1. Интегралы ФНП Диф_ур (853736), страница 23
Текст из файла (страница 23)
Z 1 d(5x)Z1d(5x) = 1 arcsin 5x + C: >5qqI==1 ; (5x)2 5 1 ; (5x)2 5 2.10. @( ZI = 1 +dx9x2 :< ; . $, 9x2 = (3x)2,% # d(3x). Z 1 d(3x)Z d(3x)13I = 1 + (3x)2 = 3 1 + (3x)2 = 13 arctg 3x + C(. 10 "0). >340 2.11. @( I=Z 2x ; parcsin xpdx:1 ; x2< 8'$' (' , #( ## $' :pZ arcsin x2xdxp; p1 ; x2 dx:I=1 ; x2 Z 2x dxI1 = p1 ; x2$, d(1 ; x2) = ;2x dx, Z ;d(1 ; x2)ZpI1 == ; (1 ; x2);1=2 d(1 ; x2) =21;x2 ;1=2+1p= ;(1;;1=x2 )+ 1 + C1 = ;2 1 ; x2 + C1:4 Z arcsin xdxI2 = p1 ; x2, (arcsin x)0 = p 1 2 1;x#d(arcsin x) = p 1 2 dx:1;xZI2 =ZpZarcsin x d(arcsin x) = (arcsin x)1=2 d(arcsin x) =1=2+1q(arcsinx)= 1=2 + 1 + C2 = 32 (arcsin x)3=2 + C2 = 23 (arcsin x)3 + C2:3418 %( qpI = I1 ; I2 = ;2 1 ; x2 ; 23 (arcsin x)3 + C($' "Z C1 C2, 3 C = C1 ; C2).
>$ #I IH 3$I" ":Z (Z 4Zp 2x dx A1) x 1 ; x dxA2) px dx 5 A3) sincos2 x4+xZZZdxdx pp3 xdxpA6)A5)4) cos2322cos x 1 + tg x A(arcsin x) 1 ; xsin xZZ ex dxZ dxdx8) pA9) e2x + 4 A7) x ln x A4 ; 9x2Z 1 + x ; x210) qdx:(1 ; x2)3qpp1) ; 13 (1 ; x2)3 + C A 2) 25 4 + x5 + C A 3) cos1 x + C A 4) 3 3 sin x + C Ap1 + tg x + C A 7) ln j ln xj + C A15) ; 21 (arcsin+CA6)22x)x8) 13 arcsin 32x A 9) 12 arctg e2 + C 10) arcsin x + p 1 2 + C:1;x||||| , &( ? ' /#0 3 #( %, 3 #$' //# #. 2.12. @( ZI = 4x2 +dx4x + 5 :342< 8 ( # $4x2 + 4x + 5 = C(2x)2 + 2 2x 1 + 1] + 4 = (2x + 1)2 + 22:, Z 1=2 d(2x + 1) 1 1dxI = (2x + 1)2 + 22 = (2x + 1)2 + 22 = 2 2 arctg 2x 2+ 1 + C == 14 arctg 2x 2+ 1 + C: > 2.13.
@( ZI = p dx:2 ; 6x ; 9x2< '-# $ ( #ZZdxdxqp=I==3 ; 1 ; 6x ; 9x23 ; (3x + 1)2Z= q p1=32d(3x + 1) 2 = 13 arcsin 3xp+ 1 + C3( 3) ; (3x + 1)($ dx 13 d(3x + 1) '$ $ "Z p 2dx 2 . 11 "0.) >a ;xZ$ #I IH 3$I( " ":Zdx1) x2 + 2x + 3 A:2) p dx8 + 6x ; 9x2Z1) p1 arctg xp+ 1 + C A222) 1 arcsin 3x ; 1 + C:33|||||343 3 . $ F 3, # . $ " : x = '(t) { //0 /#0, t = ';1 (x) { " # ( /#0, Zf (x)dx =Zf ('(t))d'(t)t=';1 (x):L#0 x = '(t) " #, " .( Z# ( x f ('(t))d'(t) #$ . % .
# (, 3' ( ( t # '(, t = ';1(x). 3.1. @( ZI = 1 + pdxx + 1 :p< 3 t = x + 1: , x = t2 ; 1 dx = 2tdt 0ZZ (t + 1) ; 1Z d(1 + t) 1Z 2tdt@AI = 1+t = 21 + t dt = 2 dt ; 1 + t =pp= 2(t ; ln j1 + tj) + C = 2( x + 1 ; ln(1 + x + 1)) + C: >='$ $' # $ "$"'" 0' ' 3 ( # ".J, ## # " x = '(t) " /#0 t = ';1 (x) ## . 3.2. @( Zpx(dxx ; 1) :< 3 t = px " $' 3 "$"'" 0'.
# 3 /' 3440# , # # "% # /, ##-" "$ ($' { #' ).Z Z 2tdt p2px(dxx ; 1) = t = x x = t dx = 2tdt = t(t2 ; 1) =Z= ;2 1 ;dt t2 = ("( 14) = p 11+t1;tx + C: >p= ;2 2 ln 1 ; t + C = ln 1 + t + C = ln 11 ;+ x 3.3. @( pxZp 2 p dx:x ; x< 8 , "p$"' 0', 3' t = x: #3 # 0# ( 3.2):pxZp 2 p dx = t = px x = t12 dx = 12t11dt =x ; xZ t6Z t14dt 12 Z t1011= t8 ; t3 12t dt = 12 t5 ; 1 = 5 t5 ; 1 5t4dt == u = t5 ; 1 du = 5t4dt t10 = (t5)2 = (u + 1)2 =Z (u + 1)2Z u2 + 2u + 11212du = 5du == 5uuZZZ du !12= 5 udu + 2du + u = (" ,3412312401212u@.
1 1 2 ) = 5 2 + 2u + ln jujA + C =0 51212(t;1)55= 5 @ 2 + 2(t ; 1) + ln jt ; 1jA + C =0 101t121555= 5 @ 2 ; t + 2 + 2t ; 2 + ln jt ; 1jA + C =3451 ; 12 2 + C $. # xj == j C1 = 125 2 5pp 5 12 p 5= 65 x5 + 125 x + 5 ln j x ; 1j + C1: > 3.4. @( Z ctg xln sin x dx:< J, (ln sin x)0 = sin1 x cos x = ctg x .. $$ . 8 #% % # 0 #, #( # ( ( $'Z ctg xln sin x dx = jt = ln sin x dt = ctg xdxj =Z dt= t = ("( . 2) == ln jtj + C = ln j ln sin xj + C: > 3.5. @( Z 2x5 ; 3x21 + 3x3 ; x6 dx:< $ $(1 + 3x3 ; x6)0 = 9x2 ; 6x5 = 3(3x2 ; 2x5) = ;3(2x5 ; 3x2)61212 ' ( 3' (;3): $', ## 3.4, # $ # ( ( # 0.Z 2x5 ; 3x23652dx=jt=1+3x;xdt=;3(2x;3x)dx361 + 3x ; xZ ; 1 dt152(2x ; 3x )dx = ; 3 dtj = 3t = ; 13 ln j1 + 3x3 ; x6j + C: >346 3.6.
@( Z1 ; sin x dx:(x + cos x)3< J, (x +cos x)0 = 1 ; sin x: = % $' ' {$ $, /#0, .( , .( $, /#0 $ t = x + cos x #" :Z 1 ; sin x(x + cos x)3 dx = jt = x + cos x dt = (1 ; sin x)dxj =Z dt= t3 = ("( , . 1) = ; 12 t12 + C =1= ; 21 (x + cosx)2 + C: >$ #I IH 3$IZdx Ap1)1+ 3 x+1pxdxZ3) px dx2) px ; px A+ px AZ344)Zp dxx 1 + x2:1) 32 (x + 1)(2=3) ; 3(x + 1)(1=3) + 3 ln j1 + (x + 1)(1=3)j + C A2) x + 65 x(5=6) + 23 x(2=3) + 2x(1=2) + 3x(1=3) + 6x(1=6) + 6 ln jx(1=6) ; 1j + C A3) 2x(1=2) ; 4x(1=4) + 4 ln(1 + x(1=4)) + C A4) ln p jxj2 + C:1+x +1||||| F / Z 0Z 0uv dx = uv ; vu dx(3:1)347ZZudv = uv ; vdu:(3:10) , u v { //0 /#0 x: 3.7 @( Zx cos xdx:< 3 u = x v0 = cos x: , u0 = 1 v = R cos xdx = sin x (# v " "$ , . (C ( ). / (3.1) ZZx cos xdx = x sin x ; sin xdx = x sin x + cos x + C: >J, % (3.1) ( # ( 3' u //0, 3' v0 ( ' ' 3).
$ u 0"$ ' 3', #( //0 ., $ v0 { 3', #( # . 3.8 @( Zxexdx:< A $ # 3.7, 3 3'u = x v0 = ex: @ ## . , '$' / (3:10): 4 3dv = exdx = dex: /#0 ex $# //0 %( # ZZxdex = udv#( $ $ ' /#0 v = ex: / (3:10) ZZZxxxxe dx = xde = xe ; exdx = xex ; ex + C: > 3.9 @( Z ;xxe dx:348< B# 3.8, e;x $# //0(e;xdx = ;de;x ). , , / (3:10) :ZZZ;x;x;xxe dx = ; x de = ;xe + e;xdx = ;xe;x ; e;x + C == ;e;x(x + 1) + C: > 3.10 @( Z 2 ;xx e dx:< 8 3 ' / , #3( $ 3 ' 3 e;x :Z 2 ;xZ 2 ;x 0Z 2 0 ;x2 ;xx e dx = x (;e ) dx = ;x e + (x ) e dx = ;x2 e;x+Z+2 xe;xdx = ;x2 e;x ; 2e;x(x + 1) + C = ;e;x(x2 + 2x + 2) + CZ( xe;xdx { $ 3.9).
> 3.11 @( Zln(x2 + 1)dx:< u = ln(x2 + 1) dv = dx :ZZZln(x2 +1)dx = x ln(x2+1); xd ln(x2+1) = x ln(x2+1); x22x+ 1 dx ="ZZ (x2 + 1) ; 1Z dx #22= x ln(x +1);2x2 + 1 dx = x ln(x +1);2 dx ; x2 + 1 == x ln(x2 + 1) ; 2x + 2 arctg x + C: > 3.12 @( Zex sin xdx:< ='$ , $(" # Zx "" { 3 I = e sin xdx: , , 3/ (3:10):I=Zsin xdex= ex sin x ;2Z xZ xxe d sin x = e sin x ; e cos xdx =349ZZ= ex sin x ; cos xdex = ex sin x ; ex cos x + exd cos x =ex sin x ; ex cos x ;Z=ex sin xdx = ex sin x ; ex cos x ; I:,# "$, I /I = ex sin x ; ex cos x ; I # ( I $ ( "" $' C1 )2I = ex sin x ; ex cos x + C1 I = 12 (ex sin x ; ex cos x) + C(C = C21 ): &',Zex sin xdx = 12 ex(sin x ; cos x) + C: >$ #I IH 3$I( ":Z1) x sin 2xdxAZ4) x3e xdxA;ZZ2) x 3xdxA3) arccos xdxAZZ5) x arctg xdxA 6) ex cos xdx:1) 14 sin 2x ; 21 x cos 2x + C Ap3) x arccos x ; 1 ; x2 + C A25) x 2+ 1 arctg x ; x2 + C Axx2) x ln3 3 ; 32 + C Aln 3324) (x ; 3x + 6x ; 6)ex + C A6) 21 ex(sin x + cos x) + C:|||||350 4 ' /#0 f (x) $# Ca b]6a = x0 < x1 < x2 < : : : < xn;1 < xn = b { $' $" $# n (6xk = xk ; xk;1 (k = 1 2 : : : n) { % $#6d = k=1maxxk { $"6:::n$k 2 Cxk;1 xk ] (k = 1 2 : : : n) { #, $' " % $#%., nX! = f ($k ) xkk=1$ ! /#0 f (x) $# Ca b].
/# " $" Ca b] $# " % # f$k g ' ! /#0( $" d.? . #( '% ! d ! 0, $.( " $" $# Ca b] " # f$k g, /#0 f (x) $ ! $# Ca b], $ ! /#0Zbf (x) % a b "$ f (x) dx:aZbaf (x) dx = dlim!0nXk=1f ($k ) xk :? /#0 f (x) $# Ca b], Zb $#. ?, # , f (x) 0 Ca b], f (x) dxa3 .' S #(( 0, ( /# /#0 y = f (x), $# Ca b] Ox $# %x = a x = b (. 4.1)Zbf (x) dx = S:a351;.
4.1Zb ( f (x) dx (b > a), a # $# Ca b] $" $# #b = x0 > x1 > : : : > xn;1 > xn = a:4 /#0 f (x), ( Ca b]ZabZbf (x) dx = ; f (x) dxa.. ( $#. --%"? /#0 f (x) $# Ca b] F (x) { $ "$% $#, #{2 bZbf (x) dx = F (x)a= F (b) ; F (a):aF / '$ % . 4.1. 8'Z4 dx:2x1 1!1< $ "$% /#0 x2 ' ; x . , / @' {E("0, Z4 dx 1 !4 1 ! 1 ! 3= ; x = ; 4 ; ; 1 = 4 : >2x11352 4.2. 8'Z91p3 x dx:3=2ppx< ,# ## 3 x dx = 3 3=2 + C = 2 x3 + C , ( $ p"$% /#0 3px 2 x3 ; 2. /-Z @' {E("0, (Z91pp3 x dx = (2 x39p3p3; 2)1 = (2 9 ; 2) ; (2 1 ; 2) = 52:>J, , " $ ## {" "$ , $' " " 3 (# ## , .
"$ ( (3 /#0 $' %( 3( # 3):(F (x) + C )jba = (F (b) + C ) ; (F (a) + C ) = F (b) ; F (a) = F (x)jba :" '$ " $ "$% ( . C = 0). ,#, p 4.2 $' . ','$ "$ 2 x3 :Z91p 39 p 3 p 3p3 x dx = 2 x = 2 9 ; 2 1 = 52:1 4.3. 8'Z;1dx :;2 (11 + 5x)3< =23Z;1dx = Z;1 1=5 d(11 + 5x) = 4 1 (11 + 5x);2 5 ;1 =5;2;2;2 (11 + 5x)3 ;2 (11 + 5x)32= 4; 1323115 ; 4; 1 5210 (11 + 5 (;1))10 (11 + 5 (;2))2 =353! 711= 10 1 ; 36 = 72 : >(8 ( " $# //0, $ /@' {E("0). 4.4. 8'Z1p dx8 + 2x ; x2:;05< 8 ( # $, Z1;05Z1Z1d(x ; 1) =dxdxqqp==8 + 2x ; x2 ;05 9 ; (x ; 1)2 ;05 32 ; (x ; 1)2('$ "( Zp 2dx 2 = arcsin xa )a ;x!1x;1= arcsin 3 = arcsin 0 ; arcsin(; 12 ) = arcsin 21 = 6 : >;05$ #I IH 3$I4"$3 " ":Zb dxZ1 dx1) x3 A 2) p3 4 (a > 0A b > 0)Axa4Z1:4) x2 +dx4x+50Z1 p1 + x dxA3)015 A1) ; 32p3 p3 b; ap32)Aab3p3) 4 23; 2 A|||||3544) arctg 1 :7/ ? /#0 u(x), v(x) % $ $#Ca b], / b b Z 00uv dx = uv ; vu dxa aab bZb Zu dv = uv ; v du:a aaZb 4.5.
8'(4:1)(4:10)=Z3x dx :2=4 sin x< / ((4:1)):=Z3=4x dxsin2 x=3=Z30= x (; ctg x) dx = x (; ctg x) + x0 ctg x dx ==4 =4=4=Z3==Z 3 cos x dxZ 3 d(sin x)1= ; 3 p + 4 1+== ; 3 ctg 3 + 4 ctg 4 +sinxsinx3=4=41!=31= 4 ; p + ln j sin xj =3 3=4! 3p3 ; 4 1 3! 11p + 2 ln 2 : >= 4 ; p + ln sin 3 ; ln sin 4 =3 312 3 4.6. 8'=Z20e2x cos x dx:< " " ' " # "":==Z 2 2xZ 2 2xe cos x dx = e d(sin x) =00355('$ /( (4:1))=e2x= e + 2=Z2Z 2 2x=2 =2xsin x0 ; sin xd(e ) = e ; 2e sin xdx =0=Z200e d(cos x) =2xe + 2e2x= e ; 2 ; 4,# "$,=Z20e cos2x 5, #',=Z20=Z20=Z2=2cos x0 ; 2 cos x d(e2x) =e2x cos x dx:x dx = e ; 2 ; 4=Z200=Z20e2x cos x dx:e2x cos x dx = e ; 2e2x cos x dx = 15 (e ; 2) : >$ #I IH 3$I4"$3 " ":Z1Zx1) x e dxA 2) x3 sin x dx:;001) 1 ; 2e 1A 2) 2 ; 6 :;|||||3562 ' /#0 f (x) $# Ca b]6/#0 x = '(t) $ C ]6t = ';1(x) { " # ( /#0 Ca b], = ';1(a) = tjx=a = ';1(b) = tjx=b:, ZbZf (x) dx = f ('(t)) d'(t):aF / 3 x = '(t) .
4.7. 8'Z9 pxpx ; 1 dx:4p< 4 $ t = x. , x = t2 { //Z9 px0 /#0 $# C4 9] px ; 1 dx =pp4j t = px6 t(4) = 4 = 26 t(9) = 9 = 36 x = t2 ) dx = 2t dt j1Z3 2t2Z3 0 t2 ; 11= t ; 1 dt = 2 @ t ; 1 + t ; 1 A dt =22Z322(t + 1) dt +332t ; 1 = (t + 1) 2 + 2(ln jt ; 1j) = 7 + 2 ln 2:2Z3 2dt2 4.8. 8'Z83px dx :1+x< =Z8 x dxpp2p=jt=1+x6x=t;16t(3)=1 + 3 = 261+x3357>pt(8) = 1 + 8 = 36 dx = 2t dt j =Z3 (t2 ; 1) 2t dt=t23210" 8 !# 3233Z3t762A@= 2 (t ; 1) dt = 2 4 3 ; t 5 = 2 (9 ; 3) ; 3 ; 2 = 3 : >22$ #I IH 3$I4"$3 " ":Z1 x dxZ1 px dx1) 1 + x A2) 1 + px :001) 2 ; 2 A2) 53 ; 2 ln 2:|||||358 5 4& " 0' /#0 R(x) 3 ' :R(x) = PPm((xx)) n Pm (x) Pn(x) { ( m n . PPm((xx)) n 1 $ , n, m < n, , m n.@' "' Pm (x) = P (x) + Ps (x) (s < n)Pn(x) m;nPn (x) Pm;n (x) { (m ; n), $( "6Ps (x) { s < n , ', "' PPs ((xx)) n'.' "' 3 $3' (% "(A Mx + N2(p; 4q < 0)k2l(x ; a)(x + px + q) A M N a p q { (' 6 k l = 1 2 :::8 $3 #3 (' a k $ Pn (x) k %A1 + A2 + ::: + Ak 6x ; a (x ; a)2(x ; a)k#3( % 3% #( #( l $Pn(x) ( .% # # % x2 + px + q) l %M1x + N1 + M2x + N2 + ::: + Ml x + Nl :x2 + px + q (x2 + px + q)2(x2 + px + q)l3594 $3 '( " (% '$ , # #//0 A1 ::: Ak ::: M1 ::: Ml N1 ::: Nl ::: $ ( 3 '( " $3 (.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
