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{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"\"\"\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$%\"xG#\"\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&*
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{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG\"\"!" }}{PARA 11 "" 1 "" 
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 " 0 "" {MPLTEXT 1 0 12 "part1:=rule;" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%&part1G,$*&#\"\"\"\"\"%F(*&%#PiGF(,(*&#\"\"&\"#=F(-%$sinG6#,&*&#F
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{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG,$*&\"\"%!\"\"%#PiG\"\"\"F*" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG,$*&\"\"#!\"\"%#PiG\"\"\"F*" }}}
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 22 "r:=sqrt((3/7)^2-3/35);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"rG,$*(\"\"#\"\"\"\"#N!\"\"\"#I#F(F'F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "xn:=[-sqrt(3/7+r),-sqrt(3/7-r),sqrt
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{MPLTEXT 1 0 54 "rule:=ce*y(xn[1])+ci*y(xn[2])+ci*y(xn[3])+ce*y(xn[4])
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 " 0 "" {MPLTEXT 1 0 12 "eq1:=rule=2;" }}{PARA 11 "" 1 "" {XPPMATH 20 
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20 "6#>%\"yGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*$)9$\"\"#\"\"\"F(F(F(
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{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq2G/,&*(\"\"#\"\"\"%#ceGF),&#\"\"$
\"\"(F)*(F(F)\"#N!\"\"\"#I#F)F(F)F)F)*(F(F)%#ciGF),&F,F)*(F(F)F0F1F2F3
F1F)F)#F(F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "solve(\{eq1,
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"> " 0 "" {MPLTEXT 1 0 10 "assign(%);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "y:=x->x^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yGf
*6#%\"xG6\"6$%)operatorG%&arrowGF(*$)9$\"\"#\"\"\"F(F(F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "simplify(rule);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6##\"\"#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 10 "y:=x->x^4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yGf*6#%\"xG6\"
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0 "" {MPLTEXT 1 0 15 "simplify(rule);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6##\"\"#\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "y:=x->x^6
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yGf*6#%\"xG6\"6$%)operatorG%&
arrowGF(*$)9$\"\"'\"\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 15 "simplify(rule);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"#\"\"(
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "y:=x->x^8;" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%\"yGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*$)9$
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fy(rule);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"$e#\"%D7" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalf([%,2/9]);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#7$$\"+\\C71@!#5$\"+AAAAAF&" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 7 "y:='y':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
137 "rule:=(B-A)*(ce/2*y((A+B)/2+(B-A)/2*xn[1])+\nci/2*y((A+B)/2+(B-A)
/2*xn[2])+\nci/2*y((A+B)/2+(B-A)/2*xn[3])+\nce/2*y((A+B)/2+(B-A)/2*xn[
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\"\"F(,**&#F(\"\"#F(*&,&F-F(*&\"#OF*\"#IF-F*F(-%\"yG6#,(*&F.F*F)F(F(*&
F.F*F'F(F(*(\"#qF*F&F(,&\"$D&F(*&F;F(F3F-F(F-F*F(F(F(*&F-F(*&,&*&F2F*F
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F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "A:=0;   B:=Pi/2;   y:
=x->sin(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG\"\"!" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"BG,$*&\"\"#!\"\"%#PiG\"\"\"F*" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"yG%$sinG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "evalf(rule);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+v
(*******!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "A:=0:   B:=P
i/4:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "part1:=rule:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "A:=Pi/4: B:=Pi/2:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "part2:=rule:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "evalf(part1+part2);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#$\"+++++5!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
44 "Gaussian intergation with a weight function:" }{MPLTEXT 1 0 0 "" }
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4 do psi[i]:=x^i-add(int(psi[i-k]*x^i*w,x=A..B)/alpha[i-k]*psi[i-k],k=
1..i):\nalpha[i]:=int(w*psi[i]^2,x=A..B): print(psi[i],alpha[i]) end d
o:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"\"F#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$,&%\"xG\"\"\"F%!\"\"F%" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6$,(*$)%\"xG\"\"#\"\"\"F(F'F(*&\"\"%F(F&F(!\"\"F*" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6$,**$)%\"xG\"\"$\"\"\"F(\"\"'!\"\"*&\"\"*F()F&\"\"#F(F
**&\"#=F(F&F(F(\"#O" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,,*$)%\"xG\"\"%
\"\"\"F(\"#CF(*&\"#;F()F&\"\"$F(!\"\"*&\"#sF()F&\"\"#F(F(*&\"#'*F(F&F(
F.\"$w&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "xn:=[fsolve(psi[
3],x)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xnG7%$\"+obudT!#5$\"+g.G
%H#!\"*$\"+$3X**G'F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "po
l:=(x-xn[2])*(x-xn[3])/(xn[1]-xn[2])/(xn[1]-xn[3])*y(xn[1])+\n(x-xn[1]
)*(x-xn[3])/(xn[2]-xn[1])/(xn[2]-xn[3])*y(xn[2])+\n(x-xn[1])*(x-xn[2])
/(xn[3]-xn[1])/(xn[3]-xn[2])*y(xn[3]);" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#>%$polG,(**$\"+E;Ni!*!#6\"\"\",&%\"xGF*$\"+g.G%H#!\"*!\"\"F*,&F,
F*$\"+$3X**G'F/F0F*-%\"yG6#$\"+obudT!#5F*F***$\"+J*)GK8F9F*,&F,F*$\"+o
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X**G'F/F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "rule:=int(po
l*exp(-x),x=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ruleG,
(*&$\"++,$46(!#5\"\"\"-%\"yG6##\"*\\&H*H\"\"*++]7$F*F**&$\"+^c#*Q5!#6F
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"" 1 "" {XPPMATH 20 "6#>%\"yGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*$)9$
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}}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++++?!\"*" }}}{EXCHG {PARA 0 "> \+
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{XPPMATH 20 "6#$\"+,+++g!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
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0 "" {MPLTEXT 1 0 5 "rule;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+-+++
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{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yGf*6#%\"xG6\"6$%)operatorG%&arrow
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "y:=x->x^6;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"yGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*$)9$\"\"'\"\"
\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "rule;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#$\"+0++So!\"(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "y:=x->1/(x+2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
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1 "" {XPPMATH 20 "6#$\"+j6l/O!#5" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "evalf(int(exp(-x)/(x+2),x=0..infinity));" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#$\"+phG8O!#5" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "(%-%%)/%%*100.;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "y:=u->1/(u/2+2):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 7 "rule/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#$\"+jq$H1#!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "evalf(int
(exp(-2*x)/(x+2),x=0..infinity));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$
\"+*\\cM1#!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "(%-%%)/%*1
00;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+b,J<D!#6" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Final example:" }{MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "restart:w:=1/sqrt(x):  \+
A:=0:  B:=1:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "for i from 0 to 4 \+
do psi[i]:=x^i-add(int(psi[i-k]*x^i*w,x=A..B)/alpha[i-k]*psi[i-k],k=1.
.i):\nalpha[i]:=int(w*psi[i]^2,x=A..B): print(psi[i],alpha[i]) end do:
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"\"\"\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$,&%\"xG\"\"\"#F%\"\"$!\"\"#\"\")\"#X" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6$,(*$)%\"xG\"\"#\"\"\"F(#\"\"$\"#NF(*&#\"\"'\"\"(F(F&F
(!\"\"#\"$G\"\"&D5\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,**$)%\"xG\"\"
$\"\"\"F(#\"\"&\"$J#!\"\"*&#\"#:\"#6F(*$)F&\"\"#F(F(F,*&#F*F0F(F&F(F(#
\"$7&\"'$p$p" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,,*$)%\"xG\"\"%\"\"\"F
(#\"\"(\"%(G\"F(*&#\"#G\"#:F(*$)F&\"\"$F(F(!\"\"*&#\"#9\"#8F(*$)F&\"\"
#F(F(F(*&#F.\"$V\"F(F&F(F3#\"&oF$\"*Do&Rq" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 20 "xn:=solve(psi[2],x);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#xnG6$,&#\"\"$\"\"(\"\"\"*(\"\"#F*\"#N!\"\"\"#I#F*F,F*,&F'F**(
F,F*F-F.F/F0F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+sub:u!#5$\"++6(e:\"F%" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "rule:=c1*y(xn[1])+c2*y(xn[2]
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ruleG,&*&%#c1G\"\"\"-%\"yG6#,
&#\"\"$\"\"(F(*(\"\"#F(\"#N!\"\"\"#I#F(F1F(F(F(*&%#c2GF(-F*6#,&F-F(*(F
1F(F2F3F4F5F3F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "y:=x->1
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "eq1:=rule=int(1/sqrt(x
),x=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq1G/,&%#c1G\"\"\"%#c
2GF(\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "y:=x->x:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "eq2:=rule=int(x/sqrt(x),x=0.
.1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq2G/,&*&%#c1G\"\"\",&#\"\"
$\"\"(F)*(\"\"#F)\"#N!\"\"\"#I#F)F/F)F)F)*&%#c2GF),&F+F)*(F/F)F0F1F2F3
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eq2\},\{c1,c2\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%#c2G,&*&\"#=!
\"\"\"#I#\"\"\"\"\"#F,F,F,/%#c1G,&*&F(F)F*F+F)F,F," }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 10 "assign(%):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "y:=x->x^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yGf
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "simplify(rule);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6##\"\"#\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 10 "y:=x->x^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yGf*6#%\"xG6\"
6$%)operatorG%&arrowGF(*$)9$\"\"$\"\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 15 "simplify(rule);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6##\"\"#\"\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "y:=x->x^4
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "simplify(rule);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6##\"$e#\"%D7" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 15 "evalf([%,2/9]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#7$$\"+\\C71@!#5$\"+AAAAAF&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
15 "y:=x->exp(x^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yGf*6#%\"xG
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalf(rule);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+M$fv_#!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 36 "evalf(int(exp(x^2)/sqrt(x),x=0..1));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+5UdUD!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 15 "(%-%%)/%%*100.;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+/VYSf!#5
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 120 "To evaluate int(exp(x^2)/sqr
t(x),x=0..1/2), we introduce u=2*x and find instead  int(exp(u^2/4)/sq
rt(u),u=0..1)/sqrt(2);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 17 "y:=u->exp(u^2/4):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "evalf(rule)/sqrt(2.);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#$\"+B()z*[\"!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "e
valf(int(exp(x^2)/sqrt(x),x=0..1/2));" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#$\"+3@8!\\\"!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "(%-%
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105" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }