t m cos t n cos tdt = R1 (sin t cos t)dt:98m>>x = n tg t >>>>>>Z=p 2 22<mdt!) R(x m + n x )dx = > dx = n cos2 t > =>p 2 2 2 m >>>>>: m + n x = tg t 47! m dtZm= R n tg t tg t n cos2 t = R1 (sin t cos t)dt:89m>>>>x = n cos t >>>>Zp 22 2<=msint) R(x n x ; m )dx = > dx = n cos2 t > =>p 2 2 2 m sin t >>>>>: n x ;m =costZ m m sin t ! m sin t Z= R n cos t cos t n cos t = R1(sin t cos t)dt:Zp) R(x ax2 + bx + c)dx:( &:1) a > 0:p 2ppax + bx + c = t ; ax- ax2 + bx + c = t2 ; 2 axt + ax2 ,pat2 + bt + cpa2;ctx = 2pat + b - dx = 2 (2pat + b)2 dt:ZpR(x ax2 + bx + c)dx =Z 0 t2 ; c pat2 + bt + cpa 1 pat2 + bt + cpaZA2pp= R @ 2pat + b dt=R1(t)dt:2 at + b(2 at + b)22) c > 0:p 2ppax + bx + c = xt + c- ax2 + bx + c = x2t2 + 2xt c + c:pct ; bpct2 ; bt + pcap2ax + b = xt2 + 2t c- x = a ; t2 - dx = 2 (a ; t2)2 dt:ZpR(x ax2 + bx + c)dx =Z 0 2pct ; b pct2 ; bt + apc 1 pct2 ; bt + pcaZA2= R @ a ; t2 dt=R1 (t)dt:a ; t2(a ; t2)23) ( x1 x2 { 2ax + bx + c = 0 :ax2 + bx + c = a(x ; x1)(x ; x2):48Zmp9 ax2 + bx + c = t(x ; x1):a(x ; x1)(x ; x2) = t2(x ; x1)2- a(x ; x2) = t2(x ; x1)2;ax2 + x1 tx = t2 ; a - dx = 2a((tx2 2;;ax)21)t dt:ZpR(x ax2 + bx + c)dx =Z 0 ;ax2 + x1t2 a(x1 ; x2)t 1 2a(x2 ; x1)tZ@A= Rt2 ; a t2 ; a t (t2 ; a)2 dt = R1(t)dt: !" #$"1.
* ! ! ! &Z p1 ; x + p1 + xpdx?22( 1 + x)32. * ! ! &ZpppR(x l ax + b m ax + b n ax + b)dxl 2 m 2 n 2{ " !1 a b { !" !.3. 5.>. %"8.Zp4. 5 ) ") ) n & 1 + xndx "! 4"?5. * ! "+! ! ??1. 5!sx x = 1 ; t2 dx = ;4tt = 11 ;+x1 + t2(1 + t2)2! & Z t(t + 1); t2 + 1 dt:2. 5! S { 8 : ! l m n. @& ! Sqt = S (ax + b) x = t a; b dx = Sa tS 1:;494. Zp1 + xndx Zx0(1 + xn )1=2dx:@& 5.>. %"8 & "! p 4" ) ! ):) n1 { , !, n = 11) n1 + 12 { , !, n = 2:50 7 , , - ) $" . ( $ f (x) : a b ] , .. f (x) 0 x 2 : a b ].
P $, $ $ y = f (x) x = a, x = b, y = 0 (. 7.1). > $ .1# { ! * . 2 < ! : a b ] n x0 = a < x1 < : : : < xk 1 < xk < : : : < xn 1 < xn = b(7:1) < , Oy (. .7.2).> ! n ,) .;;;; . 7.1 !. 7.2T xk = xk ; xk 1 (k = 1 : : :n) = 1maxT xk (7:2)k n ) :xk 1 xk ] ! ! k (k = 1 : : : n).51;;P $, * T xk f (k ). (* Sn < $ $Sn =nXk=1f (k ) T xk :U, T xk < $ ! .(< " S * $ !nXS = lim0 Sn = lim0 f (k ) T xk(7:3)!!k=1( , * ! ! : a b ] , ! k ).!) / .
( ) Ox a b F , ) F $ x , ..F = F (x). 1 ! F * a b. 2 < ! : a b ] n (7.1), ! (7.2) ) :xk 1 xk ]! ! k (k = 1 : : : n). > ! F = F (x) ) :xk 1 xk ] !) F (k ) T xk , : a bn] ! A < ) X!) F (k ) T xk .k=1( < nXA = lim0 F (k ) T xk(7:4);;!k=1( ) , *) " a b..
! ) (7.3), (7.4), 52 . E # ) , $, . < .. ( $ f (x) : a b ]. P! : a b ] n x0 = a < x1 < : : : < xk 1 < xk < : : : < xn 1 < xn = b !T xk = xk ; xk 1 (k = 1 : : :n) = 1maxT xk k n(T xk { k- , { ! # !).
) :xk 1 xk ] ! ! k (k = 1 : : : n) nXf (k ) T xk (7:5);;;;k=1 f (x) : a b ]. 7.1. 0 * (7:5) ! 0, < !! : a b ] , ! k , $ f (x) : a b ], f (x) : a b ]Zb ! f (x) dx.a> !,ZbnXf (x) dx = lim0 f (k ) T xk :(7:6)a!k=1/ 7.1. f (x) : a b ], * .. () :f (x) : a b ].
> ! ! : a b ] :xk 1 xk ] (k = 1 : : : n) :xk 1 xk ], $ f (x) ) .53;0;0! k 2 :xk 1 xk ] (k = 1 : : : n) (7.5). > 1, 2, : : : k 1, k +1, : : : n $ ! k . > (7.5) ;0;00nXk=1f (k ) T xk =kX0 1;k=1f (k ) T xk + f (k ) T xk +00nXk=k0+1f (k ) T xk :, ! A , nXf (k ) T xk = A + f (k ) (xk ; xk 1):(7:7)0k=100;> A { , $ f (x) :xk 1 xk ], * ! k 2 :xk 1 xk ], f (k ) ) ! #( ! ). (< (7.7), (7.7), ) ! #.
9 , (7.7) , $ f (x) : a b ].. $ . >, , $ 28<x ; f (x) = : 10 x ; , : a b ].2 , ! ! : a b ] , ) k , (7.5), . E ) ) k , (7.5) ! !, .. b ; a. , $ 2 (7.5) ! 0, , * ! k , , ..$ 2 : a b ].0;00;000540 .2 7.2. &"( :) f (x) : a b ]!) f (x) : a b ] * ) f (x) : a b ] , f (x) ".: a b ] , ,Zbaf (x) dx2 < # .' * $ (7.3) , $ f (x) : a b ], " S * " -ZbS = f (x) dx:a < . -E , $ (7.4) , : a b ] F (x) " a b $ZbS = F (x) dx:a #4$ # I.
f (x)Za#4$ # II.: a b ],ax = a,f (x) dx = 0: Za f (x)Zb f (x) dx = ; f (x) dx:ab9 I II .55#4$ # III. Z dx = b ; a:ba 7.1. f (x) 1, < (7.5) :nnXXf (k ) T xk = 1 T xk = b ; a:k=1k=19 ,.ZbanXdx = lim0k=1!f (k ) T xk = lim0 (b ; a) = b ; a:!9 III .#4$ # IV (4$ ! 6 6).) f (x) : a b ] c { , c f (x) : a b ] ZbaZbc f (x) dx = c f (x) dx:a!) f (x) g (x) : a b ], f (x) + g (x) : a b ] ZbaZbZbaaf (x) + g (x) dx = f (x) dx + g (x) dx:( ! ! : a b ] !! k :nnXXc f (k ) T xk = c f (k ) T xk (7:8).k=1n Xk=1k=1f (k ) + g (k ) T xk =nXk=1f (k ) T xk +nXk=1g (k ) T xk :(7:9)> f (x) g (x) : a b ], ! 0 ) $ (7.8) (7.9) *. ( ,56 $ (7.8) (7.9) ) * nnXXlimcf()Tx=climf (k ) T xk kk 0 0k=1!lim 0!n Xk=1!f (k ) + g (k ) T xk = lim0!nXk=1f (k ) T xk + lim0..
( 7.1)ZbaZbZb aac f (x) dx = c f (x) dx-k=1!nXk=1g (k ) T xk ZbZbaaf (x) + g (x) dx = f (x) dx + g (x) dx:9 IV .mX. = ! ci fi(x) i=1 : a b ] $ fi(x) : a b ] 1Zb 0 Xmm Zb@ ci fi (x)A dx = X ci fi(x) dx:i=1ai=1a#4$ # V. ) f (x) : a b ] a < c < b, f (x) ( ZbaZcZbac: a c ], : b ] -f (x) dx = f (x) dx + f (x) dx:!) f (x) ( : a c ] : b ], f (x) : a b ] ZcaZbZbcaf (x) dx + f (x) dx = f (x) dx:9 V ! ..
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