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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 26 "Vypocet urciteho integral
u" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 37 "Metoda per partes pro urcity
 integral" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student):
" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 9 "Priklad 1" }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 31 "i1:=Int((x^2+1)*ln(x), x=1..2);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%#i1G-%$IntG6$*&,&*$)%\"xG\"\"#\"\"\"\"\"\"F/F
/F/-%#lnG6#F,F//F,;F/F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "
i1=intparts(i1, ln(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*
&,&*$)%\"xG\"\"#\"\"\"\"\"\"F.F.F.-%#lnG6#F+F./F+;F.F,,&-F06#F,#\"#9\"
\"$-F%6$*&,&*$)F+F9F-#F.F9F+F.F-F+!\"\"F2!\"\"" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 17 "i1=value(lhs(%));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%$IntG6$*&,&*$)%\"xG\"\"#\"\"\"\"\"\"F.F.F.-%#lnG6#F+
F./F+;F.F,,&-F06#F,#\"#9\"\"$#!#;\"\"*F." }}}}{SECT 0 {PARA 3 "" 0 "" 
{TEXT -1 9 "Priklad 2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "i2:
=Int(x^2*cos(x), x=0..Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#i2G-%
$IntG6$*&)%\"xG\"\"#\"\"\"-%$cosG6#F*\"\"\"/F*;\"\"!%#PiG" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "i2=intparts(i2, x^2);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&)%\"xG\"\"#\"\"\"-%$cosG6#F)\"\"\"/F
);\"\"!%#PiG,$-F%6$,$*&F)F/-%$sinGF.F/F*F0!\"\"" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 25 "i2=intparts(rhs(%), 2*x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%$IntG6$*&)%\"xG\"\"#\"\"\"-%$cosG6#F)\"\"\"/F);\"\"!
%#PiG,&F3!\"#-F%6$,$F,F5F0F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 17 "i2=value(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*
&)%\"xG\"\"#\"\"\"-%$cosG6#F)\"\"\"/F);\"\"!%#PiG,$F3!\"#" }}}}{SECT 
0 {PARA 3 "" 0 "" {TEXT -1 9 "Priklad 3" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "i3:=Int(exp(x)*cos(x), x=0..Pi);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#i3G-%$IntG6$*&-%$expG6#%\"xG\"\"\"-%$cosGF+F-/F,;\"
\"!%#PiG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "intparts(i3, co
s(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%$expG6#%#PiG!\"\"F(\"\"
\"-%$IntG6$,$*&-%$sinG6#%\"xGF)-F%F1F)F(/F2;\"\"!F'F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "i3=intparts(%, sin(x));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&-%$expG6#%\"xG\"\"\"-%$cosGF*F,/
F+;\"\"!%#PiG,(-F)6#F2!\"\"F6F,-F%6$,$F'F6F/F," }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#/-%$IntG6$*&-%$expG6#%\"xG\"\"\"-%$cosGF*F,/F+;\"\"!%#PiG,(-F)6#F2!
\"\"F6F,F$F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "isolate(%,i
3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&-%$expG6#%\"xG\"\"
\"-%$cosGF*F,/F+;\"\"!%#PiG,&-F)6#F2#!\"\"\"\"#F6F," }}}}{EXCHG {PARA 
3 "" 0 "" {TEXT -1 38 "Substitucni metoda pro urcity integral" }}}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 9 "Priklad 4" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 29 "i4:=Int(x*(x^2-1)^3, x=0..1);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#i4G-%$IntG6$*&%\"xG\"\"\"),&*$)F)\"\"#\"\"\"F*!\"
\"F*\"\"$F0/F);\"\"!F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "i
4=changevar(x^2-1=t, i4, t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$In
tG6$*&%\"xG\"\"\"),&*$)F(\"\"#\"\"\"F)!\"\"F)\"\"$F//F(;\"\"!F)-F%6$,$
*$)%\"tGF1F/#F)F./F:;F0F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
17 "i4=value(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&%
\"xG\"\"\"),&*$)F(\"\"#\"\"\"F)!\"\"F)\"\"$F//F(;\"\"!F)#F0\"\")" }}}}
{SECT 0 {PARA 3 "" 0 "" {TEXT -1 9 "Priklad 5" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 40 "i5:=Int(exp(sin(x))*cos(x), x=Pi..2*Pi);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#i5G-%$IntG6$*&-%$expG6#-%$sinG6#%\"
xG\"\"\"-%$cosGF.F0/F/;%#PiG,$F5\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "i5=changevar(sin(x)=t, i5, t);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%$IntG6$*&-%$expG6#-%$sinG6#%\"xG\"\"\"-%$cosGF-F//F.
;%#PiG,$F4\"\"#-F%6$,$-F)6#%\"tG!\"\"/F<;\"\"!F@" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 17 "i5=value(rhs(%));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%$IntG6$*&-%$expG6#-%$sinG6#%\"xG\"\"\"-%$cosGF-F//F.
;%#PiG,$F4\"\"#\"\"!" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 9 "Priklad \+
6" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "i6:=Int(sin(x)*cos(x)^2
/(1+cos(x)^3)^(1/4), x=0..Pi/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
#i6G-%$IntG6$*&*&-%$sinG6#%\"xG\"\"\")-%$cosGF,\"\"#\"\"\"F3*$),&F.F.*
$)F0\"\"$F3F.#\"\"\"\"\"%F3!\"\"/F-;\"\"!,$%#PiG#F.F2" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "mezi:=i6=changevar(1+cos(x)^3=u, i6
, u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%meziG/-%$IntG6$*&*&-%$sinG
6#%\"xG\"\"\")-%$cosGF-\"\"#\"\"\"F4*$),&F/F/*$)F1\"\"$F4F/#\"\"\"\"\"
%F4!\"\"/F.;\"\"!,$%#PiG#F/F3-F'6$,$*&F4F4*$)%\"uG#\"\"\"F=F4F>#F/F:/F
K;F/F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dolni:=1+cos(0);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&dolniG\"\"#" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 19 "horni:=1+cos(Pi/2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&horniG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 23 "du:=diff(1+cos(x)^3,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#
duG,$*&)-%$cosG6#%\"xG\"\"#\"\"\"-%$sinGF*\"\"\"!\"$" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 20 "i6=value(rhs(mezi));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%$IntG6$*&*&-%$sinG6#%\"xG\"\"\")-%$cosGF+\"\"#\"\"\"
F2*$),&F-F-*$)F/\"\"$F2F-#\"\"\"\"\"%F2!\"\"/F,;\"\"!,$%#PiG#F-F1,&*$)
F1#F8F;F2#F;\"\"*#!\"%FHF-" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 9 "Pr
iklad 7" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "i7:=Int((sqrt(x)-
1)/(sqrt(x)+1), x=1..4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#i7G-%$I
ntG6$*&,&*$-%%sqrtG6#%\"xG\"\"\"\"\"\"!\"\"F0F/,&F*F0F0F0!\"\"/F.;F0\"
\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "assume(u>0);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "i7=simplify(changevar(x=u^2,
 i7, u));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&,&*$-%%sqrtG6
#%\"xG\"\"\"\"\"\"!\"\"F/F.,&F)F/F/F/!\"\"/F-;F/\"\"%,$-F%6$*&*&,&%#u|
irGF/F0F/F/F<F/F.,&F<F/F/F/F2/F<;F/\"\"#F@" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 27 "i7=simplify(value(rhs(%)));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%$IntG6$*&,&*$-%%sqrtG6#%\"xG\"\"\"\"\"\"!\"\"F/F.,&F
)F/F/F/!\"\"/F-;F/\"\"%,(F0F/-%#lnG6#\"\"$F5-F86#\"\"#!\"%" }}}}{SECT 
0 {PARA 3 "" 0 "" {TEXT -1 9 "Priklad 8" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "i8:=Int(1/(sqrt(x^2+1)-x), x=0..1);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%#i8G-%$IntG6$*&\"\"\"F),&*$-%%sqrtG6#,&*$)%\"xG\"
\"#F)\"\"\"F4F4F)F4F2!\"\"!\"\"/F2;\"\"!F4" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 12 "assume(t>0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 47 "i8=simplify(changevar(sqrt(x^2+1)-x=t, i8, t));" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(,&*$-%%sqrtG6#,&*$)%\"xG\"\"
#F(\"\"\"F3F3F(F3F1!\"\"!\"\"/F1;\"\"!F3,$-F%6$*&,&F3F3*$)%#t|irGF2F(F
3F(*$)F@\"\"$F(F5/F@;,&*$-F,6#F2F(F3F4F3F3#F3F2" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 17 "i8=value(rhs(%));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(,&*$-%%sqrtG6#,&*$)%\"xG\"\"#F(\"\"
\"F3F3F(F3F1!\"\"!\"\"/F1;\"\"!F3,$*&,*F4F3*$-F,6#F2F(F3-%#lnG6#,&F<F3
F4F3!\"$*&F?F3F=F(F2F(*$)FB\"\"#F(F5#F3F2" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 18 "i8=expand(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#/-%$IntG6$*&\"\"\"F(,&*$-%%sqrtG6#,&*$)%\"xG\"\"#F(\"\"\"F3F3F(F3F1
!\"\"!\"\"/F1;\"\"!F3,(#F3F2F3*$-F,6#F2F(F:-%#lnG6#,&F;F3F4F3#F4F2" }}
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