1. Интегралы ФНП Диф_ур (Лекции Интегралы ФНП)
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) ) .0 -! , * ( ) , , ). 0 , , * . ) #. 1 # < , { >.2 ! : "'", "3 ", "2$$ ".4 ! 5 0.6., ' .'. !# *, .3 1 1 # * : $ F (x), ! , .. $ f (x), f (x) = F (x):9 ! ! : ! $ F (x), $ f (x), ..F (x) = f (x). 1.1. 3 F (x) $ f (x) :a b], < F (x) = f (x). 1.1. 1 ! $ f (x) = x2:' ! , 0 313xF (x) = 3 @ x3 A = x2:= , $ f (x) * !, < ! 3 .
3 >, * ) F1(x) = x3 + 1 F2 (x) = x3 ; 3 ,3x!*, F (x) = 3 + C , { , 10 3x@ + C A = x2:3 1.1. F1(x) F2(x) f (x) :a b], 00000.. ! :F1 (x) = f (x) F2(x) = f (x) 8x 2 :a b]:! '(x) = F1 (x) ; F2 (x). >, ,' (x) = F1(x) ; F2(x) = f (x) ; f (x) = 0 8x 2 :a b]:4000001 , ' (x) 0 :a b] , '(x) C :a b]. 2, $ '(x) ! x 2 (a b] =) :a x], $$ :a b] , , $$ :a x]. (< =) * 2 (a x), '(x) ; '(a) == ' ( )(x ; a) = 0, ..
'(x) = '(a) 8x 2 (a b]: ! '(a) = C '(x) = C 8x 2 :a b]:> . 1.2. 9 ! $ f (x) f (x) ! Zf (x) dx: 1.3 0 $ F (x) ! $ f (x), ) F (x) + C , C { , f (x).> !, ,00Zf (x) dx = F (x) + C(1:1) F (x) = f (x).( < $ f (x) , R) f (x) dx { , { .1) ! $ f (x) f (x). : $ f (x) * ? : , .0 1.2 (). f (x) :a b], " , ," .5 () , * .1) ( $ , .., F (x) = f (x), 0Zf (x) dx = (F (x) + C ) = f (x):00(1:2)2) 2$$ ):dZf (x) dx = d(F (x) + C ) = (F (x) + C ) dx = f (x) dx:03) 1 $$ $ < $ :ZZdF (x) = F (x) + C(1:3)F (x) dx = F (x) + C:(1:4), 3) ) 0 . 1.2.Z 1.3.ZZ1p 2 dx = d (arcsin x) = arcsin x + C:1;x pZ p;x2p 2 dx = d 1 ; x = 1 ; x2 + C:1;x4) ( ) ) , ..ZZa f (x) dx = a f (x) dx a { .6(1:5)2, 1) Za f (x) dx = a f (x)0 ZZa f (x) dx = a f (x) dx = a f (x).. ) a R f (x) dx a R f (x) dx ) $ a f (x): 9, ( , ) ) ) $ ).5) 1 $ , ..Z0Z0Z(f1(x) + f2(x)) dx = f1(x) dx + f2(x) dx:(1:6)E * , (73):.ZZZ(f1 (x) + f2 (x)) dx = f1(x) + f2(x)0ZZf1(x) dx + f2(x) dx = f1 (x) + f2(x) = f1(x) + f2(x):> !, (73) ) !.
9, )Z00Z0Z(f1(x) + f2(x)) dx f1(x) dx + f2 (x) dx ! ! ) $ f1(x) + f2 (x). 5, .6) 0Zf (x) dx = F (x) + CZ(1:7)f (ax + b) dx = a1 F (ax + b) + C:2,d Z f (ax + b) dx = f (ax + b):dx7! t = ax + b ) $ , 1! 1 ddt = 1 f (ax + b)a = f (ax + b):F(ax+b)+C=(F(t))aa dtdx aG 6):0Zf (x + b) dx = F (x + b) + CZ F (x) = f (x):f (ax) dx = a1 F (ax) + C01)2)3)4)5)6)7)8)9)10)ZZ 0 dx = C+1x dx = x + 1 + C 6= ;1Z 1x dx = ln jxj + CZsin x dx = ; cos x + CZcos x dx = sin x + CZ 1cos2 x dx = tg x + CZ 1sin2 x dx = ; ctg x + CZ xe dx = ex + CxZaxa dx = ln a + CZ 11 + x2 dx = arctg x + C811)12)13)14)15)16)17)18)ZZZZZZZZZ1 dx = 1 arctg x + C a 6= 0a2 + x2aa1 dx = 1 ln 1 + x + C1 ; x22 1;x1 dx = 1 ln a + x + C a 6= 0a2 ; x22a a ; x p 1 2 dx = arcsin x + C jxj < 11;xp 21 2 dx = arcsin xa + C jxj < jaja ;xpp 21 2 dx = ln jx + x2 + a2j + C (a 6= 0)a +xpp 21 2 dx = ln jx + x2 ; a2j + C jxj > jaj > 0x ;ash x dx = ch x + Cch x dx = sh x + CZ dx20)= th x + Cch2 xZ dx21)= ; cth x + C:sh2 x9 $ ! ) ! $$ .1, $ 3)Z 1x dx = ln jxj + Ca) x > 0, (ln jxj) = (ln x) = x1 !) x < 0, (ln jxj) = (ln(;x)) = ;1x (;1) = x1 :19)00009 $ 11)Z1 x !!1 dx = 1 arctg x + Ca2 + x2aa211a1 :1arctg===2 2222aaa 1 + x a (a + x )a a + x2a2 $ 13)Z1 dx = 1 ln a + x + Ca2 ; x22a a ; x !1a+x = 1 (ln ja + xj ; ln ja ; xj) =ln2a a ; x 2a!+a+x = 1 := 21a a +1 x + a ;1 x = 21a a ;ax2 ;x2a2 ; x2 $ 15)Zp 21 2 dx = arcsin xa + Ca ;x!x1 1=p a 1=p 1 :arcsin a = vu2ua2 ; x2 aa2 ; x2t1 ; x aa2B $ 16)!p 2 21xp1+ p 2 2 =ln jx + x + a j =x + x2 + a2x +ap2 2p x2 + a2 +p x2 2 = p 21 2 :=(x + x + a ) x + ax +a4 $ ! .00000102 ) ! * $ 2) ! :2ZZx1: 1 d x = x + C 2: x dx = 2 + C #.Z3x3: x dx = 3 + C -4:2p5: px dx = 2 x + C Z 1Zp32x dx = 3 x + C -pZ dx6: x2 = ; x1 + C: !" #$"1.
y = f (x) (c. 1.1).2. , ! " "!?3. % "! " & f (x)? (c. 1.2).4. !! & .2. ( (c. 1.1).11 2 % & ! . & * . 2.1. $ " Z f (x) dx x = '(t), '(t) { ,&" & &. 'ZZf (x) dx = f ('(t)) ' (t) dt:(2:1)0() , (2.1) * !: (2.1) t ! $ t = K(x), K(x) { ! $ $ x = '(t). 2 (2.1) $$ x :d Z f (x) dx = f (x)dxZ d Z dtddx f ('(t))' (t) dt = dt f ('(t))' (t) dt dx == f ('(t))' (t) d1x = f ('(t))' (t) ' 1(t) = f ('(t)) = f (x):dt> !, , , (2.1). 2.1.
( $ ! t = (x), x = '(t). > dt = (t) dx:1,Z (x)Z dtdx=(x)t = ln jtj + C = ln j(x)j + C:12.0000000. 1) eZ2 p>Zp 2dx 2 (a > 0):a ;xx = a sin t t = arcsin xa 2Zt dt = Z dt = t + C = arcsin x + C:p 2dx 2 = aa coscos taa ;x2) eZ2 >qa ; x = a2 ; a2 sin2 t = a cos t:2x dx :1 + x2t = 1 + x2 dt = 2x dx x dx = 12 dt:x dx = 1 Z dt = 1 ln jtj + C = 1 ln(1 + x2) + C:1 + x2 2 t 22 Z 2.2.$ u(x) .
'Z.v(x){ Zu dv = uv ; v du:(2:2)( $$ d(uv) = u dv + v du:1 ZZZd(uv) = u dv + v du:13(2:3)> Zd(uv) = uv + C C ! (2.3), ae ZZuv = u dv + v du (2.2).> .Z3 (2.1) Z u dv v du. < ! ! .Z u dv $ (2.1) . 2.1. eZ>Zx sin x dx:01u=xdv=sinxdxx sin x dx = @ du = dx v = ; cos x A =Z= ;x cos x + cos x dx = ;x cos x + sin x + C: $ , $ . E , !# , , ) ! * .I. E , $ ) $ ln x arcsin xarccos x arctg x: 2 $ (2.1), u(x), # $ . 2.2.0 u = ln xn dx1Zdv=xxn+1 ln x;n+1 A =xn ln x dx = @n+1du = x1 dx v = nx + 114Zn+1n+1n+1xxx; (n + 1)x dx = n + 1 ln x ; (n + 1)2 + C: 2.3.0 u = lnk xZdv = xn dx 1nkk 1xn+1 A =x ln x dx = @klnxdu = x dx v = n + 1;n+1Z xnxk= n + 1 ln x ; n + 1 k lnk 1 x dx k 2:, Zxn lnk 1 dx) $ (2.1).
2.4.0u = arctg x dv = x dx12Zx@A2x arctg x dx == 2 arctg x;du = 1 +dxx2 v = x2;;Z2+1;12xx; 2(1 + x2) dx = 2 arctg x ; 12 x + 12 arctg x + C: 2.5.0u = arcsin x dv = dx1ZA = x arcsin x;arcsin x dx = @dx v=xdu = p1 ; x2Zp; p1 x; x2 dx = x arcsin x + 1 ; x2 + C:II.
E :ZZZ(ax + b)n sin x dx (ax + b)n cos x dx (ax + b)n e x dx a b { - n { ! ).' ! n- $ (2.1), u(x) ) ! (ax + b) * . (15) < ! ) . 2.6.0nx dx1ZZu=xdv=enxnx@Ax e dx = du = n xn 1 dx v = ex = x e ; nxn 1 ex dx:2 R xn 1 ex dx ) $ (2.1). 2.7.01nZu=xdv=sinxdxxn sin x dx = @du = n xn 1 dx v = ; cos xA =;;;; 2.8.Z= ;xnZcos x + n xn 1 cos x dx:0xn cos x dx = @;1u = xndv = cos x dxA =du = n xn 1 dxv = sin x;Z= xn sin x ; n xn 1 sin x dx:R n 1E)#,x cos x dxR n 1 x sin x dx ) $ (2.1).III.
E :Z xZ xZZe cos bx dx e sin bx dx sin(ln x) dx cos(ln x) dx:! ! < I , ae I , ) . , I ) ! C , < * I) ) ! C. 2.9.01ax Z axu=edv=cosbxdxA=I = e cos bx dx = @sinbxaxdu = a e dxv= b;;;16bx ; a Z eax sin bx dx:b Z b2 eax sin bx dx * $ (2.1). aeax sin bx a eax cos bx a Zeax cos bx dx) =;(;+eI=bbbbax sin bx aeax cos bx a2e; b2 I:(2:4)=b +b2> !, $ (2.1) I , + b sin bx eax:I = a cos bx(2:5)2a + b2 * , (2.5) C , ZI = eax cos bx dx , Z+ b sin bx eax + :I = eax cos bx dx = a cos bxa2 + b24 Z ax; b cos bx eax + :e sin bx dx = a sin bxa2 + b2 2.10.0u = sin(ln x)1Zdv=dxA=sin(ln x) dx = @1du = cos(ln x) x v = x= eax sinZ= x sin(ln x) ; cos(ln x) dx:0 u = cos(ln x)1Zdv=dxA=cos(ln x) dx = @1du = ; sin(ln x) x v = x17Z= x cos(ln x) + sin(ln x) dx:' Zsin(ln x) dx = x sin(ln x) ;2 x cos(ln x) + CZcos(ln x) dx = x sin(ln x) +2 x cos(ln x) + C:E, ! , .
( , * . 2.11.01 dx1Z xZ cos xu=xdv=2@Asin xsin2 x dx = du = dx v = ; ctg x = ;x ctg x + sin x dx =Z d sin x= ;x ctg x + sin x = ;x ctg x + ln j sin xj + C: 2.12.p2 210Zpp2 2x+adv=dxu=A=xx2 + a2 dx = @x +a ;du = p 2x 2 dx v = xx +aZZ x2 + a2 ; a2p2 2 Zp2 22; px2 + a2 dx = x x + a ; x + a dx + a pxd2 x+ a2 dx:9,ZZpp2 x2 + a2 dx = x x2 + a2 + a2 p d2 x 2 :x +a> !, Zp 2dx 2 x +aZpp 2 2 a2p122x + a dx = 2 x x + a + 2 ln jx + x2 + a2j + C:184,Zp 2 2 a2p2 21x ; a dx = 2 x x ; a ; 2 ln jx + x ; a j + C:p22 !" #$"1.
") & (c. 2.1).2. *+ &) Z, (x),(x) dx:03. & ! & (c. 2.2).4. ." ! &":Zx + 4x dx:1) 2 arctg1 + x2Z2) cos xx ;cosx xsin x dx:2.4.t = ,(x) 1Z dt = , (x) dx Z2, (x),(x) dx = t dt = t2=2 + C = (, (2x)) + C:000Z arctg xZ xZ 2 arctg x + 4x1)1 + x2 dx = 2 1 + x2 dx + 4 1 + x2 dx(c.
!! 4), 5) & &).ZZ arctg x(arctg x)2 + Cdx=(arctgx)(arctgx)dx=1 + x22(c. ! 2 32).Z xZ 2xZ (1 + x2)2dx=1=2dx=1=21 + x21 + x21 + x2 dx = 1=2 ln(1 + x ) + C(c. 2.1).,Z 2 arctg x + 4x22dx=(arctgx)+2ln(1+x) + C:21+x2)Z (x cos x)Z cos x ; sin xx cos x dx = x cos x dx = ln(x cos x) + C(c. 2.1).00019 3% ' < # !, ! $, * !) M4. !# ) . >, ! ) ) N ( < ) , ) .
1, ) N ,)* 2{5. G! ! ! , ! , 0 ! )Z ( .(! ) ! ! ! Z (, 2 : 5), 0, ! ! ! , , !) ( .E $ ), , ) ! ! ! , , ) p (, , 2). ( , ! ) R ( . < ) p (, , ;1). ' , ! ) ( .(# , ! ! i. , ) (;1), < pi2 = ;1. ) , ;1 = i. , ) * !.20 3.1.) )a + bi a b { i { .2 a b a + bi.' !:z = a + bi, Re z = a, Im z = b.1 ), , ), , ! , i2 = ;1.
3.1. 1(1 ; i)(1 + i):(1 ; i)(1 + i) = 12 ; i2 = 1 ; (;1) = 2:E a + 0i a + bi (b 6= 0):C 0 + bi (b 6= 0):' # , ax2 + bx + c = 0 , D = b2 ; 4ac < 0: < ) # . ) p , D D < 0.21 3.2. 1x2 + 2x + 2 = 0:qppp5 D = 4 ; 8 = ;4 D = ;4 = 4 (;1) = 2 ;1 = 2i x12 = ;2 2 2i = ;1 i: 3.2. G z = a + bi z = a ; bi (a b { ) .9) ! * .1 : 0 z { , z = z .2 ! z1 z2 z :2 : z1 + z2 = z1 + z23 : z1 z2 = z1 z2z ! z4 : z1 = z1 (z2 6= 0)225 : z n = zn:9 1 2 ) .2) 3 . ( z1 = a + bi z2 = c + di ( a b c d { ).
> z1 = a ; bi z 2 = c ; di z1 z2 = (a + bi)(c + di) = (ac ; bd) + i(ad + bc)z1z2 = (a ; bi)(c ; di) = (ac ; bd) ; i(ad + bc):' z1 z2 = z1 z2:2) 4 .z1 = a + bi = (a + bi)(c ; di) = (ac + bd) + (bc ; ad)i =z2 c + di (c + di)(c ; di) (c2 + d2) + (cd ; cd)i+ bd + bc ; ad i= acc2 + d2 c2 + d222z1 = a ; bi = (a ; bi)(c + di) = (ac + bd) + (ad ; bc)i =z2 c ; di (c ; di)(c + di)(c2 + d2)+ bd ; bc ; ad i:= acc2 + d2 c2 + d2' 4 .9 5 (n ; 1)- 3 :z 2 = z z = z 2 z 3 = z 2 z = z 2 z = z2 z = z 3 ::3 P (z ) P (z ) = A0 + A1z + A2z 2 + + Anz n z { ,A0 A1 An { <$$ , . !* <$$ .0 An 6= 0, P (z ) n ! Pn (z ).0 An = 0, An 1 = 0 Am+1 = 0, Am 6= 0, P (z ) = Pm (z ) { m. ( m = 0 P (z ) = P0 (z ) = A0(6= 0) { (, ).0 P (z ) 0, , .
# ! ( !), ! n ) ;Pn(z ) = (z ; z1)(z ; z2) (z ; zn)An z1 z2 zn { .& .23, Pn(zi ) = 0 (i = 1 2 n):G z1 z2 zn .0 , .0 , !O * ) , Pn(z ) = (z ; z1)k (z ; zm)km An1 z1 zm .( k1 km ,, z1 zm.( Pn(z ) n #, n, ) , , ) , ! n n , , ,k1 + + km = n:& ) n = 0, , ( 0 )., P0 (z ) = A0 6= 0.2 ) ) . 3.3. 1P2(z ) = z 2 + 2z + 2:> < z1 = ;1 + i z2 = ;1 ; i (: 3:2) z 2 + 2z + 2 = (z ; z1)(z ; z2) = (z + 1 ; i)(z + 1 + i): % <$$ ! , * * .24 3.1.