{VERSION 3 0 "IBM INTEL LINUX" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{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 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 }1 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 }1 0 0 0 6 6 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 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 }0 0 0 -1 -1 -1 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 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 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 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "Author" 0 19 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 8 8 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 21 "Integroimistekniikkaa" }} {PARA 19 "" 0 "" {TEXT -1 23 " inttekn.mws 25.1.2000" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 29 "V\344litt\366m\344sti n\344ht\344vi\344 k aavoja" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "Integraalitaulukko" }}{PARA 0 "" 0 "" {TEXT -1 147 "Sella inen syntyy hetkess\344. Voitaisiin sijoittaa exel-tyyppiseksi tauluko ksi, mutta saavat muuut tutkia ... Alla oleva syntyy monistamalla kom entoa" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "#Int(%,x)=int(%,x); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "ja laittamalla #-merkin tila lle haluttu integroitava (ja :). (Rivinvaihtoon SHIFT-ENTER)" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 278 "1/cos(x)^2:Int(%,x)=int(%,x);\n1/sin(x)^2:Int(%,x)=int(%,x);\n1/s qrt(1-x^2):Int(%,x)=int(%,x);\n1/(1+x^2):Int(%,x)=int(%,x);\n1/cosh(x) ^2:Int(%,x)=int(%,x);1/sinh(x)^2:Int(%,x)=int(%,x);\n1/sqrt(x^2+1):Int (%,x)=int(%,x);\n1/sqrt(x^2-1):Int(%,x)=int(%,x);\n1/(1-x^2):Int(%,x)= int(%,x);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(*$) -%$cosG6#%\"xG\"\"#F(!\"\"F.*&-%$sinGF-F(F+F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(*$)-%$sinG6#%\"xG\"\"#F(!\"\"F.,$*& -%$cosGF-F(F+F0!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\" \"\"F(*$-%%sqrtG6#,&\"\"\"F.*$)%\"xG\"\"#F(!\"\"F(!\"\"F1-%'arcsinG6#F 1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(,&*$)%\"xG\" \"#F(\"\"\"F.F.!\"\"F,-%'arctanG6#F," }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/-%$IntG6$*&\"\"\"F(*$)-%%coshG6#%\"xG\"\"#F(!\"\"F.*&-%%sinhGF-F(F+ F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(*$)-%%sinhG6 #%\"xG\"\"#F(!\"\"F.,$*&-%%coshGF-F(F+F0!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(*$-%%sqrtG6#,&*$)%\"xG\"\"#F(\"\"\" F2F2F(!\"\"F0-%(arcsinhG6#F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$In tG6$*&\"\"\"F(*$-%%sqrtG6#,&!\"\"\"\"\"*$)%\"xG\"\"#F(F/F(!\"\"F2-%#ln G6#,&F2F/*$-F+6#F-F(F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*& \"\"\"F(,&\"\"\"F**$)%\"xG\"\"#F(!\"\"!\"\"F--%(arctanhG6#F-" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 16 "Yleisi\344 s\344\344nt\366j\344" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "Int(diff( f(x),x)/f(x),x)=int(diff(f(x),x)/f(x),x);\nInt(f(x)^p*diff(f(x),x),x)= int(f(x)^p*diff(f(x),x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$Int G6$*&-%%diffG6$-%\"fG6#%\"xGF.\"\"\"F+!\"\"F.-%#lnG6#F+" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%$IntG6$*&)-%\"fG6#%\"xG%\"pG\"\"\"-%%diffG6$F )F,F.F,*&)F),&F-F.F.F.\"\"\"F4!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "int(1/x,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG 6#%\"xG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 153 "Maple osaa soveltaa y leisi\344 s\344\344nt\366j\344 m\344\344rittelem\344tt\366m\344\344n f unktiolausekkeeseen f(x). Se ei k\344sittele eksponentin p poikkeusarv oa p=-1, my\366sk\344\344n se ei osaa" }}{PARA 0 "" 0 "" {TEXT -1 47 " kaavojen voimassaoloalueiden mahd. rajoituksia." }}}}{SECT 1 {PARA 4 " " 0 "" {XPPEDIT 18 0 "Int(tan(x),x);" "6#-%$IntG6$-%$tanG6#%\"xGF)" } {TEXT -1 4 " ja " }{XPPEDIT 18 0 "Int(cot(x),x);" "6#-%$IntG6$-%$cotG6 #%\"xGF)" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 145 "N\344m\344 sa adaan suoraan edell\344 olevista s\344\344nn\366ist\344, kummassakin o n osoittajassa nimitt\344j\344n derivaatta (mahd. merkki\344 paitsi). \+ Anetaan Maplen laskea:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "ta n(x): Int(%,x)=int(%,x);cot(x):Int(%,x)=int(%,x);subs(cos(x)=abs(cos(x )),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%$tanG6#%\"xGF*,$ -%#lnG6#-%$cosGF)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-% $cotG6#%\"xGF*-%#lnG6#-%$sinGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% $IntG6$-%$cotG6#%\"xGF*-%#lnG6#-%$sinGF)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 133 "Kyll\344 tuossa sallitaan cos(x):n ja sin(x):n negatiivi setkin arvot, kun lis\344t\344\344n itseisarvot. Voidaan laittaa mukaa n yksinkertaisesti " }{TEXT 0 4 "subs" }{TEXT -1 5 ":lla:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "tan(x): Int(%,x)=int(%,x): subs(co s(x)=abs(cos(x)),%);\ncot(x): Int(%,x)=int(%,x): subs(sin(x)=abs(sin(x )),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%$tanG6#%\"xGF*,$ -%#lnG6#-%$absG6#-%$cosGF)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% $IntG6$-%$cotG6#%\"xGF*-%#lnG6#-%$absG6#-%$sinGF)" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 19 "Sijoitusesimerkkej\344" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 41 "Tehd\344\344n ens in \"k\344sin\" ja sitten changevar" }}{PARA 0 "" 0 "" {TEXT -1 21 "Su oritetaan sijoitus " }{XPPEDIT 18 0 "t=exp(2*x)" "6#/%\"tG-%$expG6#*& \"\"#\"\"\"%\"xGF*" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f:=1/s qrt(exp(2*x)-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*&\"\"\"F&*$ -%%sqrtG6#,&-%$expG6#,$%\"xG\"\"#\"\"\"!\"\"F2F&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "f1:=1/sqrt(t-1);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#f1G*&\"\"\"F&*$-%%sqrtG6#,&%\"tG\"\"\"!\"\"F-F&!\" \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "g:=exp(2*x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG-%$expG6#,$%\"xG\"\"#" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dt=diff(g,x)*dx;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#dtG,$*&-%$expG6#,$%\"xG\"\"#\"\"\"%#dxGF- F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Huom: yll\344 oli yht\344l \366!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "dxtyht:=solve(\{% \},dx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'dxtyhtG<#/%#dxG,$*&%#dtG \"\"\"-%$expG6#,$%\"xG\"\"#!\"\"#\"\"\"F1" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 11 "intf:=f*dx;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%i ntfG*&%#dxG\"\"\"*$-%%sqrtG6#,&-%$expG6#,$%\"xG\"\"#\"\"\"!\"\"F3F'!\" \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "intf:=subs(dxtyht,int f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%intfG,$*&%#dtG\"\"\"*&-%%sqr tG6#,&-%$expG6#,$%\"xG\"\"#\"\"\"!\"\"F4F(F.\"\"\"!\"\"#F4F3" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "Leikataan sininen output-lauseke p unaiseksi input-riviksi ja editoidaan." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "intf := 1/2/(t-1)^(1/2)*dt/t;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%intfG,$*&%#dtG \"\"\"*&-%%sqrtG6#,&%\"tG\"\"\"!\"\"F/F(F.\"\"\"!\"\"#F/\"\"#" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "P\344\344dyttiin muotoon, josta e i heti n\344hd\344, miten jatkettaisiin (kun emme ole osaavinamme osit taisintegrointia.)" }}{PARA 0 "" 0 "" {TEXT -1 23 "Annetaan Maplen las kea:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "int(1/2/(t-1)^(1/2) /t,t);subs(t=g,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'arctanG6#*$-% %sqrtG6#,&%\"tG\"\"\"!\"\"F,\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# -%'arctanG6#*$-%%sqrtG6#,&-%$expG6#,$%\"xG\"\"#\"\"\"!\"\"F1\"\"\"" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$*$-%%sqrtG6#,&-%$expG6#,$%\"xG\"\"#\"\" \"!\"\"F0F$!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "Yll\344 oleva k\344vi hieman takkuisesti. Onneksi student-pakkauksessa on helppok \344ytt\366inen ty\366kalu " }{TEXT 0 9 "changevar" }{TEXT -1 1 "." }} }{EXCHG {PARA 4 "" 0 "" {TEXT -1 18 "student[changevar]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart:with(student):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f:=1/sqrt(exp(2*x)-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*&\"\"\"F&*$-%%sqrtG6#,&-%$expG6#,$%\"xG \"\"#\"\"\"!\"\"F2F&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "i:=Int(f,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"iG-%$IntG6$*& \"\"\"F)*$-%%sqrtG6#,&-%$expG6#,$%\"xG\"\"#\"\"\"!\"\"F5F)!\"\"F3" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "changevar(t=exp(2*x),i,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*&\"\"\"F(*&-%%sqrtG6#,&% \"tG\"\"\"!\"\"F/F(F.\"\"\"!\"\"#F/\"\"#F." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 24 "changevar(t=exp(x),i,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'*&-%%sqrtG6#,&*$)%\"tG\"\"#F'\"\"\"! \"\"F1F'F/\"\"\"!\"\"F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 " changevar(t=exp(-x),i,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$ ,$*&\"\"\"F(*&-%%sqrtG6#,&*&F(F(*$)%\"tG\"\"#F(!\"\"\"\"\"!\"\"F4F(F1 \"\"\"F3F5F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "juurrettava :=1/t^2-1^(1/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,juurrettavaG,&* &\"\"\"F'*$)%\"tG\"\"#F'!\"\"\"\"\"!\"\"F-" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 35 "nim:=sqrt(expand(t^2*juurrettava));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$nimG*$-%%sqrtG6#,&\"\"\"F**$)%\"tG\"\"#\"\"\"! \"\"F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Int(-1/nim,t);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*&\"\"\"F(*$-%%sqrtG6#,&\" \"\"F.*$)%\"tG\"\"#F(!\"\"F(!\"\"F3F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%'arcs inG6#%\"tG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "subs(t=e xp(-x),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%'arcsinG6#-%$expG6#, $%\"xG!\"\"F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "diff(%,x); simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$expG6#,$%\"xG!\" \"\"\"\"*$-%%sqrtG6#,&\"\"\"F0*$)F$\"\"#F*F)F*!\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*&-%$expG6#,$%\"xG!\"\"\"\"\"*$-%%sqrtG6#,&\"\"\"F0-F %6#,$F(!\"#F)F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "sim plify(%-f,symbolic);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 22 "Neli\366ksi t\344ydent\344minen" }}{PARA 0 "" 0 "" {TEXT -1 7 "Muotoa " }{XPPEDIT 18 0 "1/p[2]" "6#*&\" \"\"\"\"\"&%\"pG6#\"\"#!\"\"" }{TEXT -1 5 " ja " }{XPPEDIT 18 0 "1/sq rt(p[2])" "6#*&\"\"\"\"\"\"-%%sqrtG6#&%\"pG6#\"\"#!\"\"" }{TEXT -1 26 " olevat lausekkeet, miss\344 " }{XPPEDIT 18 0 "p[2]" "6#&%\"pG6#\"\"# " }{TEXT -1 34 " on 2. asteen polynomi, palautuvat" }}{PARA 0 "" 0 "" {TEXT -1 39 "arctan- tai arcsin(arccos)-funktioihin." }}{PARA 0 "" 0 " " {TEXT -1 116 "Osaammehan jo koulusta neli\366ksi t\344ydent\344misen , siihenh\344n 2. asteen ratkaisukaavakin perustuu. Student osaa sen m y\366s:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart:with(stud ent):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "juurr:=completesqu are(-x^2+x+4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&juurrG,&*$),&%\"x G\"\"\"#!\"\"\"\"#F*F-\"\"\"F,#\"#<\"\"%F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "Seuraavat yritelm\344t paljastavat, ett\344 t\344ss\344 v aiheessa kyn\344/paperi on verraton yhdistelm\344." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "e2:=Int(1/sqrt('-(x-1/2)^2+17/4'),x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e2G-%$IntG6$*&\"\"\"F)*$-%%sqrtG6#, (*$)%\"xG\"\"#F)!\"\"F1\"\"\"\"\"%F4F)!\"\"F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "Vaikka yritimme hipsukoiden avulla est\344\344 Maplea \+ sievent\344m\344st\344, se sievensi vaan!" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 26 "changevar(t=(x-1/2),e2,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'*$-%%sqrtG6#,(*$),&#\"\"\"\"\"#F1%\" tGF1F2F'!\"\"#\"\"*F2F1F3F1F'!\"\"F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "expand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG 6$*&\"\"\"F'*$-%%sqrtG6#,(*$),&#\"\"\"\"\"#F1%\"tGF1F2F'!\"\"#\"\"*F2F 1F3F1F'!\"\"F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "juurr := \+ -t^2+17/4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&juurrG,&#\"#<\"\"%\" \"\"*$)%\"tG\"\"#\"\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "g:=(x-1/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG,&%\"xG\"\" \"#!\"\"\"\"#F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Otetaan viel \344 neli\366ksi t\344ydent\344minen yleisin kertoimin:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "completesquare(a*x^2+b*x+c,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&),&%\"xG\"\"\"*&%\"bG\"\"\"%\"aG! \"\"#F(\"\"#F/F+F,F(F(*&*$)F*F/F+F+F,F-#!\"\"\"\"%%\"cGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "subs(a=-1,b=1,c=4,%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,&*$),&%\"xG\"\"\"#!\"\"\"\"#F(F+\"\"\"F*#\"#<\" \"%F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 19 "Osittaisintegrointi" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Int(u(x)*diff(v(x),x),x)=int (u(x)*diff(v(x),x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*& -%\"uG6#%\"xG\"\"\"-%%diffG6$-%\"vGF*F+F,F+-%$intGF&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "Eip\344 se Maple arvannut, mit\344 tuosta halus imme. No mutta t\344m\344n se ainakin tekee:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "Diff(u(x)*v(x),x) =diff(u(x)*v(x),x);map(int,%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/- %%DiffG6$*&-%\"uG6#%\"xG\"\"\"-%\"vGF*F,F+,&*&-%%diffG6$F(F+F,F-\"\"\" F,*&F(F4-F26$F-F+F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&-%\"uG6#% \"xG\"\"\"-%\"vGF'F)-%$intG6$,&*&-%%diffG6$F%F(F)F*\"\"\"F)*&F%F4-F26$ F*F(F)F)F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "Osittaisintegrointi kaava saadaan nyt siirt\344m\344ll\344 puolelta toiselle:" }}{PARA 0 " " 0 "" {TEXT -1 26 " " }{XPPEDIT 18 0 "Int(u( x)*diff(v(x),x)) = u(x)*v(x)-Int(v(x)*diff(u(x),x),x);" "6#/-%$IntG6#* &-%\"uG6#%\"xG\"\"\"-%%diffG6$-%\"vG6#F+F+F,,&*&-F)6#F+F,-F16#F+F,F,-F %6$*&-F16#F+F,-F.6$-F)6#F+F+F,F+!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "Jostain syyst\344 tuo eka Int j\344i matikkamoodin ulkopu olelle, no en ryhdy korjailemaan." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student):" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "Esim. 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "i:=Int(x^2*exp(x),x=2..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"iG-%$IntG6$*&)%\"xG\"\"#\"\"\"-%$expG6#F*\"\"\"/F*; F+\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "i1:=intparts(i,x ^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#i1G,(-%$expG6#\"\"$\"\"*-F' 6#\"\"#!\"%-%$IntG6$,$*&%\"xG\"\"\"-F'6#F4F5F-/F4;F-F)!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "i2:=intparts(i1,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#i2G,&-%$expG6#\"\"$F)-%$IntG6$,$-F' 6#%\"xG\"\"#/F0;F1F)\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "N \344in p\344\344stiin kahdella osittaisintegroinnilla tulokseen, jonka osaamme integroida." }}{PARA 0 "" 0 "" {TEXT -1 39 "Nyt voidaan antaa Maplen jo integroida." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "v alue(i2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$expG6#\"\"$\"\"&-F%6 #\"\"#!\"#" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "Esim. 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "i: =Int(1/cos(x)^3,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"iG-%$IntG6$ *&\"\"\"F)*$)-%$cosG6#%\"xG\"\"$F)!\"\"F/" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 25 "i1:=intparts(i,1/cos(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#i1G,&*&-%$sinG6#%\"xG\"\"\"*$)-%$cosGF)\"\"#F+!\"\" \"\"\"-%$IntG6$*&*$)F'\"\"#F+F+*$)F.\"\"$F+F1F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "f:=(1-cos(x)^2)/cos(x)^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*&,&\"\"\"F'*$)-%$cosG6#%\"xG\"\"#\"\"\"!\" \"F/*$)F*\"\"$F/!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "f: =expand(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,&*&\"\"\"F'*$)-% $cosG6#%\"xG\"\"$F'!\"\"\"\"\"*&F'F'F*F/!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "i2:=1/cos(x)^2*sin(x)-Int(f,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#i2G,&*&-%$sinG6#%\"xG\"\"\"*$)-%$cosGF)\"\"#F+! \"\"\"\"\"-%$IntG6$,&*&F+F+*$)F.\"\"$F+F1F2*&F+F+F.F1!\"\"F*F<" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 201 "Saatiin yht\344l\366, jossa ratka istava integraali esiintyy kummallakin puolella. Vapautetaan muuttuja i arvostaan, koska haluamme k\344ytt\344\344 sit\344 muuttujan nimen \344 ratkaistavassa yht\344l\366ss\344 (Kts. [HAM] s. ...)." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "i:='i':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "yhtalo:=i=1/cos(x)^2*sin(x)-i+Int(1/cos(x),x) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'yhtaloG/%\"iG,(*&-%$sinG6#%\"x G\"\"\"*$)-%$cosGF+\"\"#F-!\"\"\"\"\"F&!\"\"-%$IntG6$*&F-F-F0F3F,F4" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "tulos:=solve(yhtalo,i);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&tulosG,$*&,&-%$sinG6#%\"xG\"\"\"*& -%$IntG6$*&\"\"\"F2-%$cosGF*!\"\"F+F,)F3\"\"#F2F,F2*$)F3\"\"#F2F5#F,F7 " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Palaamme my\366hemmin tyyppi \344 " }{XPPEDIT 18 0 "Int(1/cos(x),x);" "6#-%$IntG6$*&\"\"\"\"\"\"-%$ cosG6#%\"xG!\"\"F," }{TEXT -1 85 " olevien integraalien k\344sittelyyn , annamme t\344ss\344 kohdassa Maplen n\344ytt\344\344 osaamistaan." } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "tulos:=value(tulos);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&tulosG,$*&,&-%$sinG6#%\"xG\"\"\"*&- %#lnG6#,&-%$secGF*F,-%$tanGF*F,F,)-%$cosGF*\"\"#\"\"\"F,F:*$)F7\"\"#F: !\"\"#F,F9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "simplify(diff (tulos,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$*$)-%$cosG6#% \"xG\"\"$F$!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Oikein meni ( kokeile huvin vuoksi, milt\344 n\344ytt\344\344 ilman " }{TEXT 0 8 "si mplify" }{TEXT -1 11 "-komentoa)." }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 30 "Trig. funktioiden integroin tia" }}{SECT 1 {PARA 4 "" 0 "" {XPPEDIT 18 0 "Int(sin(x)^m*cos(x)^n,x) ;" "6#-%$IntG6$*&)-%$sinG6#%\"xG%\"mG\"\"\")-%$cosG6#F+%\"nGF-F+" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "Jos m tai n on pariton, \+ saadaan palautetuksi polynomiin sijoituksella " }{XPPEDIT 18 0 "u = si n(x);" "6#/%\"uG-%$sinG6#%\"xG" }{TEXT -1 5 " tai " }{XPPEDIT 18 0 "u \+ = cos(x);" "6#/%\"uG-%$cosG6#%\"xG" }{TEXT -1 2 " ." }}{PARA 0 "" 0 " " {TEXT 256 7 "Esim 1:" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "II:=Int(sin(x)^3*cos(x)^8,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#IIG-%$IntG6$*&)-%$sinG6#%\"xG\"\"$\"\"\")-%$cosGF,\" \")F/F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student):" }}{PARA 7 "" 1 "" {TEXT -1 29 "Warning, new definition for D" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "?changevar" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "III:=changevar(u=cos(x), II, u);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$IIIG-%$IntG6$,$*&,&\"\"\"F+*$)%\"uG \"\"#\"\"\"!\"\"F+)F.\"\")F0F1F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%\"u G\"#6\"\"\"#\"\"\"F'*$)F&\"\"*F(#!\"\"F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "subs(u=cos(x),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,&*$)-%$cosG6#%\"xG\"#6\"\"\"#\"\"\"F**$)F&\"\"*F+#!\"\"F0" }}}{PARA 0 "" 0 "" {TEXT 257 8 "Esim. 2:" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "II:=Int(sin(x)^4*cos(x)^7,x);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#IIG-%$IntG6$*&)-%$sinG6#%\"xG\"\"%\"\"\")-%$cosGF, \"\"(F/F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "III:=changevar (u=sin(x), II, u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$IIIG-%$IntG6$ *&)%\"uG\"\"%\"\"\"),&\"\"\"F/*$)F*\"\"#F,!\"\"\"\"$F,F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "value(%);subs(u=sin(x),%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,**$)%\"uG\"#6\"\"\"#!\"\"F'*$)F&\"\"* F(#\"\"\"\"\"$*$)F&\"\"(F(#!\"$F3*$)F&\"\"&F(#F/F8" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,**$)-%$sinG6#%\"xG\"#6\"\"\"#!\"\"F**$)F&\"\"*F+#\" \"\"\"\"$*$)F&\"\"(F+#!\"$F6*$)F&\"\"&F+#F2F;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 14 "N\344m\344 tuk evat " }{TEXT 258 15 "yleist\344 s\344\344nt\366\344" }{TEXT -1 19 ": \+ tehd\344\344n sijoitus " }{TEXT 259 37 "u=\"se, jonka potenssi on pari llinen\"." }}}{SECT 1 {PARA 4 "" 0 "" {XPPEDIT 18 0 "Int(sin(x)^n,x),I nt(cos(x)^n,x);" "6$-%$IntG6$)-%$sinG6#%\"xG%\"nGF*-F$6$)-%$cosG6#F*F+ F*" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Jos potenssit ovat parillisia, palautetaan moninkertaiste n kulmien kaavoihin." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "cos( x)^2;combine(%,trig);int(%,x);expand(%,trig);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$)-%$cosG6#%\"xG\"\"#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%$cosG6#,$%\"xG\"\"##\"\"\"F)F*F+" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,&-%$sinG6#,$%\"xG\"\"##\"\"\"\"\"%F(#F+F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%$sinG6#%\"xG\"\"\"-%$cosGF'F)#F)\"\"#F (F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "cos(x)^10;combine(%, trig);int(%,x);expand(%,trig);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$)- %$cosG6#%\"xG\"#5\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.-%$cosG6# ,$%\"xG\"#5#\"\"\"\"$7&-F%6#,$F(\"\")#\"\"&\"$c#-F%6#,$F(\"\"'#\"#XF,- F%6#,$F(\"\"%#\"#:\"#k-F%6#,$F(\"\"##\"$0\"F3#\"#jF3F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.-%$sinG6#,$%\"xG\"#5#\"\"\"\"%?^-F%6#,$F(\"\")# \"\"&\"%[?-F%6#,$F(\"\"'#\"#:\"%C5-F%6#,$F(\"\"%#F9\"$c#-F%6#,$F(\"\"# #\"$0\"\"$7&F(#\"#jF@" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,.*&-%$sinG6# %\"xG\"\"\")-%$cosGF'\"\"*\"\"\"#F)\"#5*&F%F.)F+\"\"(F.#F-\"#!)*&F%F.) F+\"\"&F.#\"#@\"$g\"*&F%F.)F+\"\"$F.#F:\"$G\"*&F%F.F+F)#\"#j\"$c#F(FB " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 46 "Jos pariton, niin voidaanpa osittaisintegroida" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student):" }}{PARA 7 "" 1 "" {TEXT -1 29 "Warning, new definition for D" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 9 "?intparts" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Int(sin(x)^5,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$)-% $sinG6#%\"xG\"\"&\"\"\"F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "intparts(%,sin(x)^4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&)-%$s inG6#%\"xG\"\"%\"\"\"-%$cosGF(\"\"\"!\"\"-%$IntG6$,$*&)F&\"\"$F+)F,\" \"#F+!\"%F)F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "changevar( u=cos(x), %, u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&)-%$sinG6#%\"x G\"\"%\"\"\"-%$cosGF(\"\"\"!\"\"-%$IntG6$,$*&,&F.F.*$)%\"uG\"\"#F+F/F. F7F+F*F8F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "value(%);subs (u=cos(x), %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&)-%$sinG6#%\"xG \"\"%\"\"\"-%$cosGF(\"\"\"!\"\"*$)%\"uG\"\"&F+#F*F3*$)F2\"\"$F+#!\"%F7 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&)-%$sinG6#%\"xG\"\"%\"\"\"-%$c osGF(\"\"\"!\"\"*$)F,\"\"&F+#F*F2*$)F,\"\"$F+#!\"%F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "diff(%,x);simplify(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,**&)-%$sinG6#%\"xG\"\"$\"\"\")-%$cosGF(\"\"#F+!\"%*$ )F&\"\"&F+\"\"\"*&F&F4)F-\"\"%F+F0*&F&F+F,F+F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&-%$sinG6#%\"xG\"\"\")-%$cosGF'\"\"%\"\"\"F)*&F%F.)F +\"\"#F.!\"#F%F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "simplif y(%-sin(x)^5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Katsotaan, miten Maplen " }{TEXT 0 4 "int " }{TEXT -1 23 "laskee ilman avustusta:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "Int(sin(x)^5,x);value(%);diff(%,x);simplify(%-sin(x)^ 5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*$)-%$sinG6#%\"xG\"\"& \"\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&)-%$sinG6#%\"xG\"\"%\" \"\"-%$cosGF(\"\"\"#!\"\"\"\"&*&)F&\"\"#F+F,F+#!\"%\"#:F,#!\")F7" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&)-%$sinG6#%\"xG\"\"$\"\"\")-%$cosG F(\"\"#F+#!\"%\"\"&*$)F&F2F+#\"\"\"F2*&F&F6F,F+#!\")\"#:*$F%F+#\"\"%F: F&#\"\")F:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "Yleisl\344\344ke: sijoitus " }{XPPEDIT 18 0 "t = tan(x/2);" "6#/%\"tG -%$tanG6#*&%\"xG\"\"\"\"\"#!\"\"" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Muotoa " }{XPPEDIT 18 0 "R(sin(x),cos(x));" "6#-%\"RG6$-%$ sinG6#%\"xG-%$cosG6#F)" }{TEXT -1 119 " oleva funktio saadaan yll\344 \+ olevalla sijoituksella palautumaan rationaalifunktioksi. T\344ss\344 \+ R(sin(x),cos(x)) tarkoittaa" }}{PARA 0 "" 0 "" {TEXT -1 46 "rationaali sta lauseketta sinist\344 ja cosinista." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart:with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "kolmio:=plot([[0,0],[1,0],[1,1/2],[0,0]]): T:=textpl ot([[1.1,0.25,\"t\"]]): Yks:=textplot([[0.5,0.03,\"1\"]]):hypot:=textp lot([[0.5,0.3,\"sqrt(1+t^2)\"]]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "display([kolmio,T,Yks,hypot]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%'CURVESG6$7&7$\"\"!F(7$$\"\"\"F (F(7$F*$\"1+++++++]!#;F'-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%%TEXTG6$7$$ \"#6F6$\"#D!\"#%\"tG-F86$7$$\"\"&F6$\"\"$F?%\"1G-F86$7$FD$FGF6%,sqrt(1 +t^2)G-%+AXESLABELSG6$%!GFQ-%%VIEWG6$%(DEFAULTGFU" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "restart:g:=tan(x/2);dt:=diff(g,x)*dx;dt:=subs(tan(x/2 )=t,dt);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG-%$tanG6#,$%\"xG#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dtG*&,&#\"\"\"\"\"#F(*$)-%$tanG6#,$%\"xGF'F)\"\"\"F' F(%#dxGF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dtG*&,&#\"\"\"\"\"#F(* $)%\"tGF)\"\"\"F'F(%#dxGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "sinx:=2*sin(x/2)*cos(x/2);sinx:=subs(sin(1/2*x)=t/sqrt(1+t^2),cos( 1/2*x)=1/sqrt(1+t^2),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sinxG,$ *&-%$sinG6#,$%\"xG#\"\"\"\"\"#F--%$cosGF)F-F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sinxG,$*&%\"tG\"\"\",&\"\"\"F**$)F'\"\"#F(F*!\"\"F- " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "cosx:=cos(x/2)^2-sin(x /2)^2;cosx:=subs(sin(1/2*x)=t/sqrt(1+t^2),cos(1/2*x)=1/sqrt(1+t^2),%); \nnormal(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%cosxG,&*$)-%$cosG6# ,$%\"xG#\"\"\"\"\"#F/\"\"\"F.*$)-%$sinGF*F/F0!\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%%cosxG,&*&\"\"\"F',&\"\"\"F)*$)%\"tG\"\"#F'F)!\"\"F )*&*$F+F'F'F(F.!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&!\"\"\" \"\"*$)%\"tG\"\"#\"\"\"F'F,,&F'F'F(F'!\"\"F&" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 5 "Esim:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Int( 1/sin(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'-%$ sinG6#%\"xG!\"\"F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with( student):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Int(1/sin(x),x );changevar(t=tan(x/2),%,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$Int G6$*&\"\"\"F'-%$sinG6#%\"xG!\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# -%$IntG6$,$*&\"\"\"F(*&-%$sinG6#,$-%'arctanG6#%\"tG\"\"#\"\"\",&\"\"\" F5*$)F1F2F(F5\"\"\"!\"\"F2F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "T \344t\344 ei Maple sievenn\344 pitemm\344lle, edell\344 esitetty \"k \344sin sievennys\" on tarpeen." }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 15 "Palautuskaavoja" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "i [0]:=-exp(-x);\nfor n to 6 do i[n]:=simplify(x^n*exp(-x)+n*i[n-1]); od ;\n#array([seq([i[k]],k=0..6)]);\n\n" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>&%\"iG6#\"\"!,$-%$expG6#,$%\"xG!\"\"F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"iG6#\"\"\",&*&%\"xGF'-%$expG6#,$F*!\"\"F'F'F+F/" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"iG6#\"\"#,(*&)%\"xGF'\"\"\"-%$ex pG6#,$F+!\"\"\"\"\"F2*&F+F2F-F,F'F-!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"iG6#\"\"$,**&)%\"xGF'\"\"\"-%$expG6#,$F+!\"\"\"\"\"F2*&)F+ \"\"#F,F-F,F'*&F+F2F-F,\"\"'F-!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >&%\"iG6#\"\"%,,*&)%\"xGF'\"\"\"-%$expG6#,$F+!\"\"\"\"\"F2*&)F+\"\"$F, F-F,F'*&)F+\"\"#F,F-F,\"#7*&F+F2F-F,\"#CF-!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"iG6#\"\"&,.*&)%\"xGF'\"\"\"-%$expG6#,$F+!\"\"\"\" \"F2*&)F+\"\"%F,F-F,F'*&)F+\"\"$F,F-F,\"#?*&)F+\"\"#F,F-F,\"#g*&F+F2F- F,\"$?\"F-!$?\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"iG6#\"\"',0*&) %\"xGF'\"\"\"-%$expG6#,$F+!\"\"\"\"\"F2*&)F+\"\"&F,F-F,F'*&)F+\"\"%F,F -F,\"#I*&)F+\"\"$F,F-F,\"$?\"*&)F+\"\"#F,F-F,\"$g$*&F+F2F-F,\"$?(F-!$? (" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 18 "[RA] Esim 10.3.2\n " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 256 "n:='n':with(student):\nII:= Int(sin(x)^n,x=0..Pi/2);\nintparts(II,sin(x)^(n-1));\n-Int(-sin(x)^(n- 2)*(n-1)*cos(x)^2,x = 0 .. 1/2*Pi);\n\nsubs(cos(x)^2=1-sin(x)^2,%);\n( n-1)*(Int(sin(x)^(n-2),x=0..Pi/2) - Int(sin(x)^n,x=0..Pi/2));\nIn=(n-1 )*(Inm2-In);\nsolve(%,In);\n" }}{PARA 7 "" 1 "" {TEXT -1 29 "Warning, \+ new definition for D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#IIG-%$IntG6 $)-%$sinG6#%\"xG%\"nG/F,;\"\"!,$%#PiG#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$IntG6$,$*&*()-%$sinG6#%\"xG,&%\"nG\"\"\"!\"\"F1F1F /F1)-%$cosGF-\"\"#\"\"\"F7F+!\"\"F2/F.;\"\"!,$%#PiG#F1F6F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$IntG6$,$*()-%$sinG6#%\"xG,&%\"nG\"\"\"! \"#F0F0,&F/F0!\"\"F0F0)-%$cosGF,\"\"#\"\"\"F3/F-;\"\"!,$%#PiG#F0F7F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$IntG6$,$*()-%$sinG6#%\"xG,&%\"n G\"\"\"!\"#F0F0,&F/F0!\"\"F0F0,&F0F0*$)F*\"\"#\"\"\"F3F0F3/F-;\"\"!,$% #PiG#F0F7F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"nG\"\"\"!\"\"F&F &,&-%$IntG6$)-%$sinG6#%\"xG,&F%F&!\"#F&/F0;\"\"!,$%#PiG#F&\"\"#F&-F*6$ )F-F%F3F'F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#InG*&,&%\"nG\"\"\"! \"\"F(F(,&%%Inm2GF(F$F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&*&,&%\" nG\"\"\"!\"\"F'F'%%Inm2GF'\"\"\"F&!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "Yll\344 piti tehd\344 v\344h\344n k\344sit\366it\344. Tut ki, mill\344 sievennyskomennoilla saataisiin vaivattomasti.\n" }}}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 18 "Rationaalifunktiot" }}{PARA 0 "" 0 "" {TEXT -1 139 "Jos osoittajan asteluku on suurempi tai yht\344suur i kuin nimitt\344j\344n, suoritetaan jako. Integroinnin kannalta j\344 \344 siten k\344sitelt\344v\344ksi tapaus " }{TEXT 261 15 "deg(P) < de g(Q)" }}{PARA 0 "" 0 "" {TEXT -1 75 "Katsotaan pieni esimerkki, jossa \+ k\344ytet\344\344n \"jakokulmana\" Maplen funktioita " }{TEXT 0 3 "quo " }{TEXT -1 4 " ja " }{TEXT 0 3 "rem" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "R:=(x^3+3 *x^2)/(x^2+1);P:=numer(R): Q:=denom(R):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG*&,&*$)%\"xG\"\"$\"\"\"\"\"\"*$)F)\"\"#F+F*F+,&F-F,F,F,!\" \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "osam:=quo(P,Q,x);jako j:=rem(P,Q,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%osamG,&%\"xG\"\" \"\"\"$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&jakojG,&%\"xG!\"\"!\"$ \"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Ruusi:=osam+jakoj /Q;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&RuusiG,(%\"xG\"\"\"\"\"$F'*& ,&F&!\"\"!\"$F'\"\"\",&*$)F&\"\"#F-F'F'F'!\"\"F'" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 10 "Tarkistus:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "normal(Ruusi);simplify(R-Ruusi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&*&)%\"xG\"\"#\"\"\",&F&\"\"\"\"\"$F*F*F(,&*$F%F(F*F*F*!\"\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "Nimitt\344j\344n asteluku = 2" }}{PARA 0 "" 0 "" {TEXT -1 49 "Neli\366ksi t\344ydent\344m\344ll\344 saadaan nimitt\344j\344 m uotoon " }{XPPEDIT 18 0 "x^2+a^2;" "6#,&*$%\"xG\"\"#\"\"\"*$%\"aG\"\"# F'" }{TEXT -1 6 " tai " }{XPPEDIT 18 0 "x^2-a^2;" "6#,&*$%\"xG\"\"#\" \"\"*$%\"aG\"\"#!\"\"" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Int(x/(x^2+a^2),x)=int(x /(x^2+a^2),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&%\"xG\" \"\",&*$)F(\"\"#F)\"\"\"*$)%\"aGF-F)F.!\"\"F(,$-%#lnG6#F*#F.F-" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Int(x/(x^2-a^2),x)=int(x/(x^ 2-a^2),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&%\"xG\"\"\", &*$)F(\"\"#F)\"\"\"*$)%\"aGF-F)!\"\"!\"\"F(,$-%#lnG6#F*#F.F-" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Int(1/(x^2+a^2),x)=int(1/(x^ 2+a^2),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(,&*$ )%\"xG\"\"#F(\"\"\"*$)%\"aGF-F(F.!\"\"F,*&-%'arctanG6#*&F,F(F1F2F(F1F2 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Int(1/(x^2-a^2),x)=int( 1/(x^2-a^2),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F (,&*$)%\"xG\"\"#F(\"\"\"*$)%\"aGF-F(!\"\"!\"\"F,,&*&-%#lnG6#,&F,F.F1F2 F(F1F3#F.F-*&-F76#,&F,F.F1F.F(F1F3#F2F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Kolme ensimm\344ist\344 on selvi\344. Mist\344 seuraa nel j\344s." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "1/(x^2-a^2)=conv ert(1/(x^2-a^2),parfrac,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&\"\" \"F%,&*$)%\"xG\"\"#F%\"\"\"*$)%\"aGF*F%!\"\"!\"\",&*&F%F%*&F.\"\"\",&F )F+F.F/\"\"\"F0#F+F**&F%F%*&F.\"\"\",&F)F+F.F+\"\"\"F0#F/F*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 262 31 "Osamurto eli \"partial fraction\" " }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 24 "Esim: [RA] 10.4.1 s. 140" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "R:=(x^4+1) /(x^3-x^2+x-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG*&,&*$)%\"xG \"\"%\"\"\"\"\"\"F,F,F+,**$)F)\"\"$F+F,*$)F)\"\"#F+!\"\"F)F,F4F,!\"\" " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Suoritetaan jako ensin \"k \344sin\":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "P:=numer(R);Q :=denom(R);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG,&*$)%\"xG\"\"%\" \"\"\"\"\"F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"QG,**$)%\"xG\"\" $\"\"\"\"\"\"*$)F(\"\"#F*!\"\"F(F+F/F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "P=(a*x+b)*Q+r;\nyht:=expand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*$)%\"xG\"\"%\"\"\"\"\"\"F*F*,&*&,&*&%\"aGF*F'F*F*% \"bGF*F*,**$)F'\"\"$F)F**$)F'\"\"#F)!\"\"F'F*F8F*F*F*%\"rGF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$yhtG/,&*$)%\"xG\"\"%\"\"\"\"\"\"F,F,,4*&% \"aGF,F(F+F,*&F/F+)F)\"\"$F+!\"\"*&F/F+)F)\"\"#F+F,*&F/F+F)F,F3*&%\"bG F,F1F+F,*&F9F+F5F+F3*&F9F+F)F+F,F9F3%\"rGF," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "kaikkivasemmalle:=lhs(yht)-rhs(yht);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%1kaikkivasemmalleG,8*$)%\"xG\"\"%\"\"\"\"\"\"F +F+*&%\"aGF+F'F*!\"\"*&F-F*)F(\"\"$F*F+*&F-F*)F(\"\"#F*F.*&F-F*F(F+F+* &%\"bGF+F0F*F.*&F7F*F3F*F+*&F7F*F(F*F.F7F+%\"rGF." }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 6 "kv:=%:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "coeff(kv,x,4)=0,coeff(kv,x,3)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/,&\"\"\"F%%\"aG!\"\"\"\"!/,&F&F%%\"bGF'F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "a:=1:b:=1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "kv;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"#\"\"\"% \"rG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "r:=2:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "R1:=a*x+b+r/Q;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#R1G,(%\"xG\"\"\"F'F'*&\"\"\"F),**$)F&\"\"$F)F '*$)F&\"\"#F)!\"\"F&F'F1F'!\"\"F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "normal(R1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*$ )%\"xG\"\"%\"\"\"\"\"\"F*F*F),**$)F'\"\"$F)F**$)F'\"\"#F)!\"\"F'F*F2F* !\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Oikein meni!" }}{PARA 0 "" 0 "" {TEXT -1 59 "Oltais saatu suoremmin valmiilla funktioilla (vrt . edell\344):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "osam:=quo( P,Q,x);jakoj:=rem(P,Q,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%osamG, &%\"xG\"\"\"F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&jakojG\"\"#" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Otetaan rationaaliosa " } {XPPEDIT 18 0 "2/Q;" "6#*&\"\"#\"\"\"%\"QG!\"\"" }{TEXT -1 0 "" } {TEXT -1 38 " k\344sittelyyn (polynomi kyll\344 hoituu.)" }}{PARA 0 " " 0 "" {TEXT -1 29 "Jaetaan nimitt\344j\344 tekij\366ihin:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Q:=factor(Q);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"QG*&,&%\"xG\"\"\"!\"\"F(F(,&*$)F'\"\"#\"\"\"F(F(F(F (" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Reseptin mukaan osamurtokehi telm\344 on muotoa:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "R2:= A/(x-1)+(B*x+C)/(x^2+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#R2G,&*& %\"AG\"\"\",&%\"xG\"\"\"!\"\"F+!\"\"F+*&,&*&%\"BGF+F*F+F+%\"CGF+F(,&*$ )F*\"\"#F(F+F+F+F-F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Kerrotaan puolittain Q:lla:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "vasen _oikea:=2-normal(Q*R2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,vasen_oi keaG,0\"\"#\"\"\"*&%\"AGF')%\"xGF&\"\"\"!\"\"F)F-*&%\"BGF'F*F,F-*&F/F, F+F'F'*&%\"CGF'F+F,F-F2F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "\{seq(coeff(vasen_oikea,x,i)=0,i=0..3)\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&/\"\"!F%/,&%\"BG!\"\"%\"AGF)F%/,&F(\"\"\"%\"CGF)F%/,( \"\"#F-F*F)F.F-F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "solve( %,\{A,B,C\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%\"BG!\"\"/%\"AG\" \"\"/%\"CGF&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "subs(%,R2); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"\"F%,&%\"xG\"\"\"!\"\"F(! \"\"F(*&,&F'F)F)F(F%,&*$)F'\"\"#F%F(F(F(F*F(" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 126 "\"Ik\344v\344 kyll\344\" kaikki yll\344 olevat hienot operaatiot voidaan korvata yhdell\344 ainoalla komennolla. Kirjoiteta an yht\344l\366n muotoon." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "R=convert(R,parfrac,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,&*$ )%\"xG\"\"%\"\"\"\"\"\"F+F+F*,**$)F(\"\"$F*F+*$)F(\"\"#F*!\"\"F(F+F3F+ !\"\",*F(F+F+F+*&F*F*,&F(F+F3F+F4F+*&,&F(F+F+F+F*,&F0F+F+F+F4F3" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 112 "Sujuuko TKK:n ulkopuolisessa kov assa maailmassa n\344in? Mist\344 voisi olla kiinni? No nimitt\344j \344n tekij\366ihin jaosta." }}{PARA 0 "" 0 "" {TEXT -1 88 "Voidaan jo utua turvautumaan numeerisiin approksimaatioihin polynomin juurille, t ottakai." }}{PARA 0 "" 0 "" {TEXT -1 134 "Palataan integraalimuunnoksi en yhteydess\344 tarkemmin, numeeriset osittaismurtoesitykset kannatta a tehd\344 Matlabilla (ainakin jos isoja)." }}}}{SECT 0 {PARA 4 "" 0 " " {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 0 " " }}}}{MARK "7 10" 0 }{VIEWOPTS 1 1 0 1 1 1803 }