{VERSION 3 0 "SUN SPARC SOLARIS" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "2 D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 0 21 "" 0 1 0 0 0 1 0 0 0 0 2 0 0 0 0 }{CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 256 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1 " 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {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 -1 107 " with a branch cut crossing the path of integration. Maple does not recognize the branch cut, an d returns " }{XPPEDIT 18 0 "F(b)-F(a)" "6#,&-%\"FG6#%\"bG\"\"\"-F%6#% \"aG!\"\"" }{TEXT -1 91 " as the value of the definite integral. Note that 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 antide rivative 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 96 "The following examples are in Maple V Release 4 and 5. They may be corrected in later releases." }}{PARA 0 "" 0 "" {TEXT -1 35 "We begin with an antiderivative of " }{XPPEDIT 18 0 "exp(I*x)/(1+x^2)" "6#*&-%$expG6#*&%\"IG\"\"\"%\"xGF)F),&\"\"\"F) *$F*\"\"#F)!\"\"" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "F:= int(exp(I*x)/(1+x^2),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"FG,$*&%\"IG\"\"\",&*&-%$expG6#\"\"\"F.-%#EiG6$F.,&*&F'F.%\"xGF.! \"\"F.F.F.#F.\"\"#*&-F,6#F5F.-F06$F.,&F3F5F5F.F.#F5F7F.F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "This is correct in the sense that differ entiation gives us back the integrand (after simplification)." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "simplify(diff(F,x));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$expG6#*&%\"IG\"\"\"%\"xGF)\"\"\"* &,&F*F)F(F)\"\"\",&F*F)F(!\"\"\"\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 15 "rationalize(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# *&-%$expG6#*&%\"IG\"\"\"%\"xGF)\"\"\",&F)F)*$)F*\"\"#F+F)!\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Here is a definite integral with t his integrand." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "evalf(int (exp(I*x)/(1+x^2), x=-2 .. 2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" +ebS8I!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "This result was obta ined by evaluating the antiderivative at the endpoints of the interval . " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "evalf(subs(x=2,F) - s ubs(x=-2,F));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ebS8I!#5" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "However, the answer is wrong. Her e is the correct value, obtained by numerical integration." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "evalf(Int(exp(I*x)/(1+x^2), x=-2 .. 2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+0z1d9!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "error:= % - %%;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%&errorG$\"+\\tsb6!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "The cause of the error is a discontinuity of the antideri vative at " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 44 ". It h as a jump of size equal to the error." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "evalf(Limit(F,x=0,left)-Limit(F,x=0,right));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+]tsb6!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "Ei(1,z)" "6#-%#EiG6$\" \"\"%\"zG" }{TEXT -1 92 " which occurs in the antiderivative has a bra nch cut along the negative real axis. Thus at " }{XPPEDIT 18 0 "x = 0 ;" "6#/%\"xG\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "-I*x-1" "6#,&*&%\" IG\"\"\"%\"xGF&!\"\"\"\"\"F(" }{TEXT -1 34 " hits the branch cut and t he term " }{XPPEDIT 18 0 "Ei(1,-I*x-1)" "6#-%#EiG6$\"\"\",&*&%\"IG\"\" \"%\"xGF*!\"\"\"\"\"F," }{TEXT -1 36 " in the antiderivative takes a j ump." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "T he 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*$)*&\"\"\"F(*&),& \"\"#\"\"\"%\"xG!\"$\"\"#F()F.\"\")F(!\"\"#F-\"\"&F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "F:= int(f,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG,$*&,&!\"#\"\"\"%\"xG\"\"$\"\"\"*$)*&)F'\"\"#F,)F *F+F,#\"\"\"\"\"&F,!\"\"#F5\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "simplify(diff(F,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$*&)*&),&!\"#\"\"\"%\"xG\"\"$\"\"#F$)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 not for negative " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 65 " (with the principal branch of the fi fth 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#,& $!+g7\")\\U!#5\"\"\"%\"IG$\"+;'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 powers" 2 "Integrals_invo lving_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" 4 }{VIEWOPTS 1 1 0 1 1 1803 }