1. Интегралы ФНП Диф_ур (853736), страница 2
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*-Pn(z ) = A0 + A1z + + Amz m + + Anz n(3:1) z & z, .. P Amz m. 5 z z. > Am zm = ( 5 ) = Amz m = (1 ) = Am z m = (3 ) = Amz m:( m = 0 1 n, , ) $ (3:1) ) z z, , 2Pn(z ) = Pn(z ) . 3.2. z0 { * Pn(z ), z0 * .. ( 3:1 Pn(z 0) = Pn(z0). 4 Pn(z0) = 0, Pn (z0) = 0 Pn(z 0) = 0. 9 , z0 { Pn(z ). > .> , <$$ ( ) ).( z0 = a + bi z0 = a ; bi (a b { ) { Pn(z ) <$$ . > ) Pn(z ) ! )(z ; z0) (z ; z 0).
' (3:2)(z ; z0)(z ; z0) = z 2 + pz + q:( p = ;(z0 + z0) = ;2a q = z0z0 = a2 + b2: , (3:2) <$$ p q. ( D = p2 ; 4q < 025 .() ), )! Pn(z ) = (z ; z0)(z ; z 0)Pn 2(z ) = (z 2 + pz + q)Pn 2(z );;(3:3) Pn 2(z ) { (n ; 2). > Pn 2 (z ) ) Pn(z ) <$$ (z 2 + pz + q) p q, <$$ Pn 2(z ) ) .0 z0, , z0 Pn 2(z ), ;;;;Pn 2 (z ) = (z ; z0)(z ; z0)Pn 4 (z ) = (z 2 + pz + q)Pn 4(z ) Pn 4(z ) { (n ; 4) <$$ .' (3:3) ;;;;Pn(z ) = (z ; z0)2(z ; z0)2Pn 4 (z ) = (z 2 + pz + q)2Pn 4 (z ):() ) (z ;z0), (z ;z 0) Pn(z ), ;;(3:4)Pn (z ) = (z ; z0)l (z ; z 0)l Pn 2l (z ) = (z 2 + pz + q)l Pn 2l (z ) Pn 2l (z ) { (n ; 2l) <$$, z0, z 0 ) .' (3:4) , ;;;* ( !* ) *.
' - ) . 5, ) <$$ .Q , !* Pn (z ) * 26*:Pn(z ) = (z ; a1)k (z ; am)km (z 2 + p1 z + q1)l (z 2 + pr z + qr )lr An: (3:5)11= ) (z ; a1) (z ; am ) a1 am (k1 km { ), ) (z 2 + p1z + q1) (z 2 + pr z + qr ) ) (l1 lr { ) ). 2 ) .! " ( Pn(z ) Qn(z ) n.0 Pn(z ) = Qn(z ) 8 z , , Pn(z ) Qn(z )) #Pn(z ) Qn(z ):9$ ).1 . 0 Pn(z ), Qn(z ) <$$ , Pn (z ) Qn (z ).2 .
0 Pn(z ) = Qn(z ) n + 1 ( z1 z2 zn+1, Pn(z ) Qn(z ).Q 1 . 9 2 ) " ".2, * z0, Pn(z0) 6= Qn(z0). P R(z ) = Pn(z ) ; Qn(z ). > R(z0) 6= 0, R(z ) { , , . E n )! ! # n. 1, , R(z1 ) = 0 R(z2 ) = 0::: R(zn+1) = 0, n + 1 ) ! ! # n. ( , * " " , z0 Pn(z0) Qn(z0) ! , .. Pn (z ) Qn(z ).Q ) #, $ . P .27 3.4. ( A, B , C )Az (z ; 1) + Bz (z ; 2) + C (z ; 1)(z ; 2) z 2 + 2:(! , ) z :(A + B + C )z 2 + (;A ; 2B ; 3C )z + 2C = z 2 + 2:G! < ! ), <$$ z 9= 1 >>== 0 >= 2: >A+B+C;A ; 2B ; 3C2CP# < , A = 3, B = ;3, C = 1, ! ).5 ! 1 .
0 2 , # *.(!, ! z1 = 0,z2 = 1, z3 = 2. >z = 0 ) 2C = 2 ) C = 1,z = 1 ) ;B = 3 ) B = ;3,z = 2 ) 2A = 6 ) A = 3.5 $ ) A, B , C . 1 * () ) ). (< # ! . !" #$"1. * ! "+! !", ", ! "? (c. 3:1).2. * !" ! "+! !" !! +? (c. 3:2 !! 1 ; 5 ).3. * & "+! !" "? * ! ! ? & ! n? (c. & " ).4. % ! & ! !" 4,! & & !? (c. 3:1).285.
5! & ! !" 4 " z0. ! ! ! z0 " 4& & ? (c. 3:2).6. * "& & ! !" 4 " " ? * ! +! 4" 4)? % +! ! ? (c. (3:5)).7. ! !! ) & . (c. ! 1 ,2 ).29 4( )* < , $ .3 , ! ) , $ (), , ) ).P $ , , Pn (x) = A0 + A1x + A2x2 + : : : + Anxn:* ) ) * ): x2 = x x, x3 = x x x ..P ! ) $ R(x) = PPm((xx)) (4:1)n Pm (x), Pn(x) { m n .% $ . #Pm (x) n 1 &, , Pn(x)m < n (.. # ) , m n. ( n = 0Pn(x) = A0 6= 0 # PPm((xx)) , *, n1 A Pm (x).
9 , < , # 0 $ .1 , ! ), * $ R(x) ) (4:1). 4.1. x3R(x) = x + x ;1 9 + xx2 ;+94 :x33xx;95 x2 + 4 = (x ; 9)(x2 + 4) :30( !* R(x) ,432223x;8x+5x; 36x + 4 x(x;9)(x+4)+(x+4)+x=R(x) =(x ; 9)(x2 + 4)x3 ; 9x2 + 4x ; 36.. R(x) = PP4((xx)) .3#$" % 4.1.
P ! A Mx + N2(p; 4q < 0),k2l(x ; a) (x + px + q) A, M , N , a, p, q { , k l { , + .( $ ) ) ! #. 4.1. $ (4:1) , Pm (x) Pn(x) { * Pn(x) = An(x ; a)k : : : (x2 + px + q)l : : : (p2 ; 4q < 0 : : :):# An { + * ( xn),(x ; a)k { , &" & a k, ,&"( ,(x2 + px + q)l { , &" ( ( l, , &"( ( ( .' (4:1) + .- & (x ; a)k k (A1A2Akx ; a + (x ; a)2 + : : : + (x ; a)k 31 & (x2 + px + q)l l (M1x + N1 + M2x + N2 + : : : + Ml x + Nl :x2 + px + q (x2 + px + q)2(x2 + px + q)l)*A1 A2 : : : Ak M1 M2 : : : Ml N1N2 : : : Nl & .2 .1 ) ! (4:1) # ( *, <$$ A1 : : : Ak M1 : : : Ml N1 : : :Nl : : : , ) !(4:1) ).
4.2. 2+ 3x + 3 :R(x) = (x4x+ 1)(x ; 1)2 4:1 ) ! B + C :R(x) = x A++ 1 x ; 1 (x ; 1)20 <$$ , ! ! ! ! .2 <$$ ) !2x ; 1) + C (x + 1)R(x) = A(x ; 1) + B(x(x++1)(1)(x ; 1)2 !, ! # ! ! ) .> , , ! !) , * :A(x ; 1)2 + B (x + 1)(x ; 1) + C (x + 1) 4x2 + 3x + 3: , ! 1:4. ( 2 ) :x = 1 ) 2C = 10 ) C = 5,x = ;1 ) 4A = 4 ) A = 1,32x = 0 ) A ; B + C = 3 ) B = 3.( A, B , C # ! <$$ , ae )4x2 + 3x + 3 = 1 ; 3 + 5 :(x + 1)(x ; 1)2 x + 1 x ; 1 (x ; 1)2 E ) ! , $ R(x) !R(x) = PPm((xx)) n Pm (x), Pn(x) { m n .0 ! (m n), ) &, ..
R(x) = Pm n (x) + PPs ((xx)) (s < n)n Pm n (x) { (m ; n), !Ps (x) { ! (! R(x)).Pn(x)S ) , Pm (x) Pn (x), "".4x 4.3. 0 R(x) = x2 + 1 ;;x2 + 1x4x4 + x2 x2 ; 1 { .;x2;x2 ; 11 { .G ! , { ! :R(x) = x2 ; 1 + x2 1+ 1 :33' $ , !* , ! ZZZ Ps (x)R(x) dx = Pm n (x) dx + P (x) dx:nS Pm n (x) ! $ . 2 !Ps (x) , !* , # ! Pn(x) # !.
4.4. Z x4x2 + 1 dx;;Zx4 dx = (: 4:3:) = Z (x2 ; 1) dx + Z dx =x2 + 1x2 + 13x= 3 ; x + arctg x + C:5 ! ! #!, !. 4.5. Z 4x2 + 3x + 3(x + 1)(x ; 1)2 dx:5 ! .(< !, ) ! # , 4:2.Z dxZ dxZZ d(x + 1)Z 4x2 + 3x + 3dx(x + 1)(x ; 1)2 dx = x + 1 ; 3 x ; 1 + 5 (x ; 1)2 = (x + 1) ;Z d(x ; 1)Z d(x ; 1);3 (x ; 1) + 5 (x ; 1)2 = ln jx + 1j ; 3 ln jx ; 1j ; x ;5 1 + C = x + 1 = ln (x ; 1)3 ; x ;5 1 + C:34(# ! < ! ! A (x ; a)k ) $$ (x ; a):ZA(x ; a)k dx = AZ89>>Alnjx;aj+Ck=1<=d(x ; a) =A(x ; a)k >: (1 ; k)(x ; a)k 1 k 6= 1 > :;2 # ! ZMx + N dx (p2 ; 4q < 0)(x2 + px + q)k , $$ .Z M (x + (p=2)) + (N ; (Mp=2))p) =d(x+:(x + (p=2))2 + (q ; (p2=4))]k2Z (x + (p=2))d(x + (p=2))= M :(x + (p=2))2 + (q ; (p2=4))]k +!ZMpd(x + (p=2))+ N; 2:(x + (p=2))2 + (q ; (p2 =4))]k ( x + p2 x) ZZdx I = (x2x+dxa2)k Ik = (x2 +a2)k(4:2)2p a = q ; 4 > 0, p2 ; 4q < 0.' I $$ x2 + a2:Z d(x2 + a2) ( 11I = 2 (x2 + a2)k = 2 ln(x2 + a2) + C k = 12359=(4:3) 2(1 ; k)(x12 + a2)k 1 + C k 6= 1 :' Ik k = 1 !Z dxI1 = x2 + a2 = a1 arctg xa + C:( k > 1 ) $, Ik ) !:2ZZ (x2 + a2) ; a2xxdxxIk = (x2 + a2)k +2k (x2 + a2)k+1 = (x2 + a2)k +2k (x2 + a2)k+1 dx =ZZdxx2= (x2 + a2)k + 2k (x2 + a2)k ; 2ka (x2 +dxa2)k+1 == (x2 +x a2)k + 2k Ik ; 2ka2Ik+1:;1 Ik+1, & Ik+1 = 2ka2(xx2 + a2)k + 22kka;21 Ik (k = 1 2 ):( < $ I2 ) I1, I3 I2 ..
4.6. Zdx :(x2 + 4)2(4:4)& I2 a2 = 4.Z dx1 I1 = x2 + 4 = 21 arctg x2 + C:5 $ (4:4) k = 1 Zdx = x + 1 1 arctg x + C ! :(x2 + 4)2 8(x2 + 4) 8 22. ' Ik ) ) * x = a tg t36 )( ) )).>, (4:6) x = 2 tg t Z1 Z cos2 t dt = 1 Z (1 + cos 2t) dt =2 dt=(4 tg2 t + 4)2 cos2 t 816 1!1 arctg x + 1 sin t cos t + C =1= 16 t + 2 sin 2t + C = 162 161 arctg x + 1 tg t + C = 1 arctg x + 1 x + C:= 162 16 1 + tg2 t162 8 4 + x2(9 , $.) !" #$"1. * "+! "? * ! ) "+! ", "? * " "+! ", "?2.
* " "+! !8? !8 (c. 4:1).3. . + ! ") 4 !8? (c. 4:2).4. * " + ! ? (c. 4:3).5. * &! ! ? (c. 4:5).6. * &! !8 (x ;Aa)k ? (c. 4:5).+ N (p2 ; 4q < 0)? (c.7. * &! !8 (x2Mx+ px + q)k ! & (4:2), (4:3) (4:4), 4:6 ).37 M 5 ,, ,./%' )*() ), )* $ , $ . 5.1. . m- ( (x y ) Pm (x y) = Lm0xm + Lm 11xm 1 y + ::: + L0mym + Lm 10xm 1 + :::+L0m 1ym 1 + ::: + L10x + L01y + L00 <$$ Lm0 ::: L00 . 5.2. / R(x y) ( ( # y) R(x y) = PQl ((xm x y ) Ql (x y) Pm (x y) { .) R(sin x cos x) , $ R(x y) x sin x, y { cos x.1, $ 32x+3xyR(x y) = x2 + 2x2y2 ++xy ++ 51 sin3 x + 3 sin x cos2 y + sin x + 5 :R(sin x cos x) = sin2x + 2 sin2 x cos2 x + cos x + 1;;;;;; " R(sin x cos x)' ) ) ! $ * :t = tg x2 x 2 (; ):382 , sin(x=2) = 2 tg(x=2) = 2t :sin x = 2 sin x2 cos x2 = 2 cos2 x2 cos(x=2) 1 + tg2(x=2) 1 + t20122 (x=2)2sin(x=2)1;tg1;t2x2 x@2 xAcos x = cos 2 ; sin 2 = cos 2 1 ; cos2(x=2) = 1 + tg2(x=2) = 1 + t2 x = 2 arctg t- dx = 1 +2 t2 dtZZ 0 2 t 1 ; t2 1 2R(sin x cos x)dx = R @ 1 + t2 1 + t2 A 1 + t2 dt =ZZ2= R1(t) 1 + t2 dt = R2(t)dt0122t1;t R1(t) R @ 1 + t2 1 + t2 A { $ ) t, R2(t) R1(t) 1 +2 t2 : 5.1.
Z8Zdx :2 + cos x9dx = <t = tg x dx = 2 dt cos x = 1 ; t2 =2 + cos x :21 + t21 + t2 2ZZ121+t= 2 + (1 ; t2)=(1 + t2) 1 + t2 dt = 2 + 2t2 + 1 ; t2 1 +2 t2 dt =Z 2= 3 + t2 dt = p2 arctg pt + C = p2 arctg tg(px=2) + C:3333. ) R(sin x cos x)dx ) , .. <$ .39& " R(sin x cos x)() Zsinx=t1: R(sin x) cos xdx = cos xdx = dt = R(t)dt:()ZZcosx=t2: R(cos x) sin xdx = ; sin xdx = dt = ; R(t)dt:() ZZZdttgx=tx=arctgt3: R(tg x)dx = dx = dt=(1 + t2) = R(t) 1 + t2 = R1(t)dt:()Ztgx=tx=arctgt224: R(sin x cos x)dx = dx = dt=(1 + t2) =1ZZ 0 t21dtA@= R 1 + t2 1 + t2 1 + t2 = R1 (t)dt:22tgxt222 sin x = cos x tg x = 1 + tg2 x = 1 + t2 cos2 x = 1 + 1tg2 x = 1 +1 t2 :Z " R(sinm x cosn x) m n { ) n { , n = 2p + 1.Z=ZZsinm x cos2p+1 xdx = sinm x(cos2 x)p cos xdx =sinm x(1 ; sin2 x)p cos xdx = 5.2.
ZZZ(sin x = t ) = Z tm(1 ; t2)pdt:cos xdx = dtcos3 xdx:Zcos xdx = cos x cos xdx = (1 ; sin2 x) cos xdx =() Z33tsinsinx=t2= cos xdx = dt = (1 ; t )dt = t ; 3 + C = sin x ; 3 x + C:3240!) m { , m = 2k + 1.ZZ= (1 ; cos22k+1sinx)k cosn x sin xdx =) m nn = 2p 0.Zx cosn xdx =((sin2 x)k cosn x sin xdx =cos x = t ) = ; Z (1 ; t2)k tndt:; sin xdx = dt{ ,sin x cos xdx =2kZ2pm = 2k 0,Z 1 ; cos 2x !k 1 + cos 2x !p2Z2dx == 2k1+p (1 ; cos 2x)k (1 + cos 2x)pdx: ) * !, , )* cos 2x , ! .
G ) t = sin 2x- )2 :, $ cos2 = 1 + cos2( <Z $ ! # , : cos(x)dx = 1 sin(x) + C:) m n { ( ( . < ) tg x = t ctg x = t. 1, n = ;2p(p > 0), ()2ktgxtgx=tcos2p x dx = cos2(p k 1) x cos2 x dx = dx= cos2 x = dt =Z sin2k xZ;Z;= t (1 + t2k2)p k;1;41Zdt = R(t)dt: ":Za) sin(x) sin(x) sin(x) cos(x)dxZ )dx) cos(x) cos(x)dx:Z' < ) * $:"#1sin(x) sin(x) = 2 cos( ; )x ; cos( + )x "#1sin(x) cos(x) = 2 sin( + )x + sin( ; )x "#1cos(x) cos(x) = 2 cos( + )x + cos( ; )x : !" #$"1. % "! ) ")?2.
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