1. Интегралы ФНП Диф_ур (853736), страница 24
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5.1, 5.2, 5.3, 5.4). B//0 $,# ## $3 . 4 0'( " 3 '$' .( .1: 8' 0 ', "' ' ( 3 ', , $ " " ' $').2: $3' $' Pn(x) 3: ((x ; a1) (x ; a2) ::: . (' # a1 a2 ::: #(x2 + p1x + q1) (x2 + p2x + q2) ::: . % 3% #( $ Pn(x).3: $3' ' "' (% "(.4: ' 0 ' ( ') (". j ' ## ( #"0 % /#0(. 2 (%"( . 3 ( %). 5.1. @(Z x5 + x4 ; 8x3 ; 4x dx:< ' "' { ' ( 5) "' $ (( 3). 8 0 ', $ ' $':x5 + x4 ; 8 x3 ; 4x; x5 ; 4x3x2 + x + 4x4 + 4x3 ; 8;x4 ; 4x24x3 + 4x2 ; 8; 4x3 ; 16x4x2 + 16x ; 8 { # 3602 + 16x ; 85 + x4 ; 84xx2,# "$, x3 ; 4x = x + x + 4 + x3 ; 4x :2$3 ' "' 4x x+3 ;164xx; 8 (, $' 3:4x2 + 16x ; 8 = 4x2 + 16x ; 8 = A1 + A2 + A3 :x3 ; 4xx(x ; 2)(x + 2) x x ; 2 x + 2 " # ".
$ # %, # :4x2 + 16x ; 8 = A1(x ; 2)(x + 2) + A2x(x + 2) + A3x(x ; 2):B//0 A1 A2 A3 3 ', x %. " (" " (' # $ %( 0'( "). 8 :x = 0 ) ;8 = ;4A16x = 2 ) 40 = 8A26x = ;2 ) ;24 = 8A36# A1 = 26 A2 = 56 A3 = ;3:@% %( :Z x5 + x4 ; 8ZZ 4x2 + 16x ; 82x3 ; 4x dx = (x + x + 4)dx +x3 ; 4x dx =Z 2Z 2dx Z 5dx Z ;3dx= (x + x + 4)dx + x + x ; 2 + x + 2 =23xx= 3 + 2 + 4x + 2 ln jxj + 5 ln jx ; 2j ; 3 ln jx + 2j + C: > 5.1. A#$' $3 '( 0'( " ( " ( #/0):3 ; 6x2 + 9x + 72 + 41x ; 91x2x") R = (x ; 2)3(x ; 5) 6) R = (x ; 1)(x + 3)(x ; 4) 6) R = x(x21+ 1) 6) R = x4 ; 3xx2 + 2 67xx+1) R = (x4 ; 1)2 :) R = x(4 + x2)2 (1 + x2) 6361< ) A $ '# (' #. 8 $3 ( " #3# # .
$3 "' :R = xA;1 1 + xA+2 3 + xA;3 4 :") A $ # x1 = 2 # 3 x2 = 5 # 1. $3 " %, .%# x1 = 2, , . # x2 = 5. 8$3 .(:A4 :3+R = xA;1 2 + (x A;22)2 + (x A; 2)3 x ; 5) =:xx pxp==x4 ; 3x2 + 2 (x2 ; 1)(x2 ; 2) (x ; 1)(x + 1)(x ; 2)(x + 2) :3 # 3, ## ), (, $3R (% "( " %:R = xA;1 1 + xA+2 1 + A3p + A4p :x; 2 x+ 2) 8 $ ' #( 3'x2 + 1, .( ('% #(. ,# ## ( , $3 ( " "+ B . , ' Axx2 + 1R = Ax1 + Ax2x2 ++ 1A3 :) 8 3 $ (4 + x2)2 $3 ( " " ' %, 3 (1+ x2) { . ,# ## (4+ x2) (1 + x2) ('% #(, .( " " ' :Ax + B Ax + B Ax + B :4 + x2 (4 + x2)2 x2 + 1362E( 3 $ x ( "'. #' :R = Ax1 + A42x++xA2 3 + A(44x++x2A)25 + A16x++xA2 7 :) =:77xxR = (x4 ; 1)2 = (x ; 1)2(x + 1)2(x2 + 1)2 =A3 + A4 + A5x + A6 + A7x + A8 : >2= xA;1 1 + (x A+; 1)2 x + 1 (x + 1)2 x2 + 1 (x2 + 1)2 5.2. @(Z dxx3 + 1 :< x3 1+ 1 { ' 0' "'61 =1A1 + A2x + A3 =32x + 1 (x + 1)(x ; x + 1) x + 1 x2 ; x + 11 = A1(x2 ; x + 1) + (A2x + A3)(x + 1):x = ;1 ) 1 = 3A1) A1 = 13 6x = 0 ) 1 = A1 + A3) A3 = 32 6x = 1 ) 1 = A1 + 2(A2 + A3) ) 1 = 13 + 2A2 + 43 ) A2 = ; 13 :1 = 1 1 + 2 ; x !:x3 + 1 3 x + 1 x2 ; x + 1$ # ' %3 #//0 $3 ( ":1 1 + 2 ; x ! = 1 x2 ; x + 1 + (2 ; x)(x + 1) =3 x + 1 x2 ; x + 1 3(x + 1)(x2 ; x + 1)221x;x+1+2x;x+2;x = 1 3 = 1 =3x3 + 13 x3 + 1 x3 + 1363..
#//0 .,' ( %( :Z dx1 Z dx ; 1 Z x ; 2 dx = (x2 ; x + 1)0 = 2x ; 1 ==x3 + 1 3 x + 1 3 x2 ; x + 1ZZ1 dx+= 31 ln jx+1j ; 16 (2xx2 ;; x1)+;13 dx = 13 ln jx+1j ; 61 x22;x ;x+12ZZ+ 21 x2 ;dxx + 1 = 13 ln jx + 1j ; 16 d(xx2 ;;xx++11) dx+Z1 ln jx + 1j ; 1 ln(x2 ; x + 1)+1=+ 2 (x ; 1=dx22) + 3=4 36+ 21 p2 arctg xp; 1=2 + C = 13 ln jx + 1j ; 16 ln(x2 ; x + 1)+33=2+ p1 arctg 2xp; 1 + C:33( '$' " " //0" " #".) > 5.3. @(Zdx :x4 + 1< J, px4 + 1 = (x4 + 2x2 + 1) ; 2x2 = (x2 + 1)2 ; ( 2x)2 =pp22= (x ; 2x + 1)(x + 2x + 1) # % 3 $3 % (' #//0, # ## %0' # , ', #. 1 = A1xp+ A2 + B1xp+ B2 :4x + 1 x2 ; 2x + 1 x2 + 2x + 1 , "x" $' ";x":1 = ;A1px + A2 + ;B1px + B2 :x4 + 1 x2 + 2x + 1 x2 ; 2x + 1364,# ## ' "' "$ ( ' %) $ (%"(, % x 3 "' :;B1px + B2 = A1xp+ A2 ;A1px + A2 = B1xp+ B2 6x2 + 2x + 1 x2 + 2x + 1x2 ; 2x + 1 x2 ; 2x + 1# B1 = ;A1 6 B2 = A2: & % :1 = A1xp+ A2 + ;A1px + A2 x4 + 1 x2 ; 2x + 1 x2 + 2x + 1pp221 = (A1x + A2)(x + 2x + 1) + (;A1 x + A2)(x ; 2x + 1): x = 0 $ , 1 = 2A2 #A2 = 12 6 x = 1, (:pp1 = (A1 + 12 )(1 + 2 + 1) + (;A1 + 12 )(1 ; 2 + 1)p1 = 2 + 2 2A1 A1 = ; p1 , # #' :2 2pp1 = ;1=(2 p2) x + 1=2 + 1=(2 2)p x + 1=2 =x4 + 1x2 ; 2x + 1x2 + 2x + 1pp 101;x+x+2= p @ 2 p+ 2 p 2 A:2 2 x ; 2x + 1 x + 2x + 1pp2Z3Z dxZ1x+2;x2 x4 + 1 = p 4 2 pdx + 2 pdx5 :2 2 xp; 2x + 1x + 2x + 1Zp 2 dx:8 2 x +x + 2x + 1pp pZx +p 2 dx = 1 Z (2x + p2) + 2 dx =2 x2 + 2x + 1x2 + 2x + 1 1 Z d(x2 + p2x + 1)ppp= d(x2 + 2x + 1) = (2x + 2)dx = 2+x2 + 2x + 1365pppZZpd(px + 2=2) =+ 22 2 pdx= 12 ln(x2+ 2x+1)+ 22x + 2x + 1(x + 2=2)2 + 1=2p ppp2x+1p2=2 + C == 2 ln(x2 + 2x + 1) + 2 2 arctg1= 2pp= 21 ln(x2 + 2x + 1) + arctg( 2x + 1) + C:J, , F 0 (x) = '(x) #% /#0( F (x) '(x), (;F (;x))0 = ;F 0 (;x) (;1) = F 0 (;x) = '(;x)..
/#0p '(;x) "$( " ;Fp(;x). ,# ##2 ; x $ /#02p+ x /#0 2 px ; 2x + 1x2 + 2x + 1#( $ "x" ";x", , #$ ,:p2p; x dx = ; " 1 ln (;x)2 ; p2x + 1 +2x2 ; 2x + 1ppp12+ arctg(; 2x + 1) + C = ; 2 ln(x ; 2x + 1) ; arctg(; 2x + 1):Z#':pp0Z1Z dxZ2;x21x+@A=pppdx+dx=422x + 1 2 2 x ; 2x + 1x + 2x + 1" 1pp1= p ; 2 ln(x2 ; 2x + 1) ; arctg(1 ; 2x)+2 2pp #12+ 2 ln(x + 2x + 1) + arctg(1 + 2x) + C =p2+px112x+1+ p arctg(1 + 2x)+= p ln 2 p4 2 x ; 2x + 1 2 2p+ arctg( 2x ; 1) + C: >366$ #I IH 3$I( ":Z x6 ; 2x4 + 3x3 ; 9x2 + 4dxA1)x5 ; 5x3 + 4xZ xdx2) x3 ; 1 :2x1) 2 + ln jxj + ln jx ; 2j ; ln jx + 2j + 12 ln jx ; 1j + 32 ln jx + 1j + C A2) 31 ln jx ; 1j ; 16 ln(x2 + x + 1) + p1 arctg 2xp+ 1 + C:33|||||= "Mx + N(x2 + px + q)k # # ZZxdxdx :J = (x2 + a2)k Jk = (x2 +a2)kJ % (x2 + a2) $# //0.J1 = a1 arctg xa + C: 4 Jk k > 1 3 '$' # /Jk = 2a2(k ; 1)(xx2 + a2)k;1 + 2a22k(k;;31) Jk;1 (k = 2 3 :::) # x = a tg t.
5.4. @(Zx3 + x ; 1 dx:(x2 + 2)233x+x;1x< (x2 + 2)2 = +(x22x+;2)x2; 1 =2x(x+1 = x ; x+1 = (x2 ++2)2)2 ; (xx2 +2)2 x2 + 2 (x2 + 2)2367.. $3 (% "( 3 "' "$. 4 :Z x3 + x ; 1Z xdxZ (x + 1)dx(x2 + 2)2 dx = x2 + 2 ; (x2 + 2)2 :8 .Z 1=2 d(x2 + 2) 1Z xdx21) x2 + 2 =x2 + 2 = 2 ln(x + 2) + C:Z xdxZ 2xdxZ d(x2 + 2)112) (x2 + 2)2 = 2 (x2 + 2)2 = 2 (x2 + 2)2 = ; 21 (x2 1+ 2) + C:= (x2 dx+ 2)2 . "$:Zp1 arctg px = x2dx+ 2 = ( / 222Z ) = x2 x+ 2 + (x22+x 2)2 dx = x2 x+ 2 +Z (x2 + 2) ; 2Z dxZx+2 (x2 + 2)2 dx = x2 + 2 + 2 x2 + 2 ; 4 (x2 dx+ 2)2 =Zx2x= x2 + 2 + p arctg p ; 4 (x2 dx+ 2)2 622 x!Zdx11x#: (x2 + 2)2 = 4 x2 + 2 + p arctg p22( $' #( / k = 2).&" , #' :Z x3 + x ; 1 11 1 ; x ;2=ln(x+2)+(x2 + 2)2 22 x2 + 2 4(x2 + 2)p; 4p1 2 arctg x 2 + C = 21 ln(x2 + 2) ; 4(xx2;+22) ;; 4p1 2 arctg px2 + C: >368$ #I IH 3$I( Z2xdx(1 + x)(1 + x2)2 :x ; 1 ; 1 ln jx + 1j + 1 ln(1 + x2) + C:2(x2 + 1) 24|||||369 6 &= ZR(sin x cos x)dx R(u v) { 0' /#0 ' % u = sin x v = cos x .' t = tg x2 # 0'( /#0 ' (( t: '$ /21;t2tsin x = 1 + t2 cos x = 1 + t2 dx = 12+dtt2 :? R(;u v) = ;R(u v) ..
' /#0 ( ' , 3 '$' #t = cos x6 R(u ;v) = ;R(u v) (' ' #), t = sin x6 R(;u ;v) = R(u v) (' # ), t = tg x t = ctg x:F # #$% % 0$ ' 3. 6.1. @(Zdx5 ; 4 sin x + 3 cos x :< ' # #xt = tg 2 $' ( # 0'(370/#0 ( , ' # # "0$" ' 3):Z1dxt = tg x 6=5 ; 4 sin x + 3 cos x 2 5 ; 4 sin x + 3 cos x =2= 5 ; 4 (2t=(1 + t2)) +1 3 (1 ; t2)=(1 + t2) = 2t2 1;+8tt + 8 6 Z2Z2dtdx = 1 + t2 = 2t2 1;+8tt + 8 1 2+dtt2 = t2 ; dt4t + 4 =Z d(t ; 2)= (t ; 2)2 = ; t ;1 2 + C = ; tg(x=12) ; 2 + C: >$ #I IH 3$I( Zdx :5 + 4 sin x!2 arctg 5 tg(x=2) + 4 + C:33 6.2. @(|||||Zsin3 xdx:< 8 R(u v) = u3 { ' /#0, '$ # t = cos x :ZZsin xdx = sin2 x sin xdx = t = cos x6 sin2 x = 1 ; cos2 x =3Z= 1 ; t26 dt = d cos x = ; sin xdx = ; (1 ; t2)dt =3x3Z 2cost= (t ; 1)dt = 3 ; t + C = 3 ; cos x + C: > 6.3.
@(Zcos5 xdx:371< J' R(u v) = v5 { /#0, ' #. & # t = sin x :ZZcos xdx = cos4 x cos xdx = jt = sin x6 cos4 x = (1 ; sin2 x)2 =5ZZ= (1;t ) 6 dt = d sin x = cos xdxj = (1;t ) dt = (t4 ;2t2 +1)dt =5353t2tsinx2sin= 5 ; 3 + t + C = 5 ; 3 x + sin x + C: > 6.4.
@(Zdx4 ; 3 cos2 x + 5 sin2 x :< 8 R(u v) = 4 ; 3v12 + 5u2 R(;u ;v) = 4 ; 3(;v)12 + 5(;u)2 = 4 ; 3v12 + 5u2 = R(u v): # t = tg x :Zdxt = tg x6 dt = 1 dx6=cos2 x4 ; 3 cos2 x + 5 sin2 x 111 =4 ; 3 cos2 x + 5 sin2 x cos2 x 4 (1= cos2 x) ; 3 + 5 tg2 x == cos12 x 4(1 + tg2 x) 1; 3 + 5 tg2 x = 9 tg21x + 1 cos12 x = Z 1Z11= 9t2 + 1 cos2 x = 9t2 + 1 cosdx2 x = 9t2dt+ 1 =Z d(3t)1= 3 (3t)2 + 1 = 13 arctg 3t + C = 13 arctg(3 tg x) + C: >2 22 2$ #I IH 3$I( ":Z1) cos3 xdxAZ2) sin5 xdxAZ3) 1 + dxcos2 x A372Z4) 1 +dxsin2 x :3351) sin x ; sin3 x + C A 2) ; cos x + 2 cos3 x ; cos5 x + C A !p1p x + C A 4) p1 arctg 2 tg x + C:3) p arctg tg222|||||= #% 3( # / 3 :cos2 x = 21 (1 + cos 2x) sin2 x = 12 (1 ; cos 2x) sin x cos x = 21 sin 2x6/ "$ $( # $% $ #6sin cos = 12 (sin( ; ) + sin( + ))sin sin = 21 (cos( ; ) ; cos( + ))cos cos = 12 (cos( ; ) + cos( + )): 6.5.
@(Zcos2 xdx:< 8'$ /( 3 (( #( ):ZZZZ 1112cos xdx = 2 (1 + cos 2x)dx = 2 dx + 4 cos 2xd(2x) == 21 x + 41 sin 2x + C: > 6.6. @(Zcos4 xdx:< ' 3 ', :!2ZZZ 1422cos xdx = (cos x) dx = 2 (1 + cos 2x) dx =373ZZZ= 14 dx + 21 cos 2xdx + 14 cos2 2xdx =ZZ 1111= 4 x + 4 cos 2xd(2x) + 4 2 (1 + cos 4x)dx =Z1 Z cos 4xd(4x) == 41 x + 41 sin 2x + 81 dx + 321 sin 4x + C == 14 x + 14 sin 2x + 81 x + 321 sin 4x + C: >= 38 x + 41 sin 2x + 32 6.7. @(Zsin2 x cos2 xdx:< 8 3 '$ / ( , $ / 3 :Z 2ZZsin x cos2 xdx = 41 sin2 2xdx = 18 (1 ; cos 4x)dx =Z1 Z cos 4xd(4x) = 1 x ; 1 sin 4x + C: >= 81 dx ; 328 32 6.8.
@(Zcos x sin 3xdx:< "$ $ sin 3x cos x :ZZ 1cos x sin 3xdx = 2 (sin 4x + sin 2x)dx =1 Z!Z11= 2 4 sin 4xd(4x) + 2 sin 2xd(2x) = ; 18 cos 4x; 14 cos 2x+C: >$ #I IH 3$IZZZZZ1) sin2 xdxA 2) sin4 xdxA 3) sin4 x cos4 xdxA 4) cos 2x cos 3xdxA 5) sin 2x sin 5xdx:3741 sin 4x + C A1) 12 x ; 14 sin 2x + C A 2) 38 x ; 14 sin 2x + 323 x ; 1 sin 4x + 1 sin 8x + C A 4) 1 sin 5x + 1 sin x + C A3) 12812810241021 sin 7x + C:5) 16 sin 3x ; 14||||| = ZppZZpR(x a2 ; x2)dx R(x a2 + x2)dx R(x x2 ; a2)dx (a > 0)! x = a sin t ( x = a cos t) x = a tg t ( x = a ctg t)!aax = cos t x = sin t # ZR1(sin t cos t)dt:J' R R1 { 0' /#0 % .4 Z0 v1uuR B@x tn ax + b CA dxcx + d a b c d { , R { 0' /#0, 3 '$' #vuu +bt = tn axcx + d # 0$ ' 3.& ' , .
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