1. Интегралы ФНП Диф_ур (853736), страница 22
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CC.CC R(t z ) = @ . A :A(t) = BB :::CB@ @fn (t y) .. @fn:::(t y) ARn (t z ).@z1@zn? "' (24.49) (( R(t z ) ( z_ = A(t)z # $ ! ( %( //0'( (24.38). #$e, "( # # y = y (( (24.38) 3 ' ( #% % R(t z )) ( ' z = 0 "3 (24.49). = . 3. 24.12. * (24.49) A(t) = A , - R(t z ) (t z ) z G = ft t0 jz j < H g:*, !, C > 0 > 0 ,jR(t z )j C jz j1+:(24:50)325/! ) z = 0 (24.49) t ! +1 t t0, ) z = 0 ! ( z_ = Az:A, $ y = y + z # # y = y (24.38) % # # z = 0 (24.49), 3 #$', y = y " # (( t ! +1 (( t t0, # ' z = 0 "3 z_ = Az: @ " #( ( ( # 3 # !(A)0 A ' ( ##( # .
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"$!! 9#! 3"? 9 " 9#! ! " (24.14)?36. 9 $ 9# (24.14). ! $ ! 1 " 9 $($ $), $?37 . L 3 x_ = a x + a y1112(24:15)y_ = a21x + a22y !" 1 aij (a11 a22 ; a21 a22 6= 0) 9 (x y) # $ ! (0A 0) f1 2g " (24.15). L! 1 33! 9& $9 (., , D2], . 96 - 100).329 / & 1 . L#0 F (x) $ /#0 f (x) # 3# X , F 0 (x) = f (x) 8x 2 X .? F (x) { "$ /#0 f (x) 3# X , 3 % "$% /#0 f (x) 3# 3 /( F (x) + C , C { , $ ! /#0 f (x):Zf (x) dx = F (x) + C:L#0 f (x) $ '( /#0(, 3f (x)dx { ' 3, x { ( .# $ ! '( /#0.' Z1: ( f (x) dx)0 = f (x):Z2: f 0 (x) dx = f (x) + C:ZZ3: af (x) dx = a f (x) dx (a ; , a 6= 0):ZZZ4: (f1(x) + f2(x)) dx = f1(x) dx + f2(x) dx:' " n+1Z nx1 : x dx = n + 1 + C (n 6= ;1):Z2: dxx = ln jxjx + C:Z x3 : a dx = lna a + C (a > 0 a 6= 1):Z x4 : e dx = ex + C:330Z5: sin x dx = ; cos x + C:Z6: cos x dx = sin x + C:Z dx7 : cos2 x = tg x + C:Z dx8 : sin2 x = ; ctg x + C:Z1 arctg x + C:9: a2 dx=+ x2 aaZ dx10: 1 + x2 = arctg x + C:Z11 : p 2dx 2 = arcsin xa + C:a ;xZ12: p dx 2 = arcsin x + C:1;xZp2=lnjx+x + j + C:13: p dxx2 +Z dx1a+x14 : a2 ; x2 = 2a ln a ; x + C:Z15 : sh x dx = ch x + C:Z16 : ch x dx = sh x + C:Z dx= th x + C:17 :ch2 xZ dx18 := ; cth x + C:sh2 x ( "0 '$ ( ( 3 4.
1.1. @( Zx dx:Z n< F { ( ( x dx (# 1"0). n = 1, 331Z21+1xxx dx = 1 + 1 + C = 2 + C: > 1.2. @( Zx2 dx:< C # 1 "0 (n = 2)32+1Z 2x dx = 2x+ 1 + C = x3 + C: > 1.3. @( Zdx:< J, ' /#0 1, # 3 ' ## x0, $3''-# ' . 1 "0 ( n = 0). 0+1ZZx0dx = x dx = 0 + 1 + C = x + C: > 1.4. @( Z dxx2 :< & .
1 "0 ( n = ;2).Z dx Z;2 dx = x;2+1 + C = x;1 + C = ; 1 + C: >=xx2;2 + 1;1x 1.5. @( Zp3x dx:< . 1 "0 (n = 13 ).Z+1pxx dx = x dx = 1 + 1 + C = 43 x4=3 + C = 34 x4 + C: >3332p3Z1= 3133 1.6. @( Z3x dx:Z xF { ( ( a dx a = 3.
& <. 3 "0xZ x3+ C:3 dx = 1.7. @( Zln 3>dx :2 + x2Z dxp< F ( ( a2 + x2 a = 2 (. 9 "0).Z dxZdx = p1 arctg px + C: >p=22+x( 2)2 + x222 1.8. @( Z dx9 ; x2 :< & . 14 "0 (a = 3)Z dxZ dx13+x + C: >==ln2229;x3 ; x 2 3 3 ; x 1.9. @( Z dxp 2:5;xp< . 11 "0 (a = 5), ZZ dxp 2 = q p dx2 2 = arcsin px + C: >5;x5( 5) ; x 1.10. @( Zp 2dx :x + ln 2333< F ( ( $ "0 . 13 = ln 2(# ## ln 2 { !). =Zpp 2dx = ln jx + x2 + ln 2 j + C: >x + ln 2$ #I IH 3$I(Z "Z p ":Zx dxA 3) 5x dxA1) x3 dxA2)ZZ5) p dx 2 A6) p dx:3;xx2 ; 74)Zdx A4 ; x2 x + 2 4xp321x51) 4 + C A 2) 3 x + C A 3) ln 5 + C A 4) 4 ln x ; 2 + C Ap5) arcsin px + C A 6) ln jx + x2 ; 7j + C:3||||| 3 .% " ( ".
#"0 % ) #3 ( ( (# 3 4 % (). 1.11. @( Z (1 ; x)2xpx dx:Z (1 ; x)2Z0 1dx = @ p12px + xp2 A dx =;x x x x x xxpx1Z 0 1p1= @ xpx ; 2 px + xA dx:( ( ( ((#"0 , #3( $ #% ":334<xpx dx =Z 1 ; 2x + x21Z 0 1@ pZ dxZ dx Z pp1Ax x ; 2 px + x dx = xpx ; 2 px + x dx =; +1; +1 x +1Z ;Z ;Zxxx dx ; 2 x dx + x dx = ; 3 + 1 ; 2 ; 1 + 1 + 1 + 1 + C =2222= ;2x; ; 4x + 23 x + C = 2x ;3p12xx ; 6 + C: > 1.12. @( Zp dx 2 :3 ; 3xZZZ dxdxdx11 arcsin x+C: >ppppp===< p3 ; 3x23 1 ; x23 1 ; x23 1.13. @( Z 1 + cos2 x1 + cos 2x dx:Zcos2 x dx = Z 1 + cos2 x dx = 1 Z 1 + cos2 x dx =< 11 ++ cos2x2 cos2 x2 cos2 x!"Z dxZ 1Z # 111= 2 cos2 x + 1 dx = 2 cos2 x + dx = 2 (tg x + x) + C:8 " .
7 1 "0 % . > 1.14. @( 321212321212tg2 x dx:Zdx< tg x dx = cos2 x dx = cos2 x dx = cos2 x ; dx == tg x ; x + C: > 1.15. @( Z (1 + 2x2)dxx2(1 + x2) :3352Z sin2 x1232ZZ12Z 1 ; cos2 xZZ (1 + 2x2)dxZ (1 + x2) + x2Z21+x< x2(1 + x2) = x2(1 + x2) dx = x2(1 + x2) dx+2Z dx Z dxZx+ x2(1 + x2) dx = x2 + 1 + x2 = ; x1 + arctg x + C:4 ' # 1 10 "0 % . >$ #I IH 3$I(":Z px"Z cos 2xZ (1 + x)23 x2;xe+x1)dxA 2) cos2 x sin2 x dxA 3) x(1 + x2) dxAx3Z4) cos 2xdx+ sin2 x :1) ; 23 xp1 x ; ex + ln jxj + C A 2) ; ctg x ; tg x + C A3) ln jxj + 2 arctg x + C A 4) tg x + C:|||||336 2 F " # ( .?Zf (u) du = F (u) + C(2:1) u = '(x) { " //0 /#0, Zf ('(x)) d'(x) = F ('(x)) + CZZf ('(x)) d'(x) = f ('(x))'0 (x) dx:, $, / (2:1) ( 3 ' " //0 /#0 .Z' " ( g(x) dx "' # /#0 u = '(x), g(x) dx = f ('(x)) d'(x)( "$ $ /#0 '(x) $#//0)., ZZg(x) dx = f ('(x)) d'(x)Z # f (u) du, # # 3#$' $3 .' "0 $ .
2.1. @( ZI = sin x cos x dx:< J' 3' cos x '( /#0 3'$' , " $# //0sin x. , ZI = sin x d(sin x)337Z # u du, #( " (. 1 n = 1). 2sinI = 2 x + C: > 2.2. @( ZI = (x + 1)15 dx:< J' $# //0 0"$ (x+1).,# ## d(x + 1) = dx, 16Z(x+1)15I = (x + 1) d(x + 1) = 16 + C:(& '$ "( 1 n = 15). > 2.3. @( ZI = (2xdx+ 3)5 :< J' # ", $# //0 (2x + 3). ,# ##d(2x + 3) = d(2x) + d(3) = 2dx + 0 = 2dxdx = 21 d(2x + 3)ZZI = 12 d(2(2xx++3)3)5 = 12 (2x + 3);5d(2x + 3) =;41(2x+3)= 2 ;4 + C = ; 8(2x1+ 3)4 + C: ', 3' 21 # $# //0 3 2, 3 " #0 ($ $# //0"""). >338 2.4.
@( Z 3I = px 4dx :x +13< 8 $, x3 dx = 14 d(x4 + 1). Z d(x4 + 1) 1 (x4 + 1);1=3+11I = 4 (x4 + 1)1=3 = 4 ;1=3 + 1 + C = 38 (x4 + 1)2=3 + C('$ "( 1 n = ; 13 ). > 2.5. @( ZI = sin3 x cos x dx:< ,# ## (sin x)0 = cos x, , //0 sin x,4Zsin3I = sin x d(sin x) = 4 x + C: > 2.6.
@( ZI = cos 3x dx:< = # ", $# //0 3x.Z1I = 3 cos 3x d3x = 13 sin 3x + C: > 2.7. @( ZI = sin(2x ; 3) dx:< J' $# //0 (2x ; 3). Z1I = 2 sin(2x ; 3) d(2x ; 3) = ; 12 cos(2x ; 3) + C: > 2.8. @( Z ex dxI = ex + 1 :339< 8 , //0 ex, Z d(ex)I = ex + 1 :4 ( 2.3), $# //0 "$ #0Z d(ex) Z d(ex + 1)I = ex + 1 = ex + 1 = ln(ex + 1) + CZ du(( /## 3 u " 2). > 2.9. @( ZI = p dx 2 :1 ; 25x< %3 "( ZI = p dx 2 = arcsin x + C:1;x, $, 25x2 = (5x)2 , //05x.
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