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{SECT 0 {SECT 0 {PARA 0 "" 0 "" {TEXT 26 7 "Advice:" }{TEXT -1 31 " Er
rors in symbolic integration" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 439 "As most calculus students know, integration ca
n be a difficult process.  Even in Maple, the difficulties have not al
l been overcome, and symbolic integration should not be considered as \+
completely reliable in all cases.  In general it is prudent to check t
he correctness of integrals, either by comparing definite integrals to
 their floating-point approximations or by comparing the derivative of
 an indefinite integral to the integrand.  " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "One common source of trouble is
 multivalued functions.  In particular, a definite integral " }
{MPLTEXT 0 21 21 "int(f(x), x = a .. b)" }{TEXT -1 78 " may give a wro
ng answer because the antiderivative is a multivalued function " }
{XPPEDIT 18 0 "F" "6#%\"FG" }{TEXT -1 105 " with a branch cut crossing
 the path of integration.  Maple may not recognize the branch cut, and
 return " }{XPPEDIT 18 0 "F(b)-F(a)" "6#,&-%\"FG6#%\"bG\"\"\"-F%6#%\"a
G!\"\"" }{TEXT -1 91 " as the value of the definite integral.  Note th
at this can happen even when the integrand " }{XPPEDIT 18 0 "f" "6#%\"
fG" }{TEXT -1 18 " is single-valued." }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 21 "In other cases, when " }{XPPEDIT 18 0 "
f" "6#%\"fG" }{TEXT -1 83 " is multivalued Maple may find an antideriv
ative for the wrong choice of branch of " }{XPPEDIT 18 0 "f" "6#%\"fG
" }{TEXT -1 2 ". " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 26 9 "Examples:" }
}{PARA 0 "" 0 "" {TEXT -1 95 "The following example is in Maple 6 as o
f January 2000.  It may be corrected in later releases." }}{PARA 0 "" 
0 "" {TEXT -1 27 "We begin with the function " }{XPPEDIT 18 0 "f = t/(
2-cos(t^2)^2);" "6#/%\"fG*&%\"tG\"\"\",&\"\"#F'*$)-%$cosG6#*$)F&F)F'F)
F'!\"\"F1" }{TEXT -1 23 " and its antiderivative" }{TEXT -1 13 ".  Not
e that " }{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT -1 38 " is continuous on \+
the whole real line." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "f:= \+
t/(2-cos(t^2)^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*&%\"tG\"\"
\",&\"\"#F'*$)-%$cosG6#*$)F&F)F'F)F'!\"\"F1" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 23 "F:= simplify(int(f,t));" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%\"FG,$*&*&-%%sqrtG6#\"\"#\"\"\",&-%'arctanG6#*&-%$tan
G6#,$*$)%\"tGF+F,#F,F+F,,&F,F,*$F(F,F,!\"\"F,-F/6#*&F2F,,&F;F,F,F<F<F,
F,F,*&F:F,F@F,F<#F,\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "This \+
is correct in the sense that the difference between " }{XPPEDIT 18 0 "
f" "6#%\"fG" }{TEXT -1 23 " and the derivative of " }{XPPEDIT 18 0 "F
" "6#%\"FG" }{TEXT -1 38 " simplifies (with a little help) to 0." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "simplify(diff(F,t)-f);" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#*&*&%\"tG\"\"\",**$)-%$cosG6#,$*$)F%\"
\"#F&#F&F0\"\"%F&F2*&F2F&)F*F0F&!\"\"*$)-F+6#F.F0F&F5F&F&F&F&*,,&!\"#F
&F6F&F&,&F&F&*$-%%sqrtG6#F0F&F&F&,(*$F4F&F0*(F0F&F4F&F?F&F&F&F&F&,&F>F
&F&F5F&,(FCF<*(F0F&F4F&F?F&F&F&F5F&F5" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "combine(%,trig);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Here is a definite integ
ral with this integrand." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 
"J1:=int(f, t= 0 .. 2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#J1G,&*&-
%%sqrtG6#\"\"#\"\"\"-%'arctanG6#*&,&F+F+*$F'F+F+F+-%$tanG6#F*F+F+#F+\"
\"%*(F5F+F'F+-F-6#*&,&F1F+F+!\"\"F+F2F+F+F+" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 10 "evalf(J1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+K
l,#\\(!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "This result is equi
valent to that obtained by evaluating the antiderivative at the endpoi
nts of the interval. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "si
mplify(eval(F,t=2)-eval(F,t=0) - J1);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 163 "However, the answer \+
is wrong (rather obviously, because the integrand is positive on this \+
interval).  Here is the correct value, obtained by numerical integrati
on." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "evalf(Int(f, t=0 .. \+
2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+:)RAZ\"!\"*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "err:= % - evalf(J1);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%$errG$\"+o9W@A!\"*" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 67 "The cause of the error is a discontinuity of the antideri
vative at " }{XPPEDIT 18 0 "t = sqrt(Pi);" "6#/%\"tG-%%sqrtG6#%#PiG" }
{TEXT -1 44 ".  It has a jump of size equal to the error." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "evalf(limit(F,t=sqrt(Pi),left)-limi
t(F,t=sqrt(Pi),right));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+p9W@A!
\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "The next example features \+
a problem with branches of a fractional power." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 31 "f:=((2-3*x)^(-2)*x^(-8))^(1/5);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"fG*$)*&\"\"\"F(*&),&\"\"#F(*&\"\"$F(%\"xGF(!
\"\"F,F()F/\"\")F(F0#F(\"\"&F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 13 "F:= int(f,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG,$*&,&!
\"#\"\"\"*&\"\"$F)%\"xGF)F)F)*$)*&)F'\"\"#F))F,F+F)#F)\"\"&F)!\"\"#F4
\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "simplify(diff(F,x)
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$*&)*&),&!\"#F$*&\"\"$F
$%\"xGF$F$\"\"#F$)F-F,F$#F$\"\"&F$F-F$!\"\"" }}}{PARA 0 "" 0 "" {TEXT 
-1 20 "This is the same as " }{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT -1 
14 " for positive " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 23 ", but n
ot for negative " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 65 " (with th
e principal branch of the fifth root, which Maple uses)." }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "evalf(subs(x=-1,diff(F,x)));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!+g7\")\\U!#5$\"+;'ow3$F&" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "evalf(subs(x=-1,f));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+3c0`_!#5" }}}}{SECT 0 {PARA 0 "" 
0 "" {TEXT 26 9 "See also:" }{TEXT -1 1 " " }{HYPERLNK 17 "int" 2 "int
" "" }{TEXT -1 2 ", " }{HYPERLNK 17 "Integrals involving fractional po
wers" 2 "Integrals_involving_fractional_powers" "" }{TEXT -1 3 ",  " }
{HYPERLNK 17 "Numerical integration" 2 "evalf/int" "" }}}{SECT 0 
{PARA 0 "" 0 "" {TEXT 26 22 "Maple Advisor Database" }{TEXT 256 18 "  \+
 R. Israel, 1998" }}}}{MARK "2 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 
1 }{PAGENUMBERS 0 1 2 33 1 1 }