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{SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 7 "Error: " }{TEXT -1 62 "(in
 int) wrong number (or type) of parameters in function iquo" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "This error is c
aused by a bug in Release 4 affecting some integrals with nested radic
als." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "int(sqrt(x+sqrt(x)),
x);" }}{PARA 8 "" 1 "" {TEXT -1 69 "Error, (in int) wrong number (or t
ype) of parameters in function iquo" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "int(sqrt(1+y^(-2/3)),y=1..2);" }}{PARA 8 "" 1 "" 
{TEXT -1 69 "Error, (in int) wrong number (or type) of parameters in f
unction iquo" }}}{PARA 0 "" 0 "" {TEXT -1 30 "The bug is fixed in Rele
ase 5." }}{PARA 0 "" 0 "" {TEXT -1 55 "A work-around is to apply a cha
nge of variables, using " }{MPLTEXT 0 21 9 "changevar" }{TEXT -1 10 " \+
from the " }{MPLTEXT 0 21 7 "student" }{TEXT -1 9 " package." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "with(student,changevar);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7#%*changevarG" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 46 "changevar(sqrt(x)=t,Int(sqrt(x+sqrt(x)),x),t);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*&,&*$%\"tG\"\"#\"\"\"F
*F,#F,F+F*F,F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(\"
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$,&*$%\"tG\"\"#\"\"\"F'F)#\"
\"$F(#F(F+*&,&F'F(F)F)F)F%#F)F(#!\"\"\"\"%-%#lnG6#,(F'F)F/F)*$F%F/F)#F
)\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "F:=subs(t=sqrt(x)
,\");" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"FG,(*$,&%\"xG\"\"\"*$F(#F
)\"\"#F)#\"\"$F,#F,F.*&,&F*F,F)F)F)F'F+#!\"\"\"\"%-%#lnG6#,(F*F)F+F)*$
F'F+F)#F)\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 96 "It's prudent to check \+
an integral by differentiating it and comparing to the original integr
and." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff(F,x);" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#,**&,&%\"xG\"\"\"*$F&#F'\"\"#F'F),&F'F'*$F&#
!\"\"F*F)F'F'*&F&F-F%F)#F.\"\"%*(,&F(F*F'F'F'F%F-F+F'#F.\"\")*&,&F,F)*
&F%F-F+F'F)F',(F(F'F)F'*$F%F)F'F.#F'F5" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "normal(\"-sqrt(x+sqrt(x)));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 22 "In the second ex
ample:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "changevar(y^(-2/3)
=t,Int(sqrt(1+y^(-2/3)),y=1..2),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#-%$IntG6$,$*&,&\"\"\"F)%\"tGF)#F)\"\"#F*#!\"&F,#\"\"$F,/F*;,$*$F,#F)F
0F+F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(\");" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$\"\"##\"\"\"F%!\"#*$,&\"\"%F'*$F%#
F'\"\"$F%F&F'*&F%F-F*F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 80 "A definite \+
integral can be checked by comparing it to a numerical approximation.
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(\");" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#$\"+i5^L8!\"*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 36 "evalf(Int(sqrt(1+y^(-2/3)),y=1..2));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#$\"+g5^L8!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 116 "Co
nfirming the necessity of checking the result, another bug produces in
correct results for very similar integrals: " }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 56 "J:=changevar(y^(-2/3)=t,Int(sqrt(1-y^(-2/3)),y=1..
2),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%$IntG6$,$*&,&\"\"\"F
+%\"tG!\"\"#F+\"\"#F,#!\"&F/#\"\"$F//F,;,$*$F/#F+F3F.F+" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(J);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,&*$,&\"\"%\"\"\"*$\"\"##F'\"\"$!\"##F'F)F'*&F)F*F%F-F-
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(\");" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#$\"+Q$QI)>!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 30 "
The numerical approximation is" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 36 "evalf(Int(sqrt(1-y^(-2/3)),y=1..2));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+WY'>]%!#5" }}}{PARA 0 "" 0 "" {TEXT -1 85 "In this \+
case the correct result would be produced by combining the fractional \+
powers." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Int(3/2*sqrt((1-t
)/t^5),t = 1/2*2^(1/3) .. 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$In
tG6$,$*$*&,&\"\"\"F*%\"tG!\"\"F*F+!\"&#F*\"\"##\"\"$F//F+;,$*$F/#F*F1F
.F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(\");" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,$*$,&*$\"\"##F'\"\"$!\"\"*$F'#\"\"\"F)F'#F)
F'#F-F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(\");" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+XY'>]%!#5" }}}}{SECT 0 {PARA 3 "" 
0 "" {TEXT 26 9 "See also:" }}{PARA 0 "" 0 "" {HYPERLNK 17 "Errors in \+
symbolic integration" 2 "Errors_in_symbolic_integration" "" }{TEXT -1 
2 ", " }{HYPERLNK 17 "int" 2 "int" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "
Integrals involving fractional powers" 2 "Integrals_involving_fraction
al_powers" "" }{TEXT -1 3 ",  " }{HYPERLNK 17 "Numerical Integration" 
2 "evalf/int" "" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 22 "Maple Advisor
 Database" }{TEXT -1 16 "  R. Israel 1998" }}}}{MARK "1 1 4" 15 }
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