restart;Grundbegriffe

Maple kann Stammfunktionen berechnen, d.h. unbestimmt integrieren:Int(x^2,x)=int(x^2,x);

NiMvLUkkSW50RzYkSSpwcm90ZWN0ZWRHRidJKF9zeXNsaWJHNiI2JCokSSJ4R0YpIiIjRiwsJCokRiwiIiQjIiIiRjA=

Maple kann bestimmt integrieren:Int(x^2,x=1..2)=int(x^2,x=1..2);

NiMvLUkkSW50RzYkSSpwcm90ZWN0ZWRHRidJKF9zeXNsaWJHNiI2JCokSSJ4R0YpIiIjL0YsOyIiIkYtIyIiKCIiJA==

Maple kann bestimmte Integrale auch numerisch auswerten:evalf(Int(x^2,x=1..2));

NiMkIitMTExMQiEiKg==

Manche Stammfunktionen k\366nnen nicht elementar angegeben werden:int(exp(-x^2),x);

NiMsJComSSNQaUdJKnByb3RlY3RlZEdGJiMiIiIiIiMtSSRlcmZHNiRGJkkoX3N5c2xpYkc2IjYjSSJ4R0YuRihGJw==

Die entsprechenden bestimmten Integrale kann Maple jedoch numerisch auswerten:int(exp(-x^2),x=0..1); evalf(%);

NiMsJComLUkkZXJmRzYkSSpwcm90ZWN0ZWRHRihJKF9zeXNsaWJHNiI2IyIiIkYsSSNQaUdGKCNGLCIiI0YuNiMkIitJOENvdSEjNQ==

Ein anderes Beispiel:int(sin(x)/x,x);

NiMtSSNTaUc2JEkqcHJvdGVjdGVkR0YmSShfc3lzbGliRzYiNiNJInhHRig=

int(sin(x)/x,x=1..2); evalf(%);

NiMsJi1JI1NpRzYkSSpwcm90ZWN0ZWRHRidJKF9zeXNsaWJHNiI2IyIiIiEiIi1GJTYjIiIjRis=NiMkIittISpIJGYnISM1

Mit Hilfe des student-Package kann man die Integration graphisch veranschaulichen:with(student):

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leftbox(x^2,x=0..4); leftsum(x^2,x=0..4); evalf(%);

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NiMtSSRTdW1HSShfc3lzbGliRzYiNiQqJEkiaUdGJiIiIy9GKTsiIiEiIiQ=NiMkIiM5IiIh

rightbox(x^2,x=0..4); rightsum(x^2,x=0..4); evalf(%);

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

NiMtSSRTdW1HSShfc3lzbGliRzYiNiQqJEkiaUdGJiIiIy9GKTsiIiIiIiU=NiMkIiNJIiIh

middlebox(x^2,x=0..4); middlesum(x^2,x=0..4); evalf(%);

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

NiMtSSRTdW1HSShfc3lzbGliRzYiNiQqJCwmSSJpR0YmIiIiI0YrIiIjRitGLS9GKjsiIiEiIiQ=NiMkIisrKysrQCEiKQ==

Nun wird die Unterteilung feiner gew\344hlt:middlebox(x^2,x=0..4,10); middlesum(x^2,x=0..4,10); evalf(%);

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

NiMsJC1JJFN1bUdJKF9zeXNsaWJHNiI2JCokLCZJImlHRicjIiIjIiImIyIiIkYuRjBGLS9GKzsiIiEiIipGLA==NiMkIisrKytHQCEiKQ==

Bei noch feinerer Unterteilung:middlebox(x^2,x=0..4,100); middlesum(x^2,x=0..4,100); evalf(%);

-%%PLOTG6cq-%'CURVESG6&7S7$$""!F+F*7$$"3Hmmmm;')=()!#>$"3$Q%pCw[&=g(!#?7$$"3RLLLe'40j"!#=$"3m$G=fuh&eEF/7$$"3mmmm;6m$[#F6$"3c8>WUDdohF/7$$"3fmmm;yYULF6$"3OV_a5"4s6"F67$$"3%HLL$eF>(>%F6$"33x010Fkh<F67$$"3Qmmm">K'*)\F6$"3'*H;"3%Hk*[#F67$$"3P*****\Kd,"eF6$"3y9^79GzvLF67$$"3-mmm"fX(emF6$"3sDSX&G*)QV%F67$$"3.*****\U7Y](F6$"3XOk%\w?>j&F67$$"3'QLLLV!pu$)F6$"30'[;a)Ra8qF67$$"3xmmm;c0T"*F6$"3Q@$*py(*)eN)F67$$"3#*******H,Q+5!#<$"3w;))\//w+5Fco7$$"3)*******\*3q3"Fco$"3a-,Qd%)e"="Fco7$$"3)*******p=\q6Fco$"3%o4Ox@^+P"Fco7$$"3mmm;fBIY7Fco$"31*Rk/dpKb"Fco7$$"3GLLLj$[kL"Fco$"3?Xj&yA%4'y"Fco7$$"3?LLL`Q"GT"Fco$"37"eqT)H/'*>Fco7$$"3!*****\s]k,:Fco$"33INwBz$\D#Fco7$$"39LLL`dF!e"Fco$"3YLKdc9F(\#Fco7$$"33++]sgam;Fco$"3ov^w6ePxFFco7$$"3/++]<ep[<Fco$"3C$*>9iq$z0$Fco7$$"3QLLLe/TM=Fco$"355FkH<1lLFco7$$"3JLL$eDBJ">Fco$"3gL.,#fS+m$Fco7$$"3immmTc-)*>Fco$"3'[\dZY1@*RFco7$$"3Mmm;f`@'3#Fco$"3-.H#[_%H_VFco7$$"3y****\nZ)H;#Fco$"3)z-PW5.&yYFco7$$"3YmmmJy*eC#Fco$"3DM]Gqq0W]Fco7$$"3')******R^bJBFco$"3;>/'3P\hV&Fco7$$"3f*****\5a`T#Fco$"3i[.a_a$R$eFco7$$"3o****\7RV'\#Fco$"3)e0![zA=KiFco7$$"3k*****\@fke#Fco$"3E9%e3Fr(*o'Fco7$$"3/LLL`4NnEFco$"3ES#[#36w9rFco7$$"3#*******\,s`FFco$"3'=-;XmuHe(Fco7$$"3[mm;zM)>$GFco$"3$p$HFE/8?!)Fco7$$"3$*******pfa<HFco$"3mRK1([u?^)Fco7$$"3#HLLeg`!)*HFco$"3w[-XBaK))*)Fco7$$"3w****\#G2A3$Fco$"3%G.'HK<++&*Fco7$$"3;LLL$)G[kJFco$"3ek4*=>&R,5!#;7$$"3#)****\7yh]KFco$"3QGUH;;lc5Fhw7$$"3xmmm')fdLLFco$"3,k!))e)GF66Fhw7$$"3bmmm,FT=MFco$"3%)*f"*)RXbo6Fhw7$$"3FLL$e#pa-NFco$"3S`kw'\$yE7Fhw7$$"3!*******Rv&)zNFco$"3-&[p1+Q:G"Fhw7$$"3ILLLGUYoOFco$"3!y7c%zHwX8Fhw7$$"3_mmm1^rZPFco$"3OaP2_o`/9Fhw7$$"34++]sI@KQFco$"37*)RI.deo9Fhw7$$"34++]2%)38RFco$"3h)3"\)3E7`"Fhw7$$""%F+$"#;F+-%&STYLEG6#%%LINEG-%*THICKNESSG6#""#-%&COLORG6&%$RGBG$"#5!""$F+Fi[lFj[l-%)POLYGONSG6$7&F)7$F*$"+++++S!#87$$Fa\l!#6F`\l7$Fd\lF*-Fd[l6&Ff[l$"1sCy?!3'>qFhw$"1#=t:!zg>!*FhwFi\l-F\\l6$7&Ff\l7$Fd\l$"+++++O!#77$$"+++++!)Fe\lFa]l7$Fe]lF*Fg\l-F\\l6$7&Fg]l7$Fe]l$"+++++5Fe\l7$$"+++++7!#5F\^l7$F_^lF*Fg\l-F\\l6$7&Fb^l7$F_^l$"++++g>Fe\l7$$"+++++;Fa^lFg^l7$Fj^lF*Fg\l-F\\l6$7&F\_l7$Fj^l$"++++SKFe\l7$$"+++++?Fa^lFa_l7$Fd_lF*Fg\l-F\\l6$7&Ff_l7$Fd_l$"++++S[Fe\l7$$"+++++CFa^lF[`l7$F^`lF*Fg\l-F\\l6$7&F``l7$F^`l$"++++gnFe\l7$$"+++++GFa^lFe`l7$Fh`lF*Fg\l-F\\l6$7&Fj`l7$Fh`l$"+++++!*Fe\l7$$"+++++KFa^lF_al7$FbalF*Fg\l-F\\l6$7&Fdal7$Fbal$"++++c6Fa^l7$$Fb]lFa^lFial7$F\blF*Fg\l-F\\l6$7&F]bl7$F\bl$"++++W9Fa^l7$$Fa\lFa^lFbbl7$FeblF*Fg\l-F\\l6$7&Ffbl7$Febl$"++++k<Fa^l7$$"+++++WFa^lF[cl7$F^clF*Fg\l-F\\l6$7&F`cl7$F^cl$"++++;@Fa^l7$$"+++++[Fa^lFecl7$FhclF*Fg\l-F\\l6$7&Fjcl7$Fhcl$"+++++DFa^l7$$"+++++_Fa^lF_dl7$FbdlF*Fg\l-F\\l6$7&Fddl7$Fbdl$"++++;HFa^l7$$"+++++cFa^lFidl7$F\elF*Fg\l-F\\l6$7&F^el7$F\el$"++++kLFa^l7$$"+++++gFa^lFcel7$FfelF*Fg\l-F\\l6$7&Fhel7$Ffel$"++++WQFa^l7$$"+++++kFa^lF]fl7$F`flF*Fg\l-F\\l6$7&Fbfl7$F`fl$"++++cVFa^l7$$"+++++oFa^lFgfl7$FjflF*Fg\l-F\\l6$7&F\gl7$Fjfl$"+++++\Fa^l7$$"+++++sFa^lFagl7$FdglF*Fg\l-F\\l6$7&Ffgl7$Fdgl$"++++waFa^l7$$"+++++wFa^lF[hl7$F^hlF*Fg\l-F\\l6$7&F`hl7$F^hl$"++++%3'Fa^l7$$Ff]lFa^lFehl7$FhhlF*Fg\l-F\\l6$7&Fihl7$Fhhl$"++++CnFa^l7$$"+++++%)Fa^lF^il7$FailF*Fg\l-F\\l6$7&Fcil7$Fail$"++++'R(Fa^l7$$"+++++))Fa^lFhil7$F[jlF*Fg\l-F\\l6$7&F]jl7$F[jl$"+++++")Fa^l7$$"+++++#*Fa^lFbjl7$FejlF*Fg\l-F\\l6$7&Fgjl7$Fejl$"++++O))Fa^l7$$"+++++'*Fa^lF\[m7$F_[mF*Fg\l-F\\l6$7&Fa[m7$F_[m$"++++/'*Fa^l7$$"""F+Ff[m7$Fi[mF*Fg\l-F\\l6$7&F[\m7$Fi[m$"+++SS5!"*7$$"++++S5Fb\mF`\m7$Fd\mF*Fg\l-F\\l6$7&Ff\m7$Fd\m$"+++gB6Fb\m7$$"++++!3"Fb\mF[]m7$F^]mF*Fg\l-F\\l6$7&F`]m7$F^]m$"++++57Fb\m7$$"++++?6Fb\mFe]m7$Fh]mF*Fg\l-F\\l6$7&Fj]m7$Fh]m$"+++g*H"Fb\m7$$"++++g6Fb\mF_^m7$Fb^mF*Fg\l-F\\l6$7&Fd^m7$Fb^m$"+++S#R"Fb\m7$$F`^lFb\mFi^m7$F\_mF*Fg\l-F\\l6$7&F]_m7$F\_m$"+++S)["Fb\m7$$"++++S7Fb\mFb_m7$Fe_mF*Fg\l-F\\l6$7&Fg_m7$Fe_m$"+++g(e"Fb\m7$$"++++!G"Fb\mF\`m7$F_`mF*Fg\l-F\\l6$7&Fa`m7$F_`m$"++++!p"Fb\m7$$"++++?8Fb\mFf`m7$Fi`mF*Fg\l-F\\l6$7&F[am7$Fi`m$"+++g&z"Fb\m7$$"++++g8Fb\mF`am7$FcamF*Fg\l-F\\l6$7&Feam7$Fcam$"+++S/>Fb\m7$$"+++++9Fb\mFjam7$F]bmF*Fg\l-F\\l6$7&F_bm7$F]bm$"+++S;?Fb\m7$$"++++S9Fb\mFdbm7$FgbmF*Fg\l-F\\l6$7&Fibm7$Fgbm$"+++gJ@Fb\m7$$"++++!["Fb\mF^cm7$FacmF*Fg\l-F\\l6$7&Fccm7$Facm$"++++]AFb\m7$$"++++?:Fb\mFhcm7$F[dmF*Fg\l-F\\l6$7&F]dm7$F[dm$"+++grBFb\m7$$"++++g:Fb\mFbdm7$FedmF*Fg\l-F\\l6$7&Fgdm7$Fedm$"+++S'\#Fb\m7$$F[_lFb\mF\em7$F_emF*Fg\l-F\\l6$7&F`em7$F_em$"+++SCEFb\m7$$"++++S;Fb\mFeem7$FhemF*Fg\l-F\\l6$7&Fjem7$Fhem$"+++gbFFb\m7$$"++++!o"Fb\mF_fm7$FbfmF*Fg\l-F\\l6$7&Fdfm7$Fbfm$"++++!*GFb\m7$$"++++?<Fb\mFifm7$F\gmF*Fg\l-F\\l6$7&F^gm7$F\gm$"+++gFIFb\m7$$"++++g<Fb\mFcgm7$FfgmF*Fg\l-F\\l6$7&Fhgm7$Ffgm$"+++SoJFb\m7$$"+++++=Fb\mF]hm7$F`hmF*Fg\l-F\\l6$7&Fbhm7$F`hm$"+++S7LFb\m7$$"++++S=Fb\mFghm7$FjhmF*Fg\l-F\\l6$7&F\im7$Fjhm$"+++gfMFb\m7$$"++++!)=Fb\mFaim7$FdimF*Fg\l-F\\l6$7&Ffim7$Fdim$"++++5OFb\m7$$"++++?>Fb\mF[jm7$F^jmF*Fg\l-F\\l6$7&F`jm7$F^jm$"+++gjPFb\m7$$Fh^lFb\mFejm7$FhjmF*Fg\l-F\\l6$7&Fijm7$Fhjm$"+++S?RFb\m7$$Fb[lF+F^[n7$Fa[nF*Fg\l-F\\l6$7&Fb[n7$Fa[n$"+++S!3%Fb\m7$$"++++S?Fb\mFg[n7$Fj[nF*Fg\l-F\\l6$7&F\\n7$Fj[n$"+++gVUFb\m7$$"++++!3#Fb\mFa\n7$Fd\nF*Fg\l-F\\l6$7&Ff\n7$Fd\n$"++++5WFb\m7$$"++++?@Fb\mF[]n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Das exakte Ergebnis (auf 10 Stellen nach dem Dezimalpunkt) lautet:evalf(int(x^2,x=0..4));

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Integrationstechniken

Partielle Integration wird korrekt ausgef\374hrt:(Bemerkung: Die Integrationskonstanten fehlen jeweils.)Int(x*exp(x),x)=int(x*exp(x),x);

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Ein Beispiel f\374r die korrekte Integration mittels Substitution:Int((ln(t))^2*1/t,t)=int((ln(t))^2*1/t,t);

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Uneigentliche Integrale

F\374r uneigentliche Integrale muss man keine neuen Maple-Befehle lernen!Ein uneigentliches Integral mit unendlichem Integrationsbereich:Int(1/(1+x^2),x=0..infinity)=int(1/(1+x^2),x=0..infinity);

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Ein uneigentliches Integral mit unbeschr\344nktem Integranden:Int(1/x^2,x=0..2)=int(1/x^2,x=0..2);

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Mehrfachintegrale

Zun\344chst wird bei einem Doppelintegral das innere Integral berechnet:Int( Int(x*y,y=x^2..x+2), x=-1..2 )=Int( int(x*y,y=x^2..x+2), x=-1..2);

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Danach wird auch noch das \344u\337ere Integral berechnet:Int( int(x*y,y=x^2..x+2), x=-1..2)=int( int(x*y,y=x^2..x+2), x=-1..2);

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Zum Schlu\337 wird das Endergebnis ausgerechnet:int( int(x*y,y=x^2..x+2), x=-1..2)= evalf(int( int(x*y,y=x^2..x+2), x=-1..2));

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