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{SECT 0 {EXCHG {PARA 265 "" 0 "" {TEXT 290 35 "Matematick\351 v\375po
\350ty se syst\351m Maple" }}}{EXCHG {PARA 266 "" 0 "" {TEXT -1 77 "au
tor:  Ing. Vladim\355r \216\341k\n  email:  zakyn@centrum.cz\n        \+
         web:   " }{URLLINK 17 "www.vladimirzak.com/maple" 4 "www.vlad
imirzak.com/maple" "" }}{PARA 265 "" 0 "" {TEXT -1 0 "" }}{PARA 265 "
" 0 "" {TEXT -1 12 "            " }{TEXT 288 9 "recenzent" }{TEXT -1 
3 ":  " }{TEXT 287 31 "Prof. RNDr. Ji\370\355 H\370eb\355\350ek, CSc.
" }}{PARA 265 "" 0 "" {TEXT -1 48 "CENTRUM BIOSTATISTIKY A ANAL\335Z (
CBA LF a P\370F MU)" }}{PARA 265 "" 0 "" {TEXT -1 43 "L\351ka\370sk
\341 a P\370\355rodov\354deck\341 fakulta MU v Brn\354" }}{PARA 265 "
" 0 "" {TEXT -1 20 "hrebicek@cba.muni.cz" }}{PARA 265 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 265 "" 0 "" {TEXT -1 478 "Tento dokument je ur
\350en jen pro v\375ukov\351 \372\350ely a pro pou\236it\355 jen osoba
mi, kter\375m byl tento dokument poskytnut p\370\355mo autorem, pop
\370. si ho st\341hli z webov\351ho s\355dla autora. Jak\351koliv \350
\341sti dokumentu nen\355 mo\236n\351 pou\236\355t bez citace zdoje, t
edy n\341zvu dokumentu a jm\351na autora s kontaktn\355mi informacemi \+
(jak je zobrazeno v hlavi\350ce), kter\341 mus\355 b\375t autorovi sd
\354lena. Pro jin\351 pou\236it\355 ne\236 pro osobn\355 pot\370eby os
ob, tedy pro distribuci \350i kop\355rov\341n\355, je nutn\351 obdr
\236et p\355semn\375 souhlas autora. " }}{PARA 265 "" 0 "" {TEXT -1 0 
"" }}{PARA 265 "" 0 "" {TEXT -1 0 "" }{TEXT 289 59 "Soubor vznikl s po
dporou grantov\351ho projektu FRV\212 3323/2006" }{TEXT -1 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 40 "Riemann
\371v a nevlastn\355 Riemann\371v integr\341l" }}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 51 "V p\370edchoz\355ch dvou kapitol\341ch byly pops\341ny \+
p\370\355kazy " }{TEXT 256 11 "int(f(x),x)" }{TEXT -1 3 " a " }{TEXT 
257 11 "Int(f(x),x)" }{TEXT -1 33 " pro primitivn\355 funkce a p\370
\355kazy " }{TEXT 258 26 "intparts(Int(h(x),x),k(x))" }{TEXT -1 3 " a \+
" }{TEXT 259 43 "changevar(relace,Int(g(x),x),nov\341 prom\354nn\341)
" }{TEXT -1 12 " z knihovny " }{TEXT 260 7 "student" }{TEXT -1 141 " p
ro v\375po\350et primitivn\355 funkce metodou per partes a metodou sub
stitu\350n\355. V\232echno, co bylo uvedeno o t\354chto p\370\355kazec
h, v\350etn\354 pou\236it\355 p\370\355kazu " }{TEXT 281 5 "value" }
{TEXT -1 26 ", se p\370en\341\232\355 i na p\370\355kazy " }{TEXT 264 
3 "int" }{TEXT -1 2 ", " }{TEXT 265 3 "Int" }{TEXT -1 2 ", " }{TEXT 
266 8 "intparts" }{TEXT -1 3 " a " }{TEXT 267 9 "changevar" }{TEXT -1 
132 " pro v\375po\350et a z\341pis Riemannova integr\341lu a nevlastn
\355ho integr\341lu. Rozd\355l je pouze v tom, \236e se v p\370\355kaz
ech dod\341 informace o mez\355ch." }}{PARA 0 "" 0 "" {TEXT -1 21 "P
\370\355kazy pak maj\355 tvar" }}{PARA 0 "" 0 "" {TEXT 261 37 "int( f(
x), x=doln\355 mez .. horn\355 mez )" }}{PARA 0 "" 0 "" {TEXT 262 37 "
Int( f(x), x=doln\355 mez .. horn\355 mez )" }}{PARA 0 "" 0 "" {TEXT 
263 53 "intparts( Int(h(x),x= doln\355 mez .. horn\355 mez), k(x) )" }
}{PARA 0 "" 0 "" {TEXT 268 72 "changevar( relace, Int( g(x),x= doln
\355 mez .. horn\355 mez), nov\341 prom\354nn\341 )" }}{PARA 0 "" 0 "
" {TEXT -1 70 "Jako meze mohou b\375t uvedeny libovoln\351 v\375razy. \+
Mohou b\375t i nevlastn\355 (-" }{TEXT 269 8 "infinity" }{TEXT -1 2 ",
 " }{TEXT 270 8 "infinity" }{TEXT -1 35 ") a integrand m\371\236e m
\355t singularity." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "resta
rt:with(student):" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "P\370\355kla
d 1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f:=x->sin(x)+cos(x)-2
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&
arrowGF(,(-%$sinG6#9$\"\"\"-%$cosGF/F1\"\"#!\"\"F(F(F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Int(f(x),x=-Pi..Pi);           # ma
tematick\375 z\341pis" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,(-%
$sinG6#%\"xG\"\"\"-%$cosGF)F+\"\"#!\"\"/F*;,$%#PiGF/F3" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "value(%);                      # vy
po\350ten\341 hodnota" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"%\"\"
\"%#PiGF&!\"\"" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "P\370\355klad 2" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 20 "Int(abs(x),x=-2..2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%$IntG6$-%$absG6#%\"xG/F);!\"#\"\"#" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#\"\"%" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 
{PARA 5 "" 0 "" {TEXT -1 9 "P\370\355klad 3" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 25 "Int(x^3*cos(x^2),x=0..1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%$IntG6$*&)%\"xG\"\"$\"\"\"-%$cosG6#*$)F(\"\"#F*F*/F(;
\"\"!F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&#\"\"\"\"\"#F&-%$cosG6#F&F&F&*&F%F
&-%$sinGF*F&F&#F&F'!\"\"" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "P\370\355klad 4" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "int(1/(x+exp(x)),x=0..1);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%$intG6$*&\"\"\"F',&%\"xGF'-%$expG6#F)F'!\"
\"/F);\"\"!F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 145 "Maple tento int
egr\341l neum\355 spo\350\355tat analyticky. My to tak\351 neum\355me.
 Proto m\355sto v\375sledku v\375po\350tu vyp\355\232e pouze matematic
k\375 z\341pis tohoto integr\341lu." }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "P\370\355klad
 5" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart:with(student):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "i:=Int(abs(x)/x,x=-1..2
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"iG-%$IntG6$*&-%$absG6#%\"xG
\"\"\"F,!\"\"/F,;F.\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 
"int(integrand(i),x);            # v\375po\350et primitivn\355 funkce
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6$7$,$%\"xG!\"\"1F(\"
\"!7$F(2F+F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(i);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 17 "Metoda per p
artes" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "P\370i v\375po\350tu pou
\236ijeme p\370\355kaz " }{TEXT 274 8 "intparts" }{TEXT -1 1 " " }}}
{SECT 1 {PARA 20 "" 0 "" {TEXT -1 9 "P\370\355klad 6" }}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 17 "Metoda per partes" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "with(student):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 24 "Int(x*cos(x),x=0..Pi/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%
$IntG6$*&%\"xG\"\"\"-%$cosG6#F'F(/F';\"\"!,$*&\"\"#!\"\"%#PiGF(F(" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Integr\341l vypo\350teme metodou p
er partes" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "intparts(%,x);
                  # za nederivovanou slo\236ku vezmeme x" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,&*&\"\"#!\"\"%#PiG\"\"\"F(-%$IntG6$-%$sinG6#%\"
xG/F/;\"\"!,$*&F%F&F'F(F(F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Te
nto integr\341l ji\236 snadno spo\350\355t\341me" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,
&*&\"\"#!\"\"%#PiG\"\"\"F(F(F&" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 9 "P\370\355klad 7" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "i:=Int(exp(x)*cos(x),x=0..Pi
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"iG-%$IntG6$*&-%$expG6#%\"xG
\"\"\"-%$cosGF+F-/F,;\"\"!%#PiG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
27 "Pou\236ijeme metodu per partes" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "intparts(i,cos(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#,(-%$expG6#%#PiG!\"\"\"\"\"F(-%$IntG6$,$*&-%$sinG6#%\"xGF)-F%F1F)F(/
F2;\"\"!F'F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Je\232t\354 jedno
u" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "intparts(%,sin(x));" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%$expG6#%#PiG!\"\"\"\"\"F(-%$IntG6
$,$*&-F%6#%\"xGF)-%$cosGF0F)F(/F1;\"\"!F'F)" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 35 "Zjednodu\232\355me a polo\236\355me do rovnice." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "i=simplify(%);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&-%$expG6#%\"xG\"\"\"-%$cosGF*F,/F+;
\"\"!%#PiG,(-F)6#F2!\"\"F,F6F$F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
28 "Integr\341l z rovnice vyj\341d\370\355me" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 13 "isolate(%,i);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#/-%$IntG6$*&-%$expG6#%\"xG\"\"\"-%$cosGF*F,/F+;\"\"!%#PiG,&*&#F,\"\"#
F,-F)6#F2F,!\"\"#F,F6F9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "
value(i);              # p\370\355m\375 v\375po\350et" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#,&*&#\"\"\"\"\"#F&-%$expG6#%#PiGF&!\"\"#F&F'F," }}
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 18 "Substit
u\350n\355 metoda" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "P\370i v\375p
o\350tu pou\236ijeme p\370\355kaz " }{TEXT 275 9 "changevar" }}}{SECT 
1 {PARA 20 "" 0 "" {TEXT -1 9 "P\370\355klad 8" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 22 "restart:with(student):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 32 "i1:=Int(x^2*sqrt(1-x^2),x=0..1);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#i1G-%$IntG6$*&)%\"xG\"\"#\"\"\",&F,F,*$F)F,!\"
\"#F,F+/F*;\"\"!F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Pou\236ijem
e substituci " }{TEXT 276 10 "x = sin(t)" }{TEXT -1 90 ". Jak je vid
\354t z n\341sleduj\355c\355ho v\375sledku operace, syst\351m Maple p
\370epo\350\355t\341 i integra\350n\355 meze." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 30 "i1s:=changevar(x=sin(t),i1,t);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%$i1sG-%$IntG6$*()-%$sinG6#%\"tG\"\"#\"\"\",&F/F/*$
F)F/!\"\"#F/F.-%$cosGF,F//F-;\"\"!,$*&F.F2%#PiGF/F/" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 22 "Po\350\355t\341me tedy integr\341l" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 " i1u:=Int(sin(t)^2*cos(t)^2,t = 0 .
. 1/2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$i1uG-%$IntG6$*&)-%$si
nG6#%\"tG\"\"#\"\"\")-%$cosGF,F.F//F-;\"\"!,$*&F.!\"\"%#PiGF/F/" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Pou\236ijeme zn\341m\351 vzorce pr
o " }{TEXT 278 3 "sin" }{TEXT -1 3 " a " }{TEXT 279 3 "cos" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "simplify(i1u,\{sin(t)^2=1/2*(1-    \+
 cos(2*t)),cos(t)^2=1/2*(1+cos(2*t))\});" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%$IntG6$,&*&#\"\"\"\"\"%F)*$)-%$cosG6#,$*&\"\"#F)%\"tGF)F)F2F)F
)!\"\"#F)F*F)/F3;\"\"!,$*&F2F4%#PiGF)F)" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 85 "V\375\232e uveden\375 integr\341l snadno spo\350\355t\341
me pomoc\355 vzorce pro dvojn\341sobn\375 argument funkce " }{TEXT 
280 3 "cos" }{TEXT -1 37 ". U\236ili jsme ho u\236 v p\370edchoz\355m \+
kroku" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,$*&\"#;!\"\"%#PiG\"\"\"F(" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 9 "P\370\355klad 9" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart:with(student):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "i2:=Int(x/sqrt(x^2+1),x=1..2
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#i2G-%$IntG6$*&%\"xG\"\"\",&*$
)F)\"\"#F*F*F*F*#!\"\"F./F);F*F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
19 "Zavedeme substituci" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "
changevar(t=x^2+1,i2,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,
$*&\"\"\"F(*&\"\"#F(%\"tG#F(F*!\"\"F(/F+;F*\"\"&" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 24 "Snadno spo\350teme integr\341l" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#,&*$\"\"&#\"\"\"\"\"#F'*$F(F&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{SECT 1 {PARA 20 "" 0 "" {TEXT -1 10 "P\370\355klad 10" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart:with(student):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "i3:=Int(x*cos(x)^2,x=-Pi..Pi);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#i3G-%$IntG6$*&%\"xG\"\"\")-%$cosG6#
F)\"\"#F*/F);,$%#PiG!\"\"F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "V
\232imn\354me si, \236e integrand je funkce lich\341" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 28 "f:=unapply(integrand(i3),x);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&9$\"
\"\")-%$cosG6#F-\"\"#F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 59 "is(f(-x)=-f(x));                   # vlastnost lich\351 funkce" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 21 "Tedy integr\341l roven 0" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "value(i3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 10 "P\370
\355klad 11" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart:with(
student):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "i4:=Int(x^3/(x
^8+1),x=-1..0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#i4G-%$IntG6$*&%
\"xG\"\"$,&*$)F)\"\")\"\"\"F/F/F/!\"\"/F);F0\"\"!" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 58 "Zavedeme substituci a po dosazen\355 vym\354n\355m
e integra\350n\355 meze" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "
changevar(x^4=u,i4,u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*
&\"\"\"F(,&*&\"\"%F()%\"uG\"\"#F(F(F+F(!\"\"F//F-;\"\"!F(" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 26 "Integr\341l se snadno spo\350\355t\341" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#,$*&\"#;!\"\"%#PiG\"\"\"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 "" }}}{SECT 1 {PARA 5 "" 
0 "" {TEXT -1 29 "Integr\341ly s prom\354nn\375mi mezemi" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 83 "Integr\341ly s prom\354nn\375mi mezemi vy
u\236ijeme pozd\354ji p\370i v\375po\350tu v\355cen\341sobn\375ch inte
gr\341l\371" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Int(x^2+1,x=
1..y-1)=int(x^2+1,x=1..y-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$In
tG6$,&*$)%\"xG\"\"#\"\"\"F,F,F,/F*;F,,&%\"yGF,F,!\"\",(*&\"\"$F1F/F4F,
#\"\"(F4F1F0F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 137 "N\341sleduj
\355c\355 p\370\355klad uk\341\236e, \236e syst\351m Maple se spr\341v
n\354 vyrovn\341 i s koliz\355 ozna\350en\355, kdy prom\354nnou v horn
\355 mezi ozna\350\355me stejn\375m p\355smenkem " }{TEXT 277 1 "x" }
{TEXT -1 27 " jako integra\350n\355 prom\354nnou." }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 20 "Int(x^2+1,x=1..x-1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%$IntG6$,&*$)%\"xG\"\"#\"\"\"F+F+F+/F);F+,&F)F+F+!\"\"
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,(*&\"\"$!\"\",&%\"xG\"\"\"F)F&F%F)#\"\"(F%F&F(F
)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 19 "Nevl
astn\355 integr\341ly" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 10 "P\370\355klad 12" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Int(1/sqrt(x-1),x=1..4);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'*$,&%\"xGF'F'!\"\"#F'\"\"
#F+/F*;F'\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"#\"\"\"\"\"$#F&F%F&" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "Jedn\341 se o nevlastn\355 integr
\341l. Integrand m\341 singularitu v bod\354 1." }}}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 20 "" 0 "" {TEXT -1 10 "P
\370\355klad 13" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Ur\350\355me, z
da dan\351 integr\341ly konverguj\355 \350i diverguj\355" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Int(1/x^3,x=1..infinity)=int(1/x^3,
x=1..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F
(*$)%\"xG\"\"$F(!\"\"/F+;F(%)infinityG#F(\"\"#" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 58 "Int(1/sqrt(x),x=1..infinity)=int(1/sqrt(x),x=1
..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(*$
%\"xG#F(\"\"#!\"\"/F*;F(%)infinityGF0" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 74 "Int(1/(x^2+1),x=-infinity..infinity)=int(1/(x^2+1),x=
-infinity..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&
\"\"\"F(,&*$)%\"xG\"\"#F(F(F(F(!\"\"/F,;,$%)infinityGF.F2%#PiG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "Int(x/sqrt(x^5+1),x=0..infin
ity)=int(x/sqrt(x^5+1),x=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%$IntG6$*&%\"xG\"\"\",&*$)F(\"\"&F)F)F)F)#!\"\"\"\"#/F(;\"\"!%
)infinityG,$*&#F)F-F)-%%BetaG6$#F)\"#5#F0F-F)F)" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 70 "Int((x^2+1)/(x^3+1),x=0..infinity)=int((x^2+1)
/(x^3+1),x=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$
*&,&*$)%\"xG\"\"#\"\"\"F-F-F-F-,&F-F-*$)F+\"\"$F-F-!\"\"/F+;\"\"!%)inf
inityGF6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 20 "" 0 "" {TEXT 
-1 10 "P\370\355klad 14" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "r
estart:with(student):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "i:
=Int(1/(1+x^3),x=0..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"
iG-%$IntG6$*&\"\"\"F),&F)F)*$)%\"xG\"\"$F)F)!\"\"/F-;\"\"!%)infinityG
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Rozlo\236\355me integrand na \+
parci\341ln\355 zlomky. Pomoc\355 p\370\355kazu " }{TEXT 271 9 "integr
and" }{TEXT -1 41 " z\355sk\341me integrand p\370edchoz\355ho integr
\341lu." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "iu:=convert(inte
grand(i),parfrac,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#iuG,&*&\"\"
\"F'*&\"\"$F',&%\"xGF'F'F'F'!\"\"F'*(F)F,,&\"\"#F'F+F,F',(*$)F+F/F'F'F
+F,F'F'F,F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Spo\350teme primit
ivn\355 funkci" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "iu1:=int(
iu,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$iu1G,(*&#\"\"\"\"\"$F(-%#
lnG6#,&%\"xGF(F(F(F(F(*&#F(\"\"'F(-F+6#,(*$)F.\"\"#F(F(F.!\"\"F(F(F(F8
*&F'F(*&F)#F(F7-%'arctanG6#,$*(F)F8,&*&F7F(F.F(F(F(F8F(F)F;F(F(F(F(" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Dle Leibnitz-Newtonovy v\354ty d
osad\355me meze. Pro " }{TEXT 272 8 "infinity" }{TEXT -1 23 " mus\355m
e po\350\355tat limitu." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "
limit(iu1,x=infinity)-simplify(subs(x=0,%));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$**\"\"#\"\"\"\"\"*!\"\"\"\"$#F&F%%#PiGF&F&" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Kontrola v\375po\350tu" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(i);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$**\"\"#\"\"\"\"\"*!\"\"\"\"$#F&F%%#PiGF&F&" }}}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 20 "" 0 "
" {TEXT -1 10 "P\370\355klad 15" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 20 "i:=Int(1/x,x=-1..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"iG
-%$IntG6$*&\"\"\"F)%\"xG!\"\"/F*;F+F)" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 168 "Probl\351mem p\370i integraci je neodstraniteln\341 sing
ularita v bod\354 0. Proto rozd\354l\355me integr\341l na dva a pokud \+
oba konverguj\355, pak jejich sou\350et je pr\341v\354 hledan\375m int
egr\341lem." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "i1:=Int(1/x,
x=-1..0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#i1G-%$IntG6$*&\"\"\"F)
%\"xG!\"\"/F*;F+\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "i2
:=Int(1/x,x=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#i2G-%$IntG6$*
&\"\"\"F)%\"xG!\"\"/F*;\"\"!F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 
"Spo\350\355t\341me limitu pro " }{TEXT 273 1 "x" }{TEXT -1 49 " jdouc
\355 k 0 z leva pro prvn\355 z d\355l\350\355ch integr\341l\371." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "limit(value(i1),x=0,left);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%)infinityG!\"\"" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 58 "Limita je nevlastn\355, a proto i p\371vodn\355
 integr\341l diverguje." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "v
alue(i);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%*undefinedG" }}}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "3 0" 0 }{VIEWOPTS 1 1 
0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }