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{SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 8 "Bug fix:" }{TEXT -1 40 "  \+
Integrals involving fractional powers\n" }}{PARA 0 "" 0 "" {TEXT -1 
274 "Maple has trouble with some integrals involving products of half-
integer powers of linear terms.  In some cases it seems the problem ha
s to do with taking different branches of multivalued functions, and t
he antiderivative may be correct in some regions but not in others. \+
\027" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "f1:= (2*x-3)^(-3/2)*
x^(1/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1G*&*$-%%sqrtG6#%\"xG
\"\"\"F+*$),&F*\"\"#\"\"$!\"\"#F0F/F+F1" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "F1:= int(f1,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
#F1G,$*&*&-%%sqrtG6#\"\"#\"\"\",&*&*,^##F,\"\"$F,-F)6#\"\"'F,-F)6#%#Pi
GF,-F)6#%\"xGF,-F)6#,&F,F,*&#F+F2F,F;F,!\"\"F,F,,&F;#F+F2F,FAFAF,*(^#F
,F,F6F,-%'arcsinG6#,$*(F(F,-F)6#F2F,F9F,F1F,F,F,F,*$-F)6#F8F,FA#FAF+" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "N:= simplify(diff(F1,x)-f
1);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"NG,$*&*&,.*(^#\"\"%\"\"\"-%
%sqrtG6#,&%\"xG\"\"'\"\"*!\"\"F,)F1\"\"#F,F,*(^#!#7F,F-F,F1F,F,*&^#F3F
,F-F,F,*(F+F,F5F,-F.6#,&F1!\"'F3F,F,F,*(\"#7F,F1F,F=F,F4*&F3F,F=F,F,F,
-F.6#F1F,F,*&-F.6#*$),&F1F6\"\"$F4\"\"(F,F,-F.6#F?F,F4F4" }}}{PARA 0 "
" 0 "" {TEXT -1 70 "This complicated function turns out to be zero (as
 it should be) when " }{XPPEDIT 18 0 "x < 3/2;" "6#2%\"xG*&\"\"$\"\"\"
\"\"#!\"\"" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
27 "assume(x<3/2); simplify(N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"
\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "assume(x>3/2); simpl
ify(N);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*(-%%sqrtG6#,&%#x|irG\"
\"'\"\"*!\"\"\"\"\"-F'6#F*F.-F'6#\"\"$F.F.*$),&F*\"\"#F3F-F7F.F-#!\"#F
3" }}}{PARA 0 "" 0 "" {TEXT -1 58 "Unfortunately we are probably most \+
interested in the case " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 3 " > \+
" }{XPPEDIT 18 0 "3/2;" "6#*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 31 ", as
 that is when the terms in " }{MPLTEXT 0 21 2 "f1" }{TEXT -1 15 " are \+
both real." }}{PARA 0 "" 0 "" {TEXT -1 53 "In other cases the result i
s wrong for all values of " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 1 "
." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "x:= 'x':\nf2:=(2*x+3)^(
1/2)*x^(-5/2);\nF2:=int(f2,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f
2G*&*$-%%sqrtG6#,&%\"xG\"\"#\"\"$\"\"\"F.F.*$)F+#\"\"&F,F.!\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#F2G,&*&*$-%%sqrtG6#,&%\"xG\"\"#\"\"
$\"\"\"F/F/*$)F,#F.F-F/!\"\"#!\"#F.*&#F-\"\"*F/*&*$F(F/F/*$-F)6#F,F/F3
F/F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "simplify(diff(F2,x)
 - f2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"\"F%*&)%\"xG#\"\"$
\"\"#F%-%%sqrtG6#,&F(F+F*F%F%!\"\"#F0F*" }}}{PARA 0 "" 0 "" {TEXT -1 
211 "A work-around is to express the integrand as a single square root
.  Note that this is not quite equivalent to the original integrand (a
lthough it is equivalent when at least one of the linear terms is posi
tive)." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "f1p:= sqrt((2*x-3)
^(-3)*x);\nF1p:= simplify(int(f1p,x));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%$f1pG*$-%%sqrtG6#*&%\"xG\"\"\"*$),&F*\"\"#\"\"$!\"\"F0F+F1F+" 
}}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$F1pG,$*&*&-%%sqrtG6#*&%\"xG\"\"\"
*$),&F,\"\"#\"\"$!\"\"F2F-F3F-,2*&-F)6#*&F0F-F,F-F-F,F-!\")*&\"#7F-F6F
-F-**\"\")F--F)6#F1F-)F,F1F--%#lnG6#F1F-F3**\"\"%F-F>F-F@F--FB6#,(*$F>
F-!\"$*(FEF-F>F-F,F-F-*&FEF-F6F-F-F-F-**\"#CF-F>F-F,F-FAF-F-**F;F-F>F-
F,F-FFF-F3*(\"#=F-F>F-FAF-F3*(\"\"*F-F>F-FFF-F-F-F-*$-F)6#F8F-F3#F-FE
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "normal(diff(F1p,x)-f1p)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 52 "f2p:= sqrt((2*x+3)/x^5);\nF2p:= simplify(int(f2p,x)
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$f2pG*$-%%sqrtG6#*&,&%\"xG\"\"
#\"\"$\"\"\"F.*$)F+\"\"&F.!\"\"F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%$F2pG,$*(-%%sqrtG6#*&,&%\"xG\"\"#\"\"$\"\"\"F/*$)F,\"\"&F/!\"\"F/F,F/
F+F/#!\"#\"\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "normal(di
ff(F2p,x)-f2p);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "
" 0 "" {TEXT -1 217 "In general it is prudent to check the correctness
 of integrals, either by comparing definite integrals to their floatin
g-point approximations or by comparing the derivative of an indefinite
 integral to the integrand. " }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 9 "S
ee also:" }{TEXT -1 1 " " }{HYPERLNK 17 "Errors in symbolic integratio
n" 2 "Errors_in_symbolic_integration" "" }{TEXT -1 1 "," }{TEXT -1 1 "
 " }{HYPERLNK 17 "int" 2 "int" "" }}}{SECT 0 {PARA 0 "" 0 "" {TEXT 26 
22 "Maple Advisor Database" }{TEXT -1 18 "   R. Israel, 1998" }}}
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1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }