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{SECT 0 {EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {MPLTEXT 0 21 51 "     ATELIER SUR L'INTEGRATION PAR UN CALCUL \+
FORMEL" }}{PARA 0 "" 0 "" {TEXT -1 114 "Michel Mizony , IREM de Lyon  \+
                                                                  Lill
e,  Avril 2005" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 174 "Rappel sur les variables, fonctions et proc\351dure
s : nous voulons par exemple \351tudier la fonction x*sin(x), avec le \+
logiciel Maple nous avons plusieurs moyens pour la nommer :" }
{MPLTEXT 0 21 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f:=x*s
in(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*&%\"xG\"\"\"-%$sinG6#
F&F'" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 112 "elle est d\351finie
 comme la variable f de param\350tre x, et pour avoir sa valeur un un \+
point a, on substitue a \340 x :" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "subs(x=a,f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"
aG\"\"\"-%$sinG6#F$F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "ff
:=x->x*sin(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ffGf*6#%\"xG6\"6$
%)operatorG%&arrowGF(*&9$\"\"\"-%$sinG6#F-F.F(F(F(" }}}{EXCHG {PARA 0 
"" 0 "" {MPLTEXT 0 21 64 "ff est une fonction de la variable x et on l
'utilise comme telle" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "ff(b
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"bG\"\"\"-%$sinG6#F$F%" }}}
{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 78 "Une fonction \351tant une pro
c\351dure simple on peut poser (de mani\350re \351quivalente)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "fff:=proc(x) x*sin(x) end;" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$fffGf*6#%\"xG6\"F(F(*&9$\"\"\"-%$
sinG6#F*F+F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "fff(c);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"cG\"\"\"-%$sinG6#F$F%" }}}
{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 62 "Enfin f \351tant une variable
, on peut la transformer en fonction" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "ffff:=unapply(f,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%%ffffGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&9$\"\"\"-%$sinG6#F-F.F(
F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "ffff := proc (x) op
tions operator, arrow; x*sin(x) end proc;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%ffffGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&9$\"\"\"-%
$sinG6#F-F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "ffff(d)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"dG\"\"\"-%$sinG6#F$F%" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {MPLTEXT 0 21 
158 "Exercice 1: des fonctions Lebesgue int\351grables et non Riemann \+
int\351grables. Cherchons une fonction g qui vaut a(x) sur les rationn
els et b(x) ailleurs. Posons:" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 54 "g:=x-> if type(x,rational) then a(x) else b(x) e
nd if;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operat
orG%&arrowGF(@%-%%typeG6$9$%)rationalG-%\"aG6#F0-%\"bGF4F(F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "g(1),g(Pi),g(2/3),g(x),g(1.5
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'-%\"aG6#\"\"\"-%\"bG6#%#PiG-F$6#
#\"\"#\"\"$-F(6#%\"xG-F(6#$\"#:!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 123 "# on remarque que le nombre d\351cimal 1.5 n'est pas
 consid\351r\351 comme un rationnel. Un d\351cimal se nommant 'float' \+
prenons alors" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "g:=x-> if \+
type(x,rational) or type(x,float) then a(x) else b(x) end if;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrow
GF(@%5-%%typeG6$9$%)rationalG-F/6$F1%&floatG-%\"aG6#F1-%\"bGF8F(F(F(" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "g(1),g(Pi),g(2/3),g(x),g(
1.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'-%\"aG6#\"\"\"-%\"bG6#%#PiG-F
$6##\"\"#\"\"$-F(6#%\"xG-F$6#$\"#:!\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{MPLTEXT 0 21 78 "Notre fonction est maintenant bien d\351finie, preno
ns son int\351grale entre 0 et y" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "A:=int(g(x),x=0..y);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%\"AG-%$intG6$-%\"bG6#%\"xG/F+;\"\"!%\"yG" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 22 "value(subs(b(x)=x,A));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$*&\"\"#!\"\"%\"yGF%\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{MPLTEXT 0 21 158 "Cette fonction g, non Riemann int\351grable, a \351
t\351 int\351gr\351e au sens de Lebesgue. (Nous avons jou\351 sur le t
ypage des nombres pour d\351finir les bor\351liens Q et R\\Q)." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff(A,y);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#-%\"bG6#%\"yG" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 
21 72 "Un divertissement : int\351grer la fonction (x-x^2)^4/(1+x^2) e
ntre 0 et 1." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "Oh:=Int((x-
x^2)^4/(1+x^2),x=0..1)=int((x-x^2)^4/(1+x^2),x=0..1);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#OhG/-%$IntG6$*&,&%\"xG\"\"\"*$)F+\"\"#F,!\"\"\"
\"%,&F,F,F-F,F0/F+;\"\"!F,,&#\"#A\"\"(F,%#PiGF0" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 15 "evalf(rhs(Oh));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#$\"(*[k7!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 181 "Exercice 2 :Int\351grer sur \+
le carr\351 [0,1]x[0,1] la fonction (x^2-y^2)/(x^2+y^2)^2. Cet exercic
e d\351licat a fait appara\356tre plusieurs r\351ponses : Pi/4, -Pi/4,
 l'infini, non d\351finie etc.." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 32 "h:=(x,y)->(x^2-y^2)/(x^2+y^2)^2;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"hGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF)*&,&*$)9$\"\"#\"\"
\"F3*$)9%F2F3!\"\"F3,&F4F3F/F3!\"#F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "I1:=int(int(h(x,y),x=0..1),y=0..1);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%#I1G,$*&\"\"%!\"\"%#PiG\"\"\"F(" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 35 "I2:=int(int(h(x,y),y=0..1),x=0..1);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#I2G,$*&\"\"%!\"\"%#PiG\"\"\"F*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "I3:=int(int(h(x,y),y=0..x),x
=0..1);#h est positive sur ce triangle" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#I3G%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 55 "A
pr\350s \351changes et discussions un consensus s'est fait :" }}{PARA 
0 "" 0 "" {TEXT -1 66 "-D'apr\350s Fubini, ou d'apr\350s I3 elle n'est
 pas Lebesgue-int\351grable." }}{PARA 0 "" 0 "" {TEXT -1 193 "-Les exp
ressions I1 et I2 n'ont pas de sens; c'est un exemple qui montre que l
'on a demand\351 au logiciel de r\351pondre faux. (Maple ne s'est pas \+
tromp\351, il a fait ce qu'on lui a demand\351 de faire)." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{MPLTEXT 0 21 98 "Comme dernier exercice (d\351licat) : int\351grer su
r [0, +infini[, la fonction sin(t)/(sqrt(t)+sin(t))." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 30 "f:=t->sin(t)/(sqrt(t)+sin(t));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"tG6\"6$%)operatorG%&arrowGF(*&-
%$sinG6#9$\"\"\",&-%%sqrtGF/F1F-F1!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 24 "int(f(t),t=0..infinity);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%$intG6$*&-%$sinG6#%\"tG\"\"\",&*$F*#F+\"\"#F+F'F+!\"
\"/F*;\"\"!%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "l
imit(f(t),t=0);# pas de probl\350me en 0." }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "A:=
asympt(f(t),t,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG,.*&-%$sinG
6#%\"tG\"\"\"*&F+F+F*!\"\"#F+\"\"#F+*&F'F/F*F-F-*&)F'\"\"$F+)F,#F3F/F+
F+*&F'\"\"%F*!\"#F-*&)F'\"\"&F+)F,#F;F/F+F+-%\"OG6#*&F+F+*$)F*F3F+F-F+
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "f1:=op(1,A);int(f1,t=0.
.infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1G*&-%$sinG6#%\"tG
\"\"\"*&F*F*F)!\"\"#F*\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"
\"#!\"\"F%#\"\"\"F%%#PiGF'F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 34 "f2:=op(2,A);int(f2,t=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#f2G,$*&-%$sinG6#%\"tG\"\"#F*!\"\"F," }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$%)infinityG!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 
0 21 179 "f1 est Riemann g\351n\351ralis\351e int\351grable mais pas L
ebesgue int\351grable; f2 n'est ni Lebesgue ni Riemann g\351n\351ralis
\351e int\351grable. De l\340 on en d\351duit que f n'est pas Lebesgue
 int\351grable." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "evalf(in
t(f(t),t=0..infinity));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!\"\"%)inf
inityG" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 81 "Cette \351valuatio
n num\351rique montre que f n'est pas Riemann g\351n\351ralis\351e int
\351grable." }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 194 "En conclusio
n, ces exercices montrent l'affirmation de ce matin : il y a une th
\351orie Maple de l'int\351gration qui n'est ni celle de Riemann, ni c
elle de Lebesgue, ni celle de Riemann g\351n\351ralis\351e.\n" }}}}
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