{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Headi ng 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Integrace - Fubibi" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 6 "Zad\341n\355" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Jest n \341m spo\350\355sti integr\341l" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Int(ln(1-a^2*x^2)/(x^2*sqrt( 1-x^2)),x=0..1);" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*(-%#lnG6#,&\"\"\"F+*&)%\"aG\"\"#F+)%\"xGF/F+! \"\"F+F1!\"#,&F+F+*$F0F+F2#F2F//F1;\"\"!F+" }}}}{SECT 0 {PARA 5 "" 0 " " {TEXT -1 11 "Konvergence" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Defin ujme" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "f := (x,a) -> ln(1- a^2*x^2)/(x^2*sqrt(1-x^2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf *6$%\"xG%\"aG6\"6$%)operatorG%&arrowGF)*(-%#lnG6#,&\"\"\"F2*&)9%\"\"#F 2)9$F6F2!\"\"F2F8!\"#-%%sqrtG6#,&F2F2*$F7F2F9F9F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Funkce " }{XPPEDIT 18 0 "f" "6#%\"fG" } {TEXT -1 23 " je definov\341na jen pro " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 39 " z intervalu (-1,1), nebo\235 mus\355 platit:" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "1-a^2*x^2>0" "6#2\"\"!,&\"\"\"F&*&%\"aG\"\" #%\"xGF)!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 6 "a tedy" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "abs(a)<1/x" "6#2-%$absG6#%\"aG*&\"\" \"F)%\"xG!\"\"" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 8 " z (0,1]" }}{PARA 0 "" 0 "" {TEXT -1 36 "Nyn\355 si na\350rtn \354me, jak se chov\341 fce " }{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT -1 41 " na intervalu v n\354m\236 ji m\341me integrovati." }}{PARA 0 "" 0 "" {TEXT -1 37 "Nejd\370\355ve si v\232ak je\232t\354 pov\232im\354m e, \236e " }{XPPEDIT 18 0 "f[a](x)=f[-a](x) " "6#/-&%\"fG6#%\"aG6#%\"x G-&F&6#,$F(!\"\"6#F*" }{TEXT -1 5 "a \236e " }{XPPEDIT 18 0 "f[0](x)=0 " "6#/-&%\"fG6#\"\"!6#%\"xGF(" }{TEXT -1 9 " pro v\232. 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Nejprve rozep\355\232eme int egrand do tvaru" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(l n(1-a^2*x^2)-ln(1-0*x^2))/(x^2*sqrt(1-x^2))=G(a)-G(0)" "6#/*&,&-%#lnG6 #,&\"\"\"F**&%\"aG\"\"#%\"xGF-!\"\"F*-F'6#,&F*F**&\"\"!F**$F.F-F*F/F/F **&F.F--%%sqrtG6#,&F*F**$F.F-F/F*F/,&-%\"GG6#F,F*-F>6#F4F/" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "kde " }{XPPEDIT 18 0 "G(a)" "6#-% \"GG6#%\"aG" }{TEXT -1 29 " je primitivn\355 funkce k k fci" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Diff(f(a,x),a)=diff(f(x,t),t );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$*(-%#lnG6#,&\"\"\"F,* &)%\"aG\"\"#F,)%\"xGF0F,!\"\"F,F/!\"#,&F,F,*$F.F,F3#F3F0F/,$**F0F,%\"t GF,,&F,F,*&)F:F0F,F1F,F3F3,&F,F,*$F1F,F3F7F3" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 34 "Integr\341l pak m\371\236eme vyj\341d\370it jako " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Int(Int(diff(f(x,t),t),t=0.. a),x=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$,$**\"\"# \"\"\"%\"tGF+,&F+F+*&)F,F*F+)%\"xGF*F+!\"\"F2,&F+F+*$F0F+F2#F2F*F2/F,; \"\"!%\"aG/F1;F8F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 111 "Nyn\355 mu s\355me ov\354\370it p\370edpoklady Fubiniho v\354ty, abychom mohli za m\354nit po\370ad\355 integrace. P\370edpokl\341dejme na chv\355li," } }{PARA 0 "" 0 "" {TEXT -1 49 "\236e je ji\236 m\341me ov\354\370eny, a zkusme spo\350\355st integr\341l" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Int(Int(diff(f(x,t),t),x=0.. 1),t=0..a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$,$**\"\"# \"\"\"%\"tGF+,&F+F+*&)F,F*F+)%\"xGF*F+!\"\"F2,&F+F+*$F0F+F2#F2F*F2/F1; \"\"!F+/F,;F8%\"aG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Pou\236ijme nejprve substituci " }{XPPEDIT 18 0 "x=sin(y) " "6#/%\"xG-%$sinG6#%\" yG" }{TEXT -1 3 ", (" }{XPPEDIT 18 0 "dx=cos(y)*dy" "6#/%#dxG*&-%$cosG 6#%\"yG\"\"\"%#dyGF*" }{TEXT -1 1 ")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Int(Int(diff(f(sin(x),t)*diff(sin(x),x),t),x=0..1),t= 0..a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$,$*,\"\"#\"\" \"%\"tGF+,&F+F+*&)F,F*F+)-%$sinG6#%\"xGF*F+!\"\"F5,&F+F+*$F0F+F5#F5F*- %$cosGF3F+F5/F4;\"\"!F+/F,;F=%\"aG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Proto\236e na intervalu [0," }{XPPEDIT 18 0 "Pi/2" "6#*&%#PiG\" \"\"\"\"#!\"\"" }{TEXT -1 5 "] je " }{XPPEDIT 18 0 "cos(y)>=0" "6#1\" \"!-%$cosG6#%\"yG" }{TEXT -1 48 ", m\371\236eme integr\341l p\370epsat do n\341sleduj\355c\355ho tvaru" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Int(-2*t*Int(1/(1-t^2*sin(y)^2),y = 0 .. Pi/2),t = 0 \+ .. a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*(\"\"#\"\"\"%\"t GF)-F$6$*&F)F),&F)F)*&)F*F(F))-%$sinG6#%\"yGF(F)!\"\"F6/F5;\"\"!,$*&F( F6%#PiGF)F)F)F6/F*;F9%\"aG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Po \+ dal\232\355 substituci, " }{XPPEDIT 18 0 "tan(y/2)=z" "6#/-%$tanG6#*&% \"yG\"\"\"\"\"#!\"\"%\"zG" }{TEXT -1 3 ", (" }{XPPEDIT 18 0 "sin(y)=2* z/(1+z^2)" "6#/-%$sinG6#%\"yG*(\"\"#\"\"\"%\"zGF*,&F*F**$F+F)F*!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "dy=2/(1+z^2)*dz" "6#/%#dyG*(\"\"#\"\" \",&F'F'*$%\"zGF&F'!\"\"%#dzGF'" }{TEXT -1 33 "), a zkr\341cen\355 dos t\341v\341me integr\341l " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "Int(-2*t*Int(1/(1+z^2*(1-t^2)),z = 0 .. infinity),t = 0 .. a);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*(\"\"#\"\"\"%\"tGF)-F$6$*& F)F),&F)F)*&)%\"zGF(F),&F)F)*$)F*F(F)!\"\"F)F)F5/F1;\"\"!%)infinityGF) F5/F*;F8%\"aG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 " Co\236 ji\236, \+ jak snadno spo\350teme, d\341v\341 integr\341l" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 171 "Int(-2*t*Int(Diff(arctan(z*sqrt(1-t^2))/sqrt( 1-t^2),z),z = 0 .. infinity),t = 0 .. a)=Int(-2*t*int(Diff(arctan(z*sq rt(1-t^2))/sqrt(1-t^2),z),z = 0 .. infinity),t = 0 ..a);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%$IntG6$,$*(\"\"#\"\"\"%\"tGF*-F%6$-%%DiffG6$* &-%'arctanG6#*&%\"zGF*,&F*F**$)F+F)F*!\"\"#F*F)F*F7#F:F)F6/F6;\"\"!%)i nfinityGF*F:/F+;F?%\"aG-F%6$,$*(F+F*%#PiGF*F7F " 0 "" {MPLTEXT 1 0 36 "int(-t*Pi/(1-t^2)^(1/2),t = 0 .. a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(%#PiG\"\"\",(*$,&F&F&*$)%\"aG\"\"#F&!\"\"#F&F-F&F&F .F*F&F&F)#F.F-F." }}}}{PARA 5 "" 0 "" {TEXT -1 33 "Ov\354\370en\355 p \370edpoklad\371 Fubiniho v\354ty" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Vr\341\235me se nyn\355 k doposud neov\354\370en\375m p\370edpokla d\371m Fubiniho v\354ty." }}{PARA 0 "" 0 "" {TEXT -1 8 "Proto\236e " } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "abs(2*t/(1-t^2*x^2)/sqrt(1 -x^2)) <= 2*t/(1-t^2*x^2)/sqrt(1-x^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#1,$*&\"\"#\"\"\"-%$absG6#*(%\"tGF',&F'F'*&)F,F&F')%\"xGF&F'!\"\" F2,&F'F'*$F0F'F2#F2F&F'F',$**F&F'F,F'F-F2F3F5F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "a proto\236e na prav\351 stran\354 stoj\355 nez\341p orn\341, m\354\370iteln\341 (spojit\341 s. v.) fce a jeliko\236 tedy e xistuje" }}}{EXCHG {PARA 257 "> " 0 "" {MPLTEXT 1 0 52 "Int(Int(2*t/(1 -t^2*x^2)/sqrt(1-x^2),t=0..a),x=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$,$**\"\"#\"\"\"%\"tGF+,&F+F+*&)F,F*F+)%\"xGF*F+!\" \"F2,&F+F+*$F0F+F2#F2F*F+/F,;\"\"!%\"aG/F1;F8F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "je fce na stran\354 prav\351 integrovatelnou majora ntou k fci na stran\354 lev\351. T\355m jsou p\370edpoklady Fubiniho v \354ty ov\354\370eny." }}{PARA 0 "" 0 "" {TEXT -1 47 "Z\341v\354rem te dy m\371\236eme zvolat \"hur\341\" a konstatovat" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "int(f(x,a),x = 0 .. 1) = Pi(sqrt(1-a^2)-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$**\"\"#!\"\"%#PiG#\"\"\"F&,$*$)% \"aGF&F*F'#F'F&,&**F&F*F(F)F-F*,&*&F*F**$F-F*F'F'F*F*F)F'*(F&F*F(F)F+F )F'F*F*-F(6#,&*$,&F*F*F,F'F)F*F*F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{MARK "16" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }