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{SECT 0 {PARA 3 "" 0 "" {TEXT -1 62 "Integration of rational functions
 using partial fractions .. V" }}{PARA 3 "" 0 "" {TEXT 264 51 " .. Int
egration of some special rational functions " }}{PARA 0 "" 0 "" {TEXT 
-1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT 
-1 18 "Version: 23.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "Int(1/(x^3+1),x)" "6#-%$IntG6$*&\"\"\"F',&*$%\"xG\"\"$F
'F'F'!\"\"F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 44 "The denominator in the integrand factors as " }
{XPPEDIT 18 0 "(x^3+1)=(x+1)*(x^2-x+1)" "6#/,&*$%\"xG\"\"$\"\"\"F(F(*&
,&F&F(F(F(F(,(*$F&\"\"#F(F&!\"\"F(F(F(" }{TEXT -1 8 ", where " }
{XPPEDIT 18 0 "x^2-x+1" "6#,(*$%\"xG\"\"#\"\"\"F%!\"\"F'F'" }{TEXT -1 
16 " is irreducible." }}{PARA 0 "" 0 "" {TEXT -1 55 "Hence we have a p
artial fraction expansion of the form " }}{PARA 256 "" 0 "" {TEXT -1 
1 " " }{XPPEDIT 18 0 "1/(x^3+1) = A/(x+1) + (B*x+C)/(x^2-x+1)" "6#/*&
\"\"\"F%,&*$%\"xG\"\"$F%F%F%!\"\",&*&%\"AGF%,&F(F%F%F%F*F%*&,&*&%\"BGF
%F(F%F%%\"CGF%F%,(*$F(\"\"#F%F(F*F%F%F*F%" }{TEXT -1 2 "  " }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "1/(x^3+1);\nconvert(%,parfrac,x);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$,&*$)%\"xG\"\"$F$F$F$F$!\"
\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"\"F%,&F%F%%\"xGF%!\"\"#F
%\"\"$*&#F%F*F%*&,&F'F%\"\"#F(F%,(*$)F'F/F%F%F'F(F%F%F(F%F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "This gives " }}{PARA 256 "" 0 "
" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/(x^3+1),x) = 1/3" "6#/-%$IntG6$
*&\"\"\"F(,&*$%\"xG\"\"$F(F(F(!\"\"F+*&F(F(F,F-" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "Int(1/(x+1),x) -1/3" "6#,&-%$IntG6$*&\"\"\"F(,&%\"xGF(F
(F(!\"\"F*F(*&F(F(\"\"$F+F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int((x-2)
/(x^2-x+1),x)" "6#-%$IntG6$*&,&%\"xG\"\"\"\"\"#!\"\"F),(*$F(F*F)F(F+F)
F)F+F(" }{TEXT -1 13 " ------- (i)." }}{PARA 0 "" 0 "" {TEXT -1 28 "Th
e first integral in (i) is" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "1/3" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "Int(1/(x+1),x) = 1/3" "6#/-%$IntG6$*&\"\"\"F(,&%\"xGF(F
(F(!\"\"F**&F(F(\"\"$F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 " ln(abs(x+1))
 + c" "6#,&-%#lnG6#-%$absG6#,&%\"xG\"\"\"F,F,F,%\"cGF," }{TEXT -1 14 "
 ------- (ii)." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int((
x-2)/(x^2-x+1),x)" "6#-%$IntG6$*&,&%\"xG\"\"\"\"\"#!\"\"F),(*$F(F*F)F(
F+F)F)F+F(" }{TEXT -1 101 " can be found by completing the square in t
he denominator of the integrand and making a substitution." }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int((x-2)/(x^2-x+1),x) = In
t((x-2)/((x-1/2)^2+3/4),x);" "6#/-%$IntG6$*&,&%\"xG\"\"\"\"\"#!\"\"F*,
(*$F)F+F*F)F,F*F*F,F)-F%6$*&,&F)F*F+F,F*,&*$,&F)F**&F*F*F+F,F,F+F**&\"
\"$F*\"\"%F,F*F,F)" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+
" }{XPPEDIT 18 0 "`` = Int((u-3/2)/(u^2+3/4),u);" "6#/%!G-%$IntG6$*&,&
%\"uG\"\"\"*&\"\"$F+\"\"#!\"\"F/F+,&*$F*F.F+*&F-F+\"\"%F/F+F/F*" }
{TEXT -1 12 ",    where  " }{XPPEDIT 18 0 "PIECEWISE([u = x-1/2, x = u
+1/2],[du = dx, ``]);" "6#-%*PIECEWISEG6$7$/%\"uG,&%\"xG\"\"\"*&F+F+\"
\"#!\"\"F./F*,&F(F+*&F+F+F-F.F+7$/%#duG%#dxG%!G" }{TEXT -1 1 " " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/2;" "6#/%!G*&
\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(2*u/(u^2+3/4),
u)-3/2;" "6#,&-%$IntG6$*(\"\"#\"\"\"%\"uGF),&*$F*F(F)*&\"\"$F)\"\"%!\"
\"F)F0F*F)*&F.F)F(F0F0" }{TEXT -1 2 "  " }{XPPEDIT 18 0 "Int(1/(u^2+3/
4),u)" "6#-%$IntG6$*&\"\"\"F',&*$%\"uG\"\"#F'*&\"\"$F'\"\"%!\"\"F'F/F*
" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/2;" "6#/%!G*&\"\"\"F&\"\"#!\"\"
" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(u^2+3/4)-3/sqrt(3);" "6#,&-%#lnG6
#,&*$%\"uG\"\"#\"\"\"*&\"\"$F+\"\"%!\"\"F+F+*&F-F+-%%sqrtG6#F-F/F/" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(2*u/sqrt(3))+c[2];" "6#,&-%'arct
anG6#*(\"\"#\"\"\"%\"uGF)-%%sqrtG6#\"\"$!\"\"F)&%\"cG6#F(F)" }{TEXT 
-1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 
1 " " }{XPPEDIT 18 0 "`` = 1/2;" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT 
-1 1 " " }{XPPEDIT 18 0 "ln(x^2-x+1)-sqrt(3);" "6#,&-%#lnG6#,(*$%\"xG
\"\"#\"\"\"F)!\"\"F+F+F+-%%sqrtG6#\"\"$F," }{TEXT -1 1 " " }{XPPEDIT 
18 0 "arctan((2*x-1)/sqrt(3))+c[2];" "6#,&-%'arctanG6#*&,&*&\"\"#\"\"
\"%\"xGF+F+F+!\"\"F+-%%sqrtG6#\"\"$F-F+&%\"cG6#F*F+" }{TEXT -1 15 " --
----- (iii)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 173 "Int((x-2)/(x^2-x+1),x);\n``=student[completesquar
e](%,x);\n``=student[changevar](x-1/2=u,rhs(%),u);\n``=map(expand,rhs(
%));\n``=value(rhs(%));\n``=simplify(subs(u=x-1/2,rhs(%)));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&,&%\"xG\"\"\"\"\"#!\"\"F),(*$)F(F
*F)F)F(F+F)F)F+F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%$IntG6$*&,&
%\"xG\"\"\"\"\"#!\"\"F+,&*$),&F*F+#F+F,F-F,F+F+#\"\"$\"\"%F+F-F*" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%$IntG6$*&,&#\"\"$\"\"#!\"\"%\"uG
\"\"\"F/,&*$)F.F,F/F/#F+\"\"%F/F-F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#/%!G-%$IntG6$,&*(\"\"$\"\"\"\"\"#!\"\",&*$)%\"uGF,F+F+#F*\"\"%F+F-F-*
&F.F-F1F+F+F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&\"\"$#\"\"\"
\"\"#-%'arctanG6#,$**F*F)F'!\"\"%\"uGF)F'F(F)F)F0*&F(F)-%#lnG6#,&*&\"
\"%F))F1F*F)F)F'F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*&\"\"
$#\"\"\"\"\"#-%'arctanG6#,$*(F'!\"\",&*&F*F)%\"xGF)F)F)F0F)F'F(F)F)F0-
%#lnG6#F*F)*&F(F)-F56#,(*$)F3F*F)F)F3F0F)F)F)F)" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 51 "Combining the results from (ii) and (iii), we have
 " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/(x^3+1),x)
 = 1/3" "6#/-%$IntG6$*&\"\"\"F(,&*$%\"xG\"\"$F(F(F(!\"\"F+*&F(F(F,F-" 
}{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/(x+1),x) -1/3" "6#,&-%$IntG6$*&
\"\"\"F(,&%\"xGF(F(F(!\"\"F*F(*&F(F(\"\"$F+F+" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "Int((x-2)/(x^2-x+1),x)" "6#-%$IntG6$*&,&%\"xG\"\"\"\"\"
#!\"\"F),(*$F(F*F)F(F+F)F)F+F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/3;" "6#/%!G*&\"\"\"F&\"\"$!\"\"
" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(abs(x+1)) - 1/6" "6#,&-%#lnG6#-%$
absG6#,&%\"xG\"\"\"F,F,F,*&F,F,\"\"'!\"\"F/" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "ln(x^2-x+1) + sqrt(3)/3" "6#,&-%#lnG6#,(*$%\"xG\"\"#\"
\"\"F)!\"\"F+F+F+*&-%%sqrtG6#\"\"$F+F1F,F+" }{TEXT -1 1 " " }{XPPEDIT 
18 0 "arctan((2*x-1)/sqrt(3)) + c" "6#,&-%'arctanG6#*&,&*&\"\"#\"\"\"%
\"xGF+F+F+!\"\"F+-%%sqrtG6#\"\"$F-F+%\"cGF+" }{TEXT -1 2 ". " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "In
t(1/(x^3+1),x);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6
$*&\"\"\"F',&*$)%\"xG\"\"$F'F'F'F'!\"\"F+" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,(-%#lnG6#,&\"\"\"F(%\"xGF(#F(\"\"$*&#F(\"\"'F(-F%6#,(*
$)F)\"\"#F(F(F)!\"\"F(F(F(F5*(F*F(-%%sqrtG6#F+F(-%'arctanG6#,$*&,&F)F4
F(F5F(F7F(F*F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 4 "Let " }}
{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "f(x) = 1/3;" "6#/-%
\"fG6#%\"xG*&\"\"\"F)\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(ab
s(x+1)) -1/6" "6#,&-%#lnG6#-%$absG6#,&%\"xG\"\"\"F,F,F,*&F,F,\"\"'!\"
\"F/" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(x^2-x+1)+1/sqrt(3);" "6#,&-%#
lnG6#,(*$%\"xG\"\"#\"\"\"F)!\"\"F+F+F+*&F+F+-%%sqrtG6#\"\"$F,F+" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "arctan((2*x-1)/sqrt(3));" "6#-%'arctanG
6#*&,&*&\"\"#\"\"\"%\"xGF*F*F*!\"\"F*-%%sqrtG6#\"\"$F," }{TEXT -1 2 " \+
 " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "`` = 1/6;" "6#/
%!G*&\"\"\"F&\"\"'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln((x^2+2*x+1
)/(x^2-x+1))+1/sqrt(3)" "6#,&-%#lnG6#*&,(*$%\"xG\"\"#\"\"\"*&F+F,F*F,F
,F,F,F,,(*$F*F+F,F*!\"\"F,F,F0F,*&F,F,-%%sqrtG6#\"\"$F0F," }{TEXT -1 
1 " " }{XPPEDIT 18 0 "arctan((2*x-1)/sqrt(3))" "6#-%'arctanG6#*&,&*&\"
\"#\"\"\"%\"xGF*F*F*!\"\"F*-%%sqrtG6#\"\"$F," }{TEXT -1 2 ". " }}
{PARA 257 "" 0 "" {TEXT -1 3 "and" }}{PARA 256 "" 0 "" {TEXT -1 2 "  \+
" }{XPPEDIT 18 0 "g(x) = 1/(x^3+1)" "6#/-%\"gG6#%\"xG*&\"\"\"F),&*$F'
\"\"$F)F)F)!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 39 "In the following picture the graph of  " 
}{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 13 " is drawn in " }
{TEXT 260 3 "red" }{TEXT -1 20 " while the graph of " }{XPPEDIT 18 0 "
g(x);" "6#-%\"gG6#%\"xG" }{TEXT -1 13 " is drawn in " }{TEXT 256 4 "bl
ue" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 45 "The picture is con
sistent with the fact that " }{XPPEDIT 18 0 "Diff([f(x)],x) = g(x);" "
6#/-%%DiffG6$7#-%\"fG6#%\"xGF+-%\"gG6#F+" }{TEXT -1 2 ". " }}{PARA 0 "
" 0 "" {TEXT -1 10 "Note that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "Limit(f(x),x=infinity)=Pi/(2*sqrt(3))" "6#/-%&LimitG6$-
%\"fG6#%\"xG/F*%)infinityG*&%#PiG\"\"\"*&\"\"#F/-%%sqrtG6#\"\"$F/!\"\"
" }{TEXT -1 6 "  and " }{XPPEDIT 18 0 "Limit(f(x),x=-infinity)=-Pi/(2*
sqrt(3))" "6#/-%&LimitG6$-%\"fG6#%\"xG/F*,$%)infinityG!\"\",$*&%#PiG\"
\"\"*&\"\"#F2-%%sqrtG6#\"\"$F2F.F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 347 "f := x -> l
n((x^2+2*x+1)/(x^2-x+1))/6+arctan((2*x-1)/sqrt(3))/sqrt(3):\n'f(x)'=f(
x);\ng := x -> 1/(x^3+1):\n'g(x)'=g(x);\np1 := plot([f(x),g(x)],x=-3..
3,-2.3..2.3,color=[red,blue],discont=true):\np2 := plots[implicitplot]
([x=-1,y=Pi/(2*sqrt(3)),y=-Pi/(2*sqrt(3))],\n   x=-3..3,y=-2.3..2.3,co
lor=COLOR(RGB,.4,.4,.4),linestyle=3):\nplots[display]([p1,p2]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&*&#\"\"\"\"\"'F+-%#lnG
6#*&,(*$)F'\"\"#F+F+*&F4F+F'F+F+F+F+F+,(F2F+F'!\"\"F+F+F7F+F+*&#F+\"\"
$F+*&F:#F+F4-%'arctanG6#,$*(F:F7,&*&F4F+F'F+F+F+F7F+F:F<F+F+F+F+" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG*&\"\"\"F),&*$)F'\"\"$F)
F)F)F)!\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 672 415 415 {PLOTDATA 2 "6)-
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!3M'f^6P(3x@F1F`eq7$7$FihpF`eqFdfq7$Fhfq7$$!3N'f^6P(3P>F1F`eq7$7$$!3;*
************z\"F1F`eqFjfq7$F^gq7$$!3M'f^6P(3(p\"F1F`eq7$7$F[jpF`eqFbgq
7$Ffgq7$$!3O'f^6P(3d9F1F`eq7$7$FdjpF`eqFhgq7$F\\hq7$$!3P'f^6P(3<7F1F`e
q7$7$F][qF`eqF^hq7$Fbhq7$$!3%Q'f^6P(3x*F-F`eq7$7$Ff[qF`eqFdhq7$Fhhq7$$
!3%R'f^6P(3P(F-F`eq7$7$F_\\qF`eqFjhq7$F^iq7$$!3-kf^6P(3(\\F-F`eq7$7$Fh
\\qF`eqF`iq7$Fdiq7$$!3cjf^6P(3d#F-F`eq7$7$Fa]qF`eqFfiq7$Fjiq7$$!3AR'f^
6P(3<FbglF`eq7$7$Fj]qF`eqF\\jq7$F`jq7$$\"3FOS[)GE\"HAF-F`eq7$7$Fc^qF`e
qFbjq7$Ffjq7$$\"3iNS[)GE\"HYF-F`eq7$7$F\\_qF`eqFhjq7$F\\[r7$$\"3aNS[)G
E\"HqF-F`eq7$7$Fe_qF`eqF^[r7$Fb[r7$$\"3XNS[)GE\"H%*F-F`eq7$7$F^`qF`eqF
d[r7$Fh[r7$$\"3_.%[)GE\"H=\"F1F`eq7$7$Fg`qF`eqFj[r7$F^\\r7$$\"3`.%[)GE
\"HU\"F1F`eq7$7$F`aqF`eqF`\\r7$Fd\\r7$$\"3_.%[)GE\"Hm\"F1F`eq7$7$F[bqF
`eqFf\\r7$Fj\\r7$$\"3t.%[)GE\"H!>F1F`eq7$7$FfbqF`eqF\\]r7$F`]r7$$\"3].
%[)GE\"H9#F1F`eq7$7$F_cqF`eqFb]r7$Ff]r7$$\"3;/%[)GE\"HQ#F1F`eq7$7$Fhcq
F`eqFh]r7$F\\^r7$$\"3$RS[)GE\"Hi#F1F`eq7$7$FadqF`eqF^^r7$Fb^r7$$\"3e/%
[)GE\"H'GF1F`eq7$7$FjdqF`eqFd^rFiepF^fp-%+AXESLABELSG6%Q\"x6\"Q!F]_r-%
%FONTG6#%(DEFAULTG-%%VIEWG6$;F(F[cm;$!#BF_io$\"#BF_io" 1 2 0 1 10 0 2 
9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3
" "Curve 4" "Curve 5" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/(x^4+1),x);" "6#-%$IntG6$*&\"\"\"
F',&*$%\"xG\"\"%F'F'F'!\"\"F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "factor(x^4+1
,sqrt(2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,(*$)%\"xG\"\"#\"\"\"F
)*&F'F)-%%sqrtG6#F(F)F)F)F)F),(F%F)F*!\"\"F)F)F)" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "The denominator in the in
tegrand factors as  " }{XPPEDIT 18 0 "x^4+1 = (x^2+x*sqrt(2)+1)*(x^2-x
*sqrt(2)+1);" "6#/,&*$%\"xG\"\"%\"\"\"F(F(*&,(*$F&\"\"#F(*&F&F(-%%sqrt
G6#F,F(F(F(F(F(,(*$F&F,F(*&F&F(-F/6#F,F(!\"\"F(F(F(" }{TEXT -1 8 ", wh
ere " }{XPPEDIT 18 0 "x^2+x*sqrt(2)+1;" "6#,(*$%\"xG\"\"#\"\"\"*&F%F'-
%%sqrtG6#F&F'F'F'F'" }{TEXT -1 7 "  and  " }{XPPEDIT 18 0 "(x^2-x*sqrt
(2)+1)" "6#,(*$%\"xG\"\"#\"\"\"*&F%F'-%%sqrtG6#F&F'!\"\"F'F'" }{TEXT 
-1 17 " are irreducible." }}{PARA 0 "" 0 "" {TEXT -1 55 "Hence we have
 a partial fraction expansion of the form " }}{PARA 256 "" 0 "" {TEXT 
-1 1 " " }{XPPEDIT 18 0 "1/(x^4+1) = (A*x+B)/(x^2+x*sqrt(2)+1)+(C*x+`D
 `)/(x^2-x*sqrt(2)+1);" "6#/*&\"\"\"F%,&*$%\"xG\"\"%F%F%F%!\"\",&*&,&*
&%\"AGF%F(F%F%%\"BGF%F%,(*$F(\"\"#F%*&F(F%-%%sqrtG6#F3F%F%F%F%F*F%*&,&
*&%\"CGF%F(F%F%%#D~GF%F%,(*$F(F3F%*&F(F%-F66#F3F%F*F%F%F*F%" }{TEXT 
-1 2 "  " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "1/(x^4+1);\nconv
ert(%,parfrac,x,sqrt(2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F
$,&*$)%\"xG\"\"%F$F$F$F$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,
&\"\"#\"\"\"*&%\"xGF'-%%sqrtG6#F&F'F'F',(*$)F)F&F'F'F(F'F'F'!\"\"#F'\"
\"%*&#F'F2F'*&,&!\"#F'F(F'F',(F.F'F(F0F'F'F0F'F0" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 11 "This gives " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "Int(1/(x^4+1),x) = 1/4;" "6#/-%$IntG6$*&\"\"\"F(,&*$%\"
xG\"\"%F(F(F(!\"\"F+*&F(F(F,F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int((s
qrt(2)*x+2)/(x^2+x*sqrt(2)+1),x)-1/4;" "6#,&-%$IntG6$*&,&*&-%%sqrtG6#
\"\"#\"\"\"%\"xGF.F.F-F.F.,(*$F/F-F.*&F/F.-F+6#F-F.F.F.F.!\"\"F/F.*&F.
F.\"\"%F5F5" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int((sqrt(2)*x-2)/(x^2-x*
sqrt(2)+1),x);" "6#-%$IntG6$*&,&*&-%%sqrtG6#\"\"#\"\"\"%\"xGF-F-F,!\"
\"F-,(*$F.F,F-*&F.F--F*6#F,F-F/F-F-F/F." }{TEXT -1 13 " ------- (i)." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "Int((sqrt(2)*x+2)/(x^2+x*sqrt(2)+1),x);" "6#-%$IntG6$*&
,&*&-%%sqrtG6#\"\"#\"\"\"%\"xGF-F-F,F-F-,(*$F.F,F-*&F.F--F*6#F,F-F-F-F
-!\"\"F." }{TEXT -1 85 " can be found by completing the square in the \+
denominator and making a substitution.\004" }}{PARA 256 "" 0 "" {TEXT 
-1 1 " " }{XPPEDIT 18 0 "Int((sqrt(2)*x+2)/(x^2+x*sqrt(2)+1),x) = Int(
(sqrt(2)*x+2)/((x+1/sqrt(2))^2+1/2),x);" "6#/-%$IntG6$*&,&*&-%%sqrtG6#
\"\"#\"\"\"%\"xGF.F.F-F.F.,(*$F/F-F.*&F/F.-F+6#F-F.F.F.F.!\"\"F/-F%6$*
&,&*&-F+6#F-F.F/F.F.F-F.F.,&*$,&F/F.*&F.F.-F+6#F-F5F.F-F.*&F.F.F-F5F.F
5F/" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "`` = Int((sqrt(2)*u+1)/(u^2+1/2),u);" "6#/%!G-%$IntG6$*&,&*&-%%sqrt
G6#\"\"#\"\"\"%\"uGF/F/F/F/F/,&*$F0F.F/*&F/F/F.!\"\"F/F4F0" }{TEXT -1 
11 ",    where " }{XPPEDIT 18 0 "PIECEWISE([u = x+1/sqrt(2), x = u-1/s
qrt(2)],[du = dx, ``]);" "6#-%*PIECEWISEG6$7$/%\"uG,&%\"xG\"\"\"*&F+F+
-%%sqrtG6#\"\"#!\"\"F+/F*,&F(F+*&F+F+-F.6#F0F1F17$/%#duG%#dxG%!G" }
{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1
/sqrt(2) " "6#/%!G*&\"\"\"F&-%%sqrtG6#\"\"#!\"\"" }{XPPEDIT 18 0 "Int(
2*u/(u^2+1/2),u)+Int(1/(u^2+1/2),u);" "6#,&-%$IntG6$*(\"\"#\"\"\"%\"uG
F),&*$F*F(F)*&F)F)F(!\"\"F)F.F*F)-F%6$*&F)F),&*$F*F(F)*&F)F)F(F.F)F.F*
F)" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 
"`` = 1/sqrt(2);" "6#/%!G*&\"\"\"F&-%%sqrtG6#\"\"#!\"\"" }{TEXT -1 1 "
 " }{XPPEDIT 18 0 "ln(u^2+1/2)+sqrt(2);" "6#,&-%#lnG6#,&*$%\"uG\"\"#\"
\"\"*&F+F+F*!\"\"F+F+-%%sqrtG6#F*F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "a
rctan(sqrt(2)*u)+c[1];" "6#,&-%'arctanG6#*&-%%sqrtG6#\"\"#\"\"\"%\"uGF
,F,&%\"cG6#F,F," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = 1/sqrt(2);" "6#/%!G*&\"\"\"F&-%%sqrtG6#\"\"#!\"\"
" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(x^2+sqrt(2)*x+1)+sqrt(2)*arctan(s
qrt(2)*x+1)+c[1];" "6#,(-%#lnG6#,(*$%\"xG\"\"#\"\"\"*&-%%sqrtG6#F*F+F)
F+F+F+F+F+*&-F.6#F*F+-%'arctanG6#,&*&-F.6#F*F+F)F+F+F+F+F+F+&%\"cG6#F+
F+" }{TEXT -1 15 "  ------- (ii)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 218 "Int((sqrt(2)*x+2)/(x^2+x*sq
rt(2)+1),x);\n``=student[completesquare](%,x);\n``=simplify(student[ch
angevar](x+1/sqrt(2)=u,rhs(%),u));\n``=expand(rhs(%));\n``=value(rhs(%
));\n``=subs(u=x+1/sqrt(2),rhs(%));\n``=simplify(rhs(%));" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#-%$IntG6$*&,&*&\"\"##\"\"\"F)%\"xGF+F+F)F+F+,(*
$)F,F)F+F+F(F+F+F+!\"\"F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%$In
tG6$*&,&*&\"\"##\"\"\"F+%\"xGF-F-F+F-F-,&*$),&F.F-*&F+!\"\"F+F,F-F+F-F
-F,F-F4F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&\"\"#\"\"\"-%$Int
G6$*&,&F(F(*&F'#F(F'%\"uGF(F(F(,&*&F'F()F0F'F(F(F(F(!\"\"F0F(F(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&\"\"#\"\"\"-%$IntG6$,&*&F(F(,&
*&F'F()%\"uGF'F(F(F(F(!\"\"F(*(F.F2F'#F(F'F1F(F(F1F(F(" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/%!G,&*&\"\"##\"\"\"F'-%'arctanG6#*&F'F(%\"uGF)F)F
)*&F(F)*&F'F(-%#lnG6#,&*&F'F))F.F'F)F)F)F)F)F)F)" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%!G,&*&\"\"##\"\"\"F'-%'arctanG6#*&F'F(,&%\"xGF)*&F'!
\"\"F'F(F)F)F)F)*&F(F)*&F'F(-%#lnG6#,&*&F'F))F.F'F)F)F)F)F)F)F)" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&#\"\"\"\"\"#F(*&F)F',(*&F)F(-%
'arctanG6#,$*(F)!\"\"F)F',&*&F)F(%\"xGF(F(*$F)F'F(F(F(F(F(-%#lnG6#F)F(
-F86#,(*$)F5F)F(F(*&F)F'F5F(F(F(F(F(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int((sqrt(2)*x-2)/(x^2-x*sqrt(2)+1),x
);" "6#-%$IntG6$*&,&*&-%%sqrtG6#\"\"#\"\"\"%\"xGF-F-F,!\"\"F-,(*$F.F,F
-*&F.F--F*6#F,F-F/F-F-F/F." }{TEXT -1 32 " can be found in a similar w
ay.\004" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int((sqrt(
2)*x-2)/(x^2-x*sqrt(2)+1),x) = Int((sqrt(2)*x-2)/((x-1/sqrt(2))^2+1/2)
,x);" "6#/-%$IntG6$*&,&*&-%%sqrtG6#\"\"#\"\"\"%\"xGF.F.F-!\"\"F.,(*$F/
F-F.*&F/F.-F+6#F-F.F0F.F.F0F/-F%6$*&,&*&-F+6#F-F.F/F.F.F-F0F.,&*$,&F/F
.*&F.F.-F+6#F-F0F0F-F.*&F.F.F-F0F.F0F/" }{TEXT -1 1 " " }}{PARA 256 "
" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = Int((sqrt(2)*u-1)/(u^2+1/2)
,u);" "6#/%!G-%$IntG6$*&,&*&-%%sqrtG6#\"\"#\"\"\"%\"uGF/F/F/!\"\"F/,&*
$F0F.F/*&F/F/F.F1F/F1F0" }{TEXT -1 11 ",    where " }{XPPEDIT 18 0 "PI
ECEWISE([u = x-1/sqrt(2), x = u+1/sqrt(2)],[du = dx, ``]);" "6#-%*PIEC
EWISEG6$7$/%\"uG,&%\"xG\"\"\"*&F+F+-%%sqrtG6#\"\"#!\"\"F1/F*,&F(F+*&F+
F+-F.6#F0F1F+7$/%#duG%#dxG%!G" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/sqrt(2);" "6#/%!G*&\"\"\"F&-%%sq
rtG6#\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(2*u/(u^2+1/2),u)-
Int(1/(u^2+1/2),u);" "6#,&-%$IntG6$*(\"\"#\"\"\"%\"uGF),&*$F*F(F)*&F)F
)F(!\"\"F)F.F*F)-F%6$*&F)F),&*$F*F(F)*&F)F)F(F.F)F.F*F." }{TEXT -1 1 "
 " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/sqrt(2);
" "6#/%!G*&\"\"\"F&-%%sqrtG6#\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 
0 "ln(u^2+1/2)-sqrt(2);" "6#,&-%#lnG6#,&*$%\"uG\"\"#\"\"\"*&F+F+F*!\"
\"F+F+-%%sqrtG6#F*F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(sqrt(2)*u
)+c[2];" "6#,&-%'arctanG6#*&-%%sqrtG6#\"\"#\"\"\"%\"uGF,F,&%\"cG6#F+F,
" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`
` = 1/sqrt(2);" "6#/%!G*&\"\"\"F&-%%sqrtG6#\"\"#!\"\"" }{TEXT -1 1 " \+
" }{XPPEDIT 18 0 "ln(x^2-sqrt(2)*x+1)-sqrt(2);" "6#,&-%#lnG6#,(*$%\"xG
\"\"#\"\"\"*&-%%sqrtG6#F*F+F)F+!\"\"F+F+F+-F.6#F*F0" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "arctan(sqrt(2)*x-1)+c[2];" "6#,&-%'arctanG6#,&*&-%%sqrt
G6#\"\"#\"\"\"%\"xGF-F-F-!\"\"F-&%\"cG6#F,F-" }{TEXT -1 15 " ------- (
iii)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 218 "Int((sqrt(2)*x-2)/(x^2-x*sqrt(2)+1),x);\n``=student[
completesquare](%,x);\n``=simplify(student[changevar](x-1/sqrt(2)=u,rh
s(%),u));\n``=expand(rhs(%));\n``=value(rhs(%));\n``=subs(u=x-1/sqrt(2
),rhs(%));\n``=simplify(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%
$IntG6$*&,&*&\"\"##\"\"\"F)%\"xGF+F+F)!\"\"F+,(*$)F,F)F+F+F(F-F+F+F-F,
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%$IntG6$*&,&*&\"\"##\"\"\"F+%
\"xGF-F-F+!\"\"F-,&*$),&F.F-*&F+F/F+F,F/F+F-F-F,F-F/F." }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/%!G,$*&\"\"#\"\"\"-%$IntG6$*&,&F(!\"\"*&F'#F(F'%
\"uGF(F(F(,&*&F'F()F1F'F(F(F(F(F.F1F(F(" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/%!G,$*&\"\"#\"\"\"-%$IntG6$,&*&F(F(,&*&F'F()%\"uGF'F(F(F(F(!\"
\"F2*(F.F2F'#F(F'F1F(F(F1F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,
&*&\"\"##\"\"\"F'-%'arctanG6#*&F'F(%\"uGF)F)!\"\"*&F(F)*&F'F(-%#lnG6#,
&*&F'F))F.F'F)F)F)F)F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&
\"\"##\"\"\"F'-%'arctanG6#*&F'F(,&%\"xGF)*&F'!\"\"F'F(F1F)F)F1*&F(F)*&
F'F(-%#lnG6#,&*&F'F))F.F'F)F)F)F)F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/%!G,$*&#\"\"\"\"\"#F(*&F)F',(*&F)F(-%'arctanG6#,$*(F)!\"\"F)F',
&*&F)F(%\"xGF(F(*$F)F'F2F(F(F(F2-%#lnG6#F)F(-F86#,(*$)F5F)F(F(*&F)F'F5
F(F2F(F(F(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 51 "Combining the results from (ii) and (iii), we h
ave " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/(x^4+1)
,x) = 1/4;" "6#/-%$IntG6$*&\"\"\"F(,&*$%\"xG\"\"%F(F(F(!\"\"F+*&F(F(F,
F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int((sqrt(2)*x+2)/(x^2+x*sqrt(2)+1
),x)-1/4;" "6#,&-%$IntG6$*&,&*&-%%sqrtG6#\"\"#\"\"\"%\"xGF.F.F-F.F.,(*
$F/F-F.*&F/F.-F+6#F-F.F.F.F.!\"\"F/F.*&F.F.\"\"%F5F5" }{TEXT -1 1 " " 
}{XPPEDIT 18 0 "Int((sqrt(2)*x-2)/(x^2-x*sqrt(2)+1),x);" "6#-%$IntG6$*
&,&*&-%%sqrtG6#\"\"#\"\"\"%\"xGF-F-F,!\"\"F-,(*$F.F,F-*&F.F--F*6#F,F-F
/F-F-F/F." }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = 1/(4*sqrt(2));" "6#/%!G*&\"\"\"F&*&\"\"%F&-%%sqrtG
6#\"\"#F&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(x^2+sqrt(2)*x+1)+sq
rt(2)/4" "6#,&-%#lnG6#,(*$%\"xG\"\"#\"\"\"*&-%%sqrtG6#F*F+F)F+F+F+F+F+
*&-F.6#F*F+\"\"%!\"\"F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(sqrt(2
)*x+1) - 1/(4*sqrt(2))" "6#,&-%'arctanG6#,&*&-%%sqrtG6#\"\"#\"\"\"%\"x
GF-F-F-F-F-*&F-F-*&\"\"%F--F*6#F,F-!\"\"F4" }{TEXT -1 1 " " }{XPPEDIT 
18 0 "ln(x^2-sqrt(2)*x+1)+sqrt(2)/4" "6#,&-%#lnG6#,(*$%\"xG\"\"#\"\"\"
*&-%%sqrtG6#F*F+F)F+!\"\"F+F+F+*&-F.6#F*F+\"\"%F0F+" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "arctan(sqrt(2)*x-1)+c;" "6#,&-%'arctanG6#,&*&-%%sqrtG6#
\"\"#\"\"\"%\"xGF-F-F-!\"\"F-%\"cGF-" }{TEXT -1 1 " " }}{PARA 256 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/(4*sqrt(2));" "6#/%!G*&\"\"
\"F&*&\"\"%F&-%%sqrtG6#\"\"#F&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "l
n((x^2+sqrt(2)*x+1)/(x^2-sqrt(2)*x+1))+1/(2*sqrt(2));" "6#,&-%#lnG6#*&
,(*$%\"xG\"\"#\"\"\"*&-%%sqrtG6#F+F,F*F,F,F,F,F,,(*$F*F+F,*&-F/6#F+F,F
*F,!\"\"F,F,F6F,*&F,F,*&F+F,-F/6#F+F,F6F," }{XPPEDIT 18 0 "``(arctan(s
qrt(2)*x+1)+arctan(sqrt(2)*x-1))+c;" "6#,&-%!G6#,&-%'arctanG6#,&*&-%%s
qrtG6#\"\"#\"\"\"%\"xGF1F1F1F1F1-F)6#,&*&-F.6#F0F1F2F1F1F1!\"\"F1F1%\"
cGF1" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 262 
42 "__________________________________________" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 42 "Maple's integration procedure gives . . . " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
27 "Int(1/(x^4+1),x);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%
$IntG6$*&\"\"\"F',&F'F'*$)%\"xG\"\"%F'F'!\"\"F+" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,(*&#\"\"\"\"\")F&*&\"\"##F&F)-%#lnG6#*&,(*$)%\"xGF)F&F
&*&F)F*F2F&F&F&F&F&,(F0F&F3!\"\"F&F&F5F&F&F&*&#F&\"\"%F&*&F)F*-%'arcta
nG6#,&F3F&F&F&F&F&F&*&F7F&*&F)F*-F;6#,&F3F&F&F5F&F&F&" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 257 "" 0 "" {TEXT -1 4 "Let " }}{PARA 256 "" 0 "" {TEXT -1 1 "
 " }{XPPEDIT 18 0 "f(x) = 1/(4*sqrt(2));" "6#/-%\"fG6#%\"xG*&\"\"\"F)*
&\"\"%F)-%%sqrtG6#\"\"#F)!\"\"" }{TEXT -1 2 " l" }{XPPEDIT 18 0 "n((x^
2+sqrt(2)*x+1)/(x^2-sqrt(2)*x+1))+1/(2*sqrt(2));" "6#,&-%\"nG6#*&,(*$%
\"xG\"\"#\"\"\"*&-%%sqrtG6#F+F,F*F,F,F,F,F,,(*$F*F+F,*&-F/6#F+F,F*F,!
\"\"F,F,F6F,*&F,F,*&F+F,-F/6#F+F,F6F," }{XPPEDIT 18 0 "``(arctan(sqrt(
2)*x+1)+arctan(sqrt(2)*x-1));" "6#-%!G6#,&-%'arctanG6#,&*&-%%sqrtG6#\"
\"#\"\"\"%\"xGF0F0F0F0F0-F(6#,&*&-F-6#F/F0F1F0F0F0!\"\"F0" }{TEXT -1 
1 " " }}{PARA 257 "" 0 "" {TEXT -1 3 "and" }}{PARA 256 "" 0 "" {TEXT 
-1 2 "  " }{XPPEDIT 18 0 "g(x) = 1/(x^4+1);" "6#/-%\"gG6#%\"xG*&\"\"\"
F),&*$F'\"\"%F)F)F)!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "In the following picture the graph
 of  " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 13 " is drawn
 in " }{TEXT 260 3 "red" }{TEXT -1 20 " while the graph of " }
{XPPEDIT 18 0 "g(x);" "6#-%\"gG6#%\"xG" }{TEXT -1 13 " is drawn in " }
{TEXT 256 4 "blue" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 45 "The
 picture is consistent with the fact that " }{XPPEDIT 18 0 "Diff([f(x)
],x) = g(x);" "6#/-%%DiffG6$7#-%\"fG6#%\"xGF+-%\"gG6#F+" }{TEXT -1 2 "
. " }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(f(x),x = infinity) = Pi/(2*sqrt(2
));" "6#/-%&LimitG6$-%\"fG6#%\"xG/F*%)infinityG*&%#PiG\"\"\"*&\"\"#F/-
%%sqrtG6#F1F/!\"\"" }{TEXT -1 6 "  and " }{XPPEDIT 18 0 "Limit(f(x),x \+
= -infinity) = -Pi/(2*sqrt(2));" "6#/-%&LimitG6$-%\"fG6#%\"xG/F*,$%)in
finityG!\"\",$*&%#PiG\"\"\"*&\"\"#F2-%%sqrtG6#F4F2F.F." }{TEXT -1 1 ".
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 406 "f := x -> ln((x^2+sqrt(2)*x+1)/(x^2-sqrt(2)*x+1))/(4*sqrt(2))
+                          (arctan(sqrt(2)*x+1)+arctan(sqrt(2)*x-1))/(
2*sqrt(2)):\n'f(x)'=f(x);\ng := x -> 1/(x^4+1):\n'g(x)'=g(x);\np1 := p
lot([f(x),g(x)],x=-3..3,-2.3..2.3,color=[red,blue],discont=true):\np2 \+
:= plot([Pi/(2*sqrt(2)),-Pi/(2*sqrt(2))],x=-3..3,-2.3..2.3,\n         \+
       color=COLOR(RGB,.4,.4,.4),linestyle=3):\nplots[display]([p1,p2]
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,&*&#\"\"\"\"\")F+
*&\"\"##F+F.-%#lnG6#*&,(*$)F'F.F+F+*&F.F/F'F+F+F+F+F+,(F5F+F7!\"\"F+F+
F9F+F+F+*&#F+\"\"%F+*&,&-%'arctanG6#,&F7F+F+F+F+-F@6#,&F7F+F+F9F+F+F.F
/F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG*&\"\"\"F),&F)F
)*$)F'\"\"%F)F)!\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 672 415 415 
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gz-F$6%7S7$F($!3_\"fRXt?26\"F-7$F/Ff^m7$F4Ff^m7$F9Ff^m7$F>Ff^m7$FCFf^m
7$FHFf^m7$FMFf^m7$FRFf^m7$FWFf^m7$FfnFf^m7$F[oFf^m7$F`oFf^m7$FeoFf^m7$
F[pFf^m7$F`pFf^m7$FepFf^m7$FjpFf^m7$F_qFf^m7$FdqFf^m7$FiqFf^m7$F^rFf^m
7$FcrFf^m7$FhrFf^m7$F]sFf^m7$FcsFf^m7$FhsFf^m7$F]tFf^m7$FbtFf^m7$FgtFf
^m7$F\\uFf^m7$FauFf^m7$FfuFf^m7$F[vFf^m7$F`vFf^m7$FevFf^m7$FjvFf^m7$F_
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FgyFf^m7$F\\zFf^m7$FazFf^m7$FfzFf^mFi]mF_^m-%+AXESLABELSG6%Q\"x6\"Q!F
\\bm-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F(Ffz;$!#BF^^m$\"#BF^^m" 1 2 0 1 
10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "C
urve 3" "Curve 4" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 1 " \+
" }{XPPEDIT 18 0 "Int(1/(x^5+1),x);" "6#-%$IntG6$*&\"\"\"F',&*$%\"xG\"
\"&F'F'F'!\"\"F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "factor(x^5+1,sqrt(5));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(,**$)%\"xG\"\"#\"\"\"F)F(!\"\"*&F(
F*-%%sqrtG6#\"\"&F*F+F)F*F*,*F&F)F(F+F,F*F)F*F*,&F*F*F(F*F*#F*\"\"%" }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "The den
ominator in the integrand factors as  " }{XPPEDIT 18 0 "x^5+1 = (x+1)*
(x^2+x*(sqrt(5)-1)/2+1)*(x^2-x*(sqrt(5)+1)/2+1);" "6#/,&*$%\"xG\"\"&\"
\"\"F(F(*(,&F&F(F(F(F(,(*$F&\"\"#F(*(F&F(,&-%%sqrtG6#F'F(F(!\"\"F(F-F3
F(F(F(F(,(*$F&F-F(*(F&F(,&-F16#F'F(F(F(F(F-F3F3F(F(F(" }{TEXT -1 8 ", \+
where " }{XPPEDIT 18 0 "x^2+x*(sqrt(5)-1)/2+1;" "6#,(*$%\"xG\"\"#\"\"
\"*(F%F',&-%%sqrtG6#\"\"&F'F'!\"\"F'F&F.F'F'F'" }{TEXT -1 7 "  and  " 
}{XPPEDIT 18 0 "x^2-x*(sqrt(5)+1)/2+1;" "6#,(*$%\"xG\"\"#\"\"\"*(F%F',
&-%%sqrtG6#\"\"&F'F'F'F'F&!\"\"F.F'F'" }{TEXT -1 17 " are irreducible.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "(x+1)*(x^2+x*(sqrt(5)-1)/2+1
)*(x^2-x*(sqrt(5)+1)/2+1);\n``=expand(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*(,&\"\"\"F%%\"xGF%F%,(*$)F&\"\"#F%F%*(#F%F*F%F&F%,&*$-
%%sqrtG6#\"\"&F%F%F%!\"\"F%F%F%F%F%,(F(F%*&#F%F*F%*&F&F%,&F.F%F%F%F%F%
F3F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*$)%\"xG\"\"&\"\"\"F
*F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 
"Hence we have a partial fraction expansion of the form " }}{PARA 256 
"" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/(x^5+1) = P/(x+1)+(Q*x+R)/(x^
2+x*(sqrt(5)-1)/2+1)+(S*x+T)/(x^2-x*(sqrt(5)+1)/2+1);" "6#/*&\"\"\"F%,
&*$%\"xG\"\"&F%F%F%!\"\",(*&%\"PGF%,&F(F%F%F%F*F%*&,&*&%\"QGF%F(F%F%%
\"RGF%F%,(*$F(\"\"#F%*(F(F%,&-%%sqrtG6#F)F%F%F*F%F6F*F%F%F%F*F%*&,&*&%
\"SGF%F(F%F%%\"TGF%F%,(*$F(F6F%*(F(F%,&-F:6#F)F%F%F%F%F6F*F*F%F%F*F%" 
}{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 173 "unassign('x','P','Q','R','S','T'):\neq := 1/(
x^5+1)=P/(x+1)+(Q*x+R)/(x^2+x*(sqrt(5)-1)/2+1)+(S*x+T)/(x^2-x*(sqrt(5)
+1)/2+1);\nsolve(identity(eq,x),\{P,Q,R,S,T\});\nassign(%);\neq;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG/*&\"\"\"F',&*$)%\"xG\"\"&F'F'F'
F'!\"\",(*&%\"PGF',&F'F'F+F'F-F'*&,&*&%\"QGF'F+F'F'%\"RGF'F',(*$)F+\"
\"#F'F'*(#F'F:F'F+F',&*$-%%sqrtG6#F,F'F'F'F-F'F'F'F'F-F'*&,&*&%\"SGF'F
+F'F'%\"TGF'F',(F8F'*&#F'F:F'*&F+F',&F>F'F'F'F'F'F-F'F'F-F'" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#<'/%\"PG#\"\"\"\"\"&/%\"TG#\"\"#F(/%\"RGF+/%
\"SG,&*$-%%sqrtG6#F(F'#!\"\"\"#5#F'F8F7/%\"QG,&F6F'*&#F'F8F'F3F'F'" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&\"\"\"F%,&*$)%\"xG\"\"&F%F%F%F%!\"
\",(*&F%F%,&F%F%F)F%F+#F%F**&,&*&,&#F+\"#5F%*&#F%F5F%-%%sqrtG6#F*F%F%F
%F)F%F%#\"\"#F*F%F%,(*$)F)F<F%F%*(#F%F<F%F)F%,&*$F8F%F%F%F+F%F%F%F%F+F
%*&,&*&,&FCF4#F%F5F+F%F)F%F%F;F%F%,(F>F%*&#F%F<F%*&F)F%,&FCF%F%F%F%F%F
+F%F%F+F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "This gives " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/(x^5+1),x) = 1/
5;" "6#/-%$IntG6$*&\"\"\"F(,&*$%\"xG\"\"&F(F(F(!\"\"F+*&F(F(F,F-" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/(x+1),x)+1/5;" "6#,&-%$IntG6$*&\"
\"\"F(,&%\"xGF(F(F(!\"\"F*F(*&F(F(\"\"&F+F(" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "Int(((sqrt(5)-1)*x/2+2)/(x^2+x*(sqrt(5)-1)/2+1),x)-1/5;
" "6#,&-%$IntG6$*&,&*(,&-%%sqrtG6#\"\"&\"\"\"F/!\"\"F/%\"xGF/\"\"#F0F/
F2F/F/,(*$F1F2F/*(F1F/,&-F,6#F.F/F/F0F/F2F0F/F/F/F0F1F/*&F/F/F.F0F0" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(((sqrt(5)+1)*x/2-2)/(x^2-x*(sqrt(5)
+1)/2+1),x)" "6#-%$IntG6$*&,&*(,&-%%sqrtG6#\"\"&\"\"\"F.F.F.%\"xGF.\"
\"#!\"\"F.F0F1F.,(*$F/F0F.*(F/F.,&-F+6#F-F.F.F.F.F0F1F1F.F.F1F/" }
{TEXT -1 13 " ------- (i)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/(x+1),x) = ln(abs(x+1
))+c[1];" "6#/-%$IntG6$*&\"\"\"F(,&%\"xGF(F(F(!\"\"F*,&-%#lnG6#-%$absG
6#,&F*F(F(F(F(&%\"cG6#F(F(" }{TEXT -1 14 " ------- (ii)." }}{PARA 0 "
" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(((sqrt(5)-1)*x/2+2)/(x^2+x*(
sqrt(5)-1)/2+1),x);" "6#-%$IntG6$*&,&*(,&-%%sqrtG6#\"\"&\"\"\"F.!\"\"F
.%\"xGF.\"\"#F/F.F1F.F.,(*$F0F1F.*(F0F.,&-F+6#F-F.F.F/F.F1F/F.F.F.F/F0
" }{TEXT -1 85 " can be found by completing the square in the denomina
tor and making a substitution.\004" }}{PARA 256 "" 0 "" {TEXT -1 1 " \+
" }{XPPEDIT 18 0 "Int(((sqrt(5)-1)*x/2+2)/(x^2+x*(sqrt(5)-1)/2+1),x) =
 Int((x*(sqrt(5)-1)/2+2)/((x+(sqrt(5)-1)/4)^2+(5+sqrt(5))/8),x);" "6#/
-%$IntG6$*&,&*(,&-%%sqrtG6#\"\"&\"\"\"F/!\"\"F/%\"xGF/\"\"#F0F/F2F/F/,
(*$F1F2F/*(F1F/,&-F,6#F.F/F/F0F/F2F0F/F/F/F0F1-F%6$*&,&*(F1F/,&-F,6#F.
F/F/F0F/F2F0F/F2F/F/,&*$,&F1F/*&,&-F,6#F.F/F/F0F/\"\"%F0F/F2F/*&,&F.F/
-F,6#F.F/F/\"\")F0F/F0F1" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT 
-1 1 " " }{XPPEDIT 18 0 "`` = Int(((sqrt(5)-1)/2*u+(5+sqrt(5))/4)/(u^2
+(5+sqrt(5))/8),u);" "6#/%!G-%$IntG6$*&,&*(,&-%%sqrtG6#\"\"&\"\"\"F0!
\"\"F0\"\"#F1%\"uGF0F0*&,&F/F0-F-6#F/F0F0\"\"%F1F0F0,&*$F3F2F0*&,&F/F0
-F-6#F/F0F0\"\")F1F0F1F3" }{TEXT -1 13 ",    where   " }{XPPEDIT 18 0 
"PIECEWISE([u = x+(sqrt(5)-1)/4, x = u-(sqrt(5)-1)/4],[du = dx, ``]);
" "6#-%*PIECEWISEG6$7$/%\"uG,&%\"xG\"\"\"*&,&-%%sqrtG6#\"\"&F+F+!\"\"F
+\"\"%F2F+/F*,&F(F+*&,&-F/6#F1F+F+F2F+F3F2F27$/%#duG%#dxG%!G" }{TEXT 
-1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (sqrt
(5)-1)/4;" "6#/%!G*&,&-%%sqrtG6#\"\"&\"\"\"F+!\"\"F+\"\"%F," }{TEXT 
-1 1 " " }{XPPEDIT 18 0 "Int(2*u/(u^2+(5+sqrt(5))/8),u)+(5+sqrt(5))/4;
" "6#,&-%$IntG6$*(\"\"#\"\"\"%\"uGF),&*$F*F(F)*&,&\"\"&F)-%%sqrtG6#F/F
)F)\"\")!\"\"F)F4F*F)*&,&F/F)-F16#F/F)F)\"\"%F4F)" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "Int(1/(u^2+(5+sqrt(5))/8),u)" "6#-%$IntG6$*&\"\"\"F',&*
$%\"uG\"\"#F'*&,&\"\"&F'-%%sqrtG6#F.F'F'\"\")!\"\"F'F3F*" }{TEXT -1 3 
"   " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 "
 " }{XPPEDIT 18 0 "`` = (sqrt(5)-1)/4;" "6#/%!G*&,&-%%sqrtG6#\"\"&\"\"
\"F+!\"\"F+\"\"%F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(u^2+(5+sqrt(5))
/8)+sqrt(10+2*sqrt(5))/2;" "6#,&-%#lnG6#,&*$%\"uG\"\"#\"\"\"*&,&\"\"&F
+-%%sqrtG6#F.F+F+\"\")!\"\"F+F+*&-F06#,&\"#5F+*&F*F+-F06#F.F+F+F+F*F3F
+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(4*u/sqrt(10+2*sqrt(5)))+c[2]
" "6#,&-%'arctanG6#*(\"\"%\"\"\"%\"uGF)-%%sqrtG6#,&\"#5F)*&\"\"#F)-F,6
#\"\"&F)F)!\"\"F)&%\"cG6#F1F)" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = \+
(sqrt(5)-1)/4;" "6#/%!G*&,&-%%sqrtG6#\"\"&\"\"\"F+!\"\"F+\"\"%F," }
{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(x^2+x*(sqrt(5)-1)/2+1)+sqrt(10+2*sqr
t(5))/2;" "6#,&-%#lnG6#,(*$%\"xG\"\"#\"\"\"*(F)F+,&-%%sqrtG6#\"\"&F+F+
!\"\"F+F*F2F+F+F+F+*&-F/6#,&\"#5F+*&F*F+-F/6#F1F+F+F+F*F2F+" }{TEXT 
-1 1 " " }{XPPEDIT 18 0 "arctan((4*x+sqrt(5)-1)/sqrt(10+2*sqrt(5)))+c[
2]" "6#,&-%'arctanG6#*&,(*&\"\"%\"\"\"%\"xGF+F+-%%sqrtG6#\"\"&F+F+!\"
\"F+-F.6#,&\"#5F+*&\"\"#F+-F.6#F0F+F+F1F+&%\"cG6#F7F+" }{TEXT -1 16 " \+
 ------- (iii)." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(
((sqrt(5)+1)*x/2-2)/(x^2-x*(sqrt(5)+1)/2+1),x);" "6#-%$IntG6$*&,&*(,&-
%%sqrtG6#\"\"&\"\"\"F.F.F.%\"xGF.\"\"#!\"\"F.F0F1F.,(*$F/F0F.*(F/F.,&-
F+6#F-F.F.F.F.F0F1F1F.F.F1F/" }{TEXT -1 32 " can be found in a similar
 way.\004" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(((sq
rt(5)+1)*x/2-2)/(x^2-x*(sqrt(5)+1)/2+1),x) = Int((x*(sqrt(5)+1)/2-2)/(
(x-(sqrt(5)+1)/4)^2+(5-sqrt(5))/8),x);" "6#/-%$IntG6$*&,&*(,&-%%sqrtG6
#\"\"&\"\"\"F/F/F/%\"xGF/\"\"#!\"\"F/F1F2F/,(*$F0F1F/*(F0F/,&-F,6#F.F/
F/F/F/F1F2F2F/F/F2F0-F%6$*&,&*(F0F/,&-F,6#F.F/F/F/F/F1F2F/F1F2F/,&*$,&
F0F/*&,&-F,6#F.F/F/F/F/\"\"%F2F2F1F/*&,&F.F/-F,6#F.F2F/\"\")F2F/F2F0" 
}{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` \+
= Int(((sqrt(5)+1)*u/2-(5-sqrt(5))/4)/(u^2+(5-sqrt(5))/8),u);" "6#/%!G
-%$IntG6$*&,&*(,&-%%sqrtG6#\"\"&\"\"\"F0F0F0%\"uGF0\"\"#!\"\"F0*&,&F/F
0-F-6#F/F3F0\"\"%F3F3F0,&*$F1F2F0*&,&F/F0-F-6#F/F3F0\"\")F3F0F3F1" }
{TEXT -1 13 ",    where   " }{XPPEDIT 18 0 "PIECEWISE([u = x-(sqrt(5)+
1)/4, x = u+(sqrt(5)+1)/4],[du = dx, ``]);" "6#-%*PIECEWISEG6$7$/%\"uG
,&%\"xG\"\"\"*&,&-%%sqrtG6#\"\"&F+F+F+F+\"\"%!\"\"F3/F*,&F(F+*&,&-F/6#
F1F+F+F+F+F2F3F+7$/%#duG%#dxG%!G" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (sqrt(5)+1)/4;" "6#/%!G*&,&-%%sqrt
G6#\"\"&\"\"\"F+F+F+\"\"%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(2*
u/(u^2+(5-sqrt(5))/8),u)-(5-sqrt(5))/4;" "6#,&-%$IntG6$*(\"\"#\"\"\"%
\"uGF),&*$F*F(F)*&,&\"\"&F)-%%sqrtG6#F/!\"\"F)\"\")F3F)F3F*F)*&,&F/F)-
F16#F/F3F)\"\"%F3F3" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/(u^2+(5-sqr
t(5))/8),u);" "6#-%$IntG6$*&\"\"\"F',&*$%\"uG\"\"#F'*&,&\"\"&F'-%%sqrt
G6#F.!\"\"F'\"\")F2F'F2F*" }{TEXT -1 3 "   " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (sqrt(
5)+1)/4;" "6#/%!G*&,&-%%sqrtG6#\"\"&\"\"\"F+F+F+\"\"%!\"\"" }{TEXT -1 
1 " " }{XPPEDIT 18 0 "ln(u^2+(5-sqrt(5))/8)-sqrt(10-2*sqrt(5))/2;" "6#
,&-%#lnG6#,&*$%\"uG\"\"#\"\"\"*&,&\"\"&F+-%%sqrtG6#F.!\"\"F+\"\")F2F+F
+*&-F06#,&\"#5F+*&F*F+-F06#F.F+F2F+F*F2F2" }{TEXT -1 1 " " }{XPPEDIT 
18 0 "arctan(4*u/sqrt(10-2*sqrt(5)))+c[3]" "6#,&-%'arctanG6#*(\"\"%\"
\"\"%\"uGF)-%%sqrtG6#,&\"#5F)*&\"\"#F)-F,6#\"\"&F)!\"\"F5F)&%\"cG6#\"
\"$F)" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "
" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (sqrt(5)+1)/4;" "6#/%!G*&,&
-%%sqrtG6#\"\"&\"\"\"F+F+F+\"\"%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 
"ln(x^2-x*(sqrt(5)+1)/2+1)-sqrt(10-2*sqrt(5))/2;" "6#,&-%#lnG6#,(*$%\"
xG\"\"#\"\"\"*(F)F+,&-%%sqrtG6#\"\"&F+F+F+F+F*!\"\"F2F+F+F+*&-F/6#,&\"
#5F+*&F*F+-F/6#F1F+F2F+F*F2F2" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arctan(
(4*x-sqrt(5)-1)/sqrt(10-2*sqrt(5)))+c[3]" "6#,&-%'arctanG6#*&,(*&\"\"%
\"\"\"%\"xGF+F+-%%sqrtG6#\"\"&!\"\"F+F1F+-F.6#,&\"#5F+*&\"\"#F+-F.6#F0
F+F1F1F+&%\"cG6#\"\"$F+" }{TEXT -1 14 " ------- (iv)." }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Combining the results \+
from (ii), (iii) and (iv), we have " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+
" }{XPPEDIT 18 0 "Int(1/(x^5+1),x) = 1/5;" "6#/-%$IntG6$*&\"\"\"F(,&*$
%\"xG\"\"&F(F(F(!\"\"F+*&F(F(F,F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int
(1/(x+1),x)+1/5;" "6#,&-%$IntG6$*&\"\"\"F(,&%\"xGF(F(F(!\"\"F*F(*&F(F(
\"\"&F+F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(((sqrt(5)-1)*x/2+2)/(x^
2+x*(sqrt(5)-1)/2+1),x)-1/5;" "6#,&-%$IntG6$*&,&*(,&-%%sqrtG6#\"\"&\"
\"\"F/!\"\"F/%\"xGF/\"\"#F0F/F2F/F/,(*$F1F2F/*(F1F/,&-F,6#F.F/F/F0F/F2
F0F/F/F/F0F1F/*&F/F/F.F0F0" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(((sqrt
(5)+1)*x/2-2)/(x^2-x*(sqrt(5)+1)/2+1),x)" "6#-%$IntG6$*&,&*(,&-%%sqrtG
6#\"\"&\"\"\"F.F.F.%\"xGF.\"\"#!\"\"F.F0F1F.,(*$F/F0F.*(F/F.,&-F+6#F-F
.F.F.F.F0F1F1F.F.F1F/" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/5;" "6#/%!
G*&\"\"\"F&\"\"&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(abs(x+1)) + \+
(sqrt(5)-1)/20" "6#,&-%#lnG6#-%$absG6#,&%\"xG\"\"\"F,F,F,*&,&-%%sqrtG6
#\"\"&F,F,!\"\"F,\"#?F3F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(x^2+x*(s
qrt(5)-1)/2+1)+sqrt(10+2*sqrt(5))/10;" "6#,&-%#lnG6#,(*$%\"xG\"\"#\"\"
\"*(F)F+,&-%%sqrtG6#\"\"&F+F+!\"\"F+F*F2F+F+F+F+*&-F/6#,&\"#5F+*&F*F+-
F/6#F1F+F+F+F7F2F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arctan((4*x+sqrt(5
)-1)/sqrt(10+2*sqrt(5)))" "6#-%'arctanG6#*&,(*&\"\"%\"\"\"%\"xGF*F*-%%
sqrtG6#\"\"&F*F*!\"\"F*-F-6#,&\"#5F**&\"\"#F*-F-6#F/F*F*F0" }{TEXT -1 
1 " " }}{PARA 256 "" 0 "" {TEXT -1 7 "       " }{XPPEDIT 18 0 "-(sqrt(
5)+1)/20;" "6#,$*&,&-%%sqrtG6#\"\"&\"\"\"F*F*F*\"#?!\"\"F," }{TEXT -1 
1 " " }{XPPEDIT 18 0 "ln(x^2-x*(sqrt(5)+1)/2+1)+sqrt(10-2*sqrt(5))/10;
" "6#,&-%#lnG6#,(*$%\"xG\"\"#\"\"\"*(F)F+,&-%%sqrtG6#\"\"&F+F+F+F+F*!
\"\"F2F+F+F+*&-F/6#,&\"#5F+*&F*F+-F/6#F1F+F2F+F7F2F+" }{TEXT -1 1 " " 
}{XPPEDIT 18 0 "arctan((4*x-sqrt(5)-1)/sqrt(10-2*sqrt(5)))+c;" "6#,&-%
'arctanG6#*&,(*&\"\"%\"\"\"%\"xGF+F+-%%sqrtG6#\"\"&!\"\"F+F1F+-F.6#,&
\"#5F+*&\"\"#F+-F.6#F0F+F1F1F+%\"cGF+" }{TEXT -1 2 "  " }}{PARA 256 "
" 0 "" {TEXT -1 1 " " }{TEXT 263 46 "_________________________________
_____________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 231 "f:=x->ln(abs(x+1))+(sqrt(5)
-1)/20*ln(x^2+x*(sqrt(5)-1)/2+1)+sqrt(10+2*sqrt(5))/10*arctan((4*x+sqr
t(5)-1)/sqrt(10+2*sqrt(5)))-(sqrt(5)+1)/20*ln(x^2-x*(sqrt(5)+1)/2+1)+s
qrt(10-2*sqrt(5))/10*arctan((4*x-sqrt(5)-1)/sqrt(10-2*sqrt(5)));" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 42 "Maple's integration procedure gives . . . " }}
{PARA 0 "" 0 "" {TEXT -1 1 "\004" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 27 "Int(1/(x^5+1),x);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#-%$IntG6$*&\"\"\"F',&*$)%\"xG\"\"&F'F'F'F'!\"\"F+" }}{PARA 12 "" 1 "
" {XPPMATH 20 "6#,4-%#lnG6#,&\"\"\"F(%\"xGF(#F(\"\"&*&#F(\"#?F(*&-F%6#
,**$)F)\"\"#F(F5F)!\"\"*&F)F(-%%sqrtG6#F+F(F6F5F(F(F8F(F(F6*&#F(F.F(F0
F(F6*&-%'arctanG6#*&,(F)\"\"%F(F6*$F8F(F6F(*$-F96#,&\"#5F(*&F5F(F8F(F6
F(F6F(*$-F96#FHF(F6F(*&#F(F+F(*&*&F>F(F8F(F(*$-F96#FHF(F6F(F6*(#F(F.F(
-F%6#,*F3F5F)F6F7F(F5F(F(F8F(F(*&#F(F.F(FWF(F6*&-F?6#*&,(F)FCF(F6FDF(F
(*$-F96#,&FIF(*&F5F(F8F(F(F(F6F(*$-F96#F^oF(F6F(*&*(F*F(FgnF(F8F(F(*$-
F96#F^oF(F6F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = 1/5;" "6#/-%\"
fG6#%\"xG*&\"\"\"F)\"\"&!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(abs(
x+1)) + (sqrt(5)-1)/20" "6#,&-%#lnG6#-%$absG6#,&%\"xG\"\"\"F,F,F,*&,&-
%%sqrtG6#\"\"&F,F,!\"\"F,\"#?F3F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(
x^2+x*(sqrt(5)-1)/2+1)+sqrt(10+2*sqrt(5))/10;" "6#,&-%#lnG6#,(*$%\"xG
\"\"#\"\"\"*(F)F+,&-%%sqrtG6#\"\"&F+F+!\"\"F+F*F2F+F+F+F+*&-F/6#,&\"#5
F+*&F*F+-F/6#F1F+F+F+F7F2F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arctan((4
*x+sqrt(5)-1)/sqrt(10+2*sqrt(5)))" "6#-%'arctanG6#*&,(*&\"\"%\"\"\"%\"
xGF*F*-%%sqrtG6#\"\"&F*F*!\"\"F*-F-6#,&\"#5F**&\"\"#F*-F-6#F/F*F*F0" }
{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 7 "       " }{XPPEDIT 18 
0 "-(sqrt(5)+1)/20;" "6#,$*&,&-%%sqrtG6#\"\"&\"\"\"F*F*F*\"#?!\"\"F," 
}{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(x^2-x*(sqrt(5)+1)/2+1)+sqrt(10-2*sq
rt(5))/10;" "6#,&-%#lnG6#,(*$%\"xG\"\"#\"\"\"*(F)F+,&-%%sqrtG6#\"\"&F+
F+F+F+F*!\"\"F2F+F+F+*&-F/6#,&\"#5F+*&F*F+-F/6#F1F+F2F+F7F2F+" }{TEXT 
-1 1 " " }{XPPEDIT 18 0 "arctan((4*x-sqrt(5)-1)/sqrt(10-2*sqrt(5)));" 
"6#-%'arctanG6#*&,(*&\"\"%\"\"\"%\"xGF*F*-%%sqrtG6#\"\"&!\"\"F*F0F*-F-
6#,&\"#5F**&\"\"#F*-F-6#F/F*F0F0" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" 
{TEXT -1 3 "and" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "g
(x) = 1/(x^5+1);" "6#/-%\"gG6#%\"xG*&\"\"\"F),&*$F'\"\"&F)F)F)!\"\"" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 39 "In the following picture the graph of  " }{XPPEDIT 18 0 "
f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 13 " is drawn in " }{TEXT 260 3 "red
" }{TEXT -1 20 " while the graph of " }{XPPEDIT 18 0 "g(x);" "6#-%\"gG
6#%\"xG" }{TEXT -1 13 " is drawn in " }{TEXT 256 4 "blue" }{TEXT -1 2 
". " }}{PARA 0 "" 0 "" {TEXT -1 45 "The picture is consistent with the
 fact that " }{XPPEDIT 18 0 "Diff([f(x)],x) = g(x);" "6#/-%%DiffG6$7#-
%\"fG6#%\"xGF+-%\"gG6#F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 477 "f:=x->ln(abs(x+1))+(
sqrt(5)-1)/20*ln(x^2+x*(sqrt(5)-1)/2+1)+sqrt(10+2*sqrt(5))/10*arctan((
4*x+sqrt(5)-1)/sqrt(10+2*sqrt(5)))-(sqrt(5)+1)/20*ln(x^2-x*(sqrt(5)+1)
/2+1)+sqrt(10-2*sqrt(5))/10*arctan((4*x-sqrt(5)-1)/sqrt(10-2*sqrt(5)))
:\n'f(x)'=f(x);\ng := x -> 1/(x^5+1):\n'g(x)'=g(x);\np1 := plot([f(x),
g(x)],x=-3..3,y=-2.3..2.3,color=[red,blue],discont=true):\np2 := plots
[implicitplot]([x=-1],x=-3..3,y=-2.3..2.3,\n     color=COLOR(RGB,.4,.4
,.4),linestyle=3):\nplots[display]([p1,p2]);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#/-%\"fG6#%\"xG,,-%#lnG6#-%$absG6#,&F'\"\"\"F0F0F0*&#F0
\"#?F0*&,&*$\"\"&#F0\"\"#F0F0!\"\"F0-F*6#,(*$)F'F9F0F0*(F9F:F'F0F5F0F0
F0F0F0F0F0*&#F0\"#5F0*&,&FCF0*&F9F0F7F8F0F8-%'arctanG6#*&,(*&\"\"%F0F'
F0F0F6F0F0F:F0FE#F:F9F0F0F0*&#F0F3F0*&,&F6F0F0F0F0-F*6#,(F>F0*(F9F:F'F
0FRF0F:F0F0F0F0F:*&FBF0*&,&FCF0*&F9F0F7F8F:F8-FH6#*&,(*&FMF0F'F0F0F6F:
F0F:F0FYFNF0F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG*&\"
\"\"F),&*$)F'\"\"&F)F)F)F)!\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 515 313 
313 {PLOTDATA 2 "6'-%'CURVESG6%7gn7$$!\"$\"\"!$!3'*Q<gV8daT!#=7$$!3s\\
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