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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 62 "Integration of ``rational
'' functions in the logarithmic case." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "restart; theta:=log(x^2+2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&thetaG-%#lnG6#,&*$)%\"xG\"\"#\"\"\"\"\"\"F,F." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "F:=(theta^4+theta^3)/(theta^
2-x) - 2*log(theta^2-x)+3*log(theta-1);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"FG,(*&,&*$)-%#lnG6#,&*$)%\"xG\"\"#\"\"\"\"\"\"F1F3\"\"%F2F3*
$)F*\"\"$F2F3F2,&*$)F*F1F2F3F0!\"\"!\"\"F3-F+6#F8!\"#-F+6#,&F*F3F;F3F7
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "f:=normal(diff(F,x));" 
}}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"fG*&,@*$)-%#lnG6#,&*$)%\"xG\"\"#
\"\"\"\"\"\"F0F2\"\"$F1F0*$)F)\"\"&F1F0*$)F)F0F1!\"%F/\"\"%*&F)F2F/F2F
9*&)F/F3F1F)F1!\"#*&F5F1F.F1F2*&)F)F:F1F.F1!\")*&FAF1F/F1F9*&F(F1F/F1
\"\")*&F.F1F)F1FB*&F(F1F.F1F3*$F=F1FE*&)F)\"\"'F1F/F1F:*&F5F1F/F1F>F1*
(),&F7F2F/!\"\"\"\"#F1F,\"\"\",&F)F2FPF2\"\"\"!\"\"" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 39 "indets(f);  # contains just x and theta" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$%\"xG-%#lnG6#,&*$)F$\"\"#\"\"\"\"
\"\"F+F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "denom(f);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#*(),&*$)-%#lnG6#,&*$)%\"xG\"\"#\"\"\"
\"\"\"F/F1F/F0F1F.!\"\"F/F0F+F1,&F(F1F2F1F1" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 189 "Note that we view this as a polynomial in theta, we do
n't care about the factor x^2+2. All that we care about is that we hav
e a square-free factorization where theta is viewed as a variable" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "subs(theta=Theta,%); # make \+
theta into a variable" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(),&*$)%&The
taG\"\"#\"\"\"\"\"\"%\"xG!\"\"F)F*,&*$)F,F)F*F+F)F+F+,&F(F+F-F+F+" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "sqrfree(%,Theta);  # conside
r only the variable Theta, view x as some \"constant\"" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#7$,&*$)%\"xG\"\"#\"\"\"\"\"\"F(F*7$7$,&*$)%&ThetaG
F(F)F*F'!\"\"F(7$,&F0F*F1F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 5 "%[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$7$,&*$)%&ThetaG\"\"#\"
\"\"\"\"\"%\"xG!\"\"F)7$,&F(F+F-F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 24 "Highest pole order is 2." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 14 "[Theta^2-x,2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$,&*$)%&The
taG\"\"#\"\"\"\"\"\"%\"xG!\"\"F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "d, pole_order := %[1], %[2];" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>6$%\"dG%+pole_orderG6$,&*$)%&ThetaG\"\"#\"\"\"\"\"\"%
\"xG!\"\"F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Like before, we wr
ite:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "A1:=add(c[i]*Theta^
i,i=0..degree(d,Theta)-1) / d^(pole_order-1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#A1G*&,&&%\"cG6#\"\"!\"\"\"*&&F(6#F+F+%&ThetaGF+F+\"
\"\",&*$)F/\"\"#F0F+%\"xG!\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 50 "A1:=subs(Theta=theta,A1); dd:=subs(Theta=theta,d);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A1G*&,&&%\"cG6#\"\"!\"\"\"*&&F(6#F
+F+-%#lnG6#,&*$)%\"xG\"\"#\"\"\"F+F6F+F+F+F7,&*$)F/F6F7F+F5!\"\"!\"\"
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ddG,&*$)-%#lnG6#,&*$)%\"xG\"\"#
\"\"\"\"\"\"F/F1F/F0F1F.!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 41 "normal( (f-diff(A1,x)) * dd^pole_order );" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#*&,Z*&&%\"cG6#\"\"!\"\"\"-%#lnG6#,&*$)%\"xG\"\"#\"\"\"F
*F2F*F*!\"#*&&F'6#F*F*)F+F2F3F4*(F6F3F1F*F8F3F4F1\"\"%*(F&F3F0F3F+F3!
\"\"*$F8F3!\"%*$)F1\"\"$F3\"\")*$)F+FAF3F2*&)F+\"\"&F3F1F3F4*&)F+F:F3F
1F3F>*&FDF3F0F3FA*&FIF3F0F3!\")*&FDF3F1F3FB*&F0F3F+F3FL*&F@F3F+F3F4*&F
+F3F1F3F>*(F6F3F1F3FDF3F2*(F6F3F0F3F+F3FA*(F&F3F+F3F1F3F>*(F&F3F8F3F1F
3F:*(F6F3F0F3F8F3F<*&)F+\"\"'F3F1F3F:*&FFF3F0F3F**$FFF3F2*&F6F3F0F3F4*
&F&F3F0F3F*F&F2*&F6F3F+F3F2F3*&,&F+F*F<F*\"\"\"F.\"\"\"!\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "numer(%);" }}{PARA 12 "" 1 "
" {XPPMATH 20 "6#,Z*&&%\"cG6#\"\"!\"\"\"-%#lnG6#,&*$)%\"xG\"\"#\"\"\"F
)F1F)F)!\"#*&&F&6#F)F))F*F1F2F3*(F5F2F0F)F7F2F3F0\"\"%*(F%F2F/F2F*F2!
\"\"*$F7F2!\"%*$)F0\"\"$F2\"\")*$)F*F@F2F1*&)F*\"\"&F2F0F2F3*&)F*F9F2F
0F2F=*&FCF2F/F2F@*&FHF2F/F2!\")*&FCF2F0F2FA*&F/F2F*F2FK*&F?F2F*F2F3*&F
*F2F0F2F=*(F5F2F0F2FCF2F1*(F5F2F/F2F*F2F@*(F%F2F*F2F0F2F=*(F%F2F7F2F0F
2F9*(F5F2F/F2F7F2F;*&)F*\"\"'F2F0F2F9*&FEF2F/F2F)*$FEF2F1*&F5F2F/F2F3*
&F%F2F/F2F)F%F1*&F5F2F*F2F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
78 "subs(theta=Theta,%);  # make theta into a variable again so that w
e can do rem" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,Z*&)%&ThetaG\"\"$\"\"
\"%\"xG\"\"\"\"\")*&F&F*F)F(!\"%*$F%F(\"\"#*&)F&\"\"'F(F)F(\"\"%*&)F&
\"\"&F()F)F/F(F**&)F&F3F(F7F(!\")*&F7F(F&F(F:*&)F)F'F(F&F(!\"#*&&%\"cG
6#F*F*)F&F/F(F>*&F5F(F)F(F>*&F9F(F)F(F-*&F%F(F7F(F'*$F5F(F/*&&FA6#\"\"
!F*F&F(F>F)F3*$F=F(F+*(F@F(F7F(FCF(!\"\"*(FIF(FCF(F)F(F3*(FIF(F&F(F)F(
F-*(F@F(F7F(F&F(F'*(F@F(F)F(F%F(F/*(FIF(F7F(F&F(FN*(F@F(F)F(FCF(F>*$FC
F(F-*&F@F(F&F(F/*&F@F(F7F(F>*&FIF(F7F(F*FIF/" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 15 "rem(%,d,Theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#,2*&,4*&&%\"cG6#\"\"!\"\"\"%\"xGF+!\"%*&F'\"\"\")F,\"\"#F/!\"\"*$F0
F/F1F,!\"#F'F4*&&F(6#F+F+F0F/\"\"&F6F1*$)F,\"\"$F/F2*$)F,\"\"%F/F+F+%&
ThetaGF+F+F5F-F.F8F'F1F9F>*&F6F/F:F/F2*&F6F/F,F/F4F<F-" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "eqns:=\{coeffs(%,Theta)\};" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqnsG<$,4*&&%\"cG6#\"\"!\"\"\"%\"xG
F,!\"%*&F(\"\"\")F-\"\"#F0!\"\"*$F1F0F2F-!\"#F(F5*&&F)6#F,F,F1F0\"\"&F
7F2*$)F-\"\"$F0F3*$)F-\"\"%F0F,,0F6F.F/F9F(F2F:F?*&F7F0F;F0F3*&F7F0F-F
0F5F=F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "vars:=indets(eqn
s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%varsG<%%\"xG&%\"cG6#\"\"!&F(
6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "vars:=vars minu
s \{x\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%varsG<$&%\"cG6#\"\"!&F'
6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "solve(eqns,vars
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/&%\"cG6#\"\"!*$)%\"xG\"\"#\"
\"\"/&F&6#\"\"\"F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "A1:=s
ubs(%,A1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A1G*&,&*$)%\"xG\"\"#
\"\"\"\"\"\"*&-%#lnG6#,&F'F,F*F,F,F)F,F,F+,&*$)F.F*F+F,F)!\"\"!\"\"" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "new_f:=normal(f-diff(A1,x)
);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&new_fG*&,@*&)-%#lnG6#,&*$)%\"
xG\"\"#\"\"\"\"\"\"F0F2\"\"%F1F/F2F3*$)F)\"\"$F1F0*&F5F1F.F1F2*&F5F1F/
F1!\"#*&F.F1)F)F0F1!\"&*&F;F1F/F1!\"%*$F;F1F9*&F.F1F)F2F3F)F3*&F)F1F/F
1\"\"'*&)F/F6F1F)F1!\"\"F-!\"'F>F2*$FDF1F2F/F0F1*(,&F)F2FEF2\"\"\"F,\"
\"\",&F?F2F/FE\"\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 
"denom(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,&-%#lnG6#,&*$)%\"xG\"
\"#\"\"\"\"\"\"F,F.F.!\"\"F.F.F(F.,&*$)F%F,F-F.F+F/F." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "sqrfree(subs(theta=Theta,%),Theta);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$,&*$)%\"xG\"\"#\"\"\"\"\"\"F(F*7
$7$,&%&ThetaGF*!\"\"F*F*7$,&*$)F.F(F)F*F'F/F*" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 41 "d, pole_order:=(Theta-1)*(Theta^2-x),  1;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>6$%\"dG%+pole_orderG6$*&,&%&ThetaG\"
\"\"!\"\"F+F+,&*$)F*\"\"#\"\"\"F+%\"xGF,F+F+" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 109 "Now we have pole_order=1 so we'll need a logarithmic e
xtension. We'll use the resultant to find the residues." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "a,b:=numer(new_f),denom(new_f);" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>6$%\"aG%\"bG6$,@*&)-%#lnG6#,&*$)%\"xG
\"\"#\"\"\"\"\"\"F2F4\"\"%F3F1F4F5*$)F+\"\"$F3F2*&F7F3F0F3F4*&F7F3F1F3
!\"#*&F0F3)F+F2F3!\"&*&F=F3F1F3!\"%*$F=F3F;*&F0F3F+F4F5F+F5*&F+F3F1F3
\"\"'*&)F1F8F3F+F3!\"\"F/!\"'F@F4*$FFF3F4F1F2*(,&F+F4FGF4F4F.F4,&FAF4F
1FGF4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "b_prime:=diff(b,x)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(b_primeG,(*&%\"xG\"\"\",&*$)-%
#lnG6#,&*$)F'\"\"#\"\"\"F(F2F(F2F3F(F'!\"\"F(F2*(,&F,F(F4F(F(F'F3F)F3F
2*(F6F3F/F(,&*&*&F,F(F'F3F3F/!\"\"\"\"%F4F(F(F(" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 114 "a,b,b_prime:=op(subs(theta=Theta,[a,b,b_prime
])); # replace theta by a variable Theta so we can take the resultant
" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>6%%\"aG%\"bG%(b_primeG6%,@*&)%&Th
etaG\"\"%\"\"\"%\"xG\"\"\"F-*$)F,\"\"$F.\"\"#*&F2F.)F/F4F.F0*&F2F.F/F.
!\"#*&F6F.)F,F4F.!\"&*&F:F.F/F.!\"%*$F:F.F8*&F6F.F,F0F-F,F-*&F,F.F/F.
\"\"'*&)F/F3F.F,F.!\"\"*$F6F.!\"'F=F0*$FCF.F0F/F4*(,&F,F0FDF0F0,&FEF0F
4F0F0,&F>F0F/FDF0,(*&F/F.FKF.F4*(FIF.F/F.FKF.F4*(FIF.FJF.,&*&*&F,F.F/F
.F.FJ!\"\"F-FDF0F0F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "res
ultant(a-b_prime*z,b,Theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**,*
*&%\"zG\"\"\")%\"xG\"\"#\"\"\"F+*$F)F,!\"'F*\"\"'*&F'F,F*F(!\"#F(),&F-
F(F+F(\"\"%F,),&F'F(F+F(F+F,,.*$)F*F4F,!#<*$)F*\"\"$F,\"#?*$)F*\"\"&F,
F(F*F4F-!\"%FBF(F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "
factor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*.%\"xG\"\"\",&%\"zGF&
!\"$F&F&),&F%F&!\"\"F&\"\"#\"\"\"),&*$)F%F-F.F&F-F&\"\"%F.),&F(F&F-F&F
-F.,**$)F%F3F.F&*$)F%\"\"$F.!#;F1F3F3F&F&!\"#" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 23 "residues:=\{solve(%,z)\};" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%)residuesG<$!\"#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 42 "add(r*log(gcd(a-b_prime*r,b)),r=residues);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,&-%#lnG6#,&*$)%&ThetaG\"\"#\"\"\"!\"\"%\"xG
\"\"\"!\"#-F%6#,&F*F/F-F/\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 24 "A2:=subs(Theta=theta,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
#A2G,&-%#lnG6#,&*$)-F'6#,&*$)%\"xG\"\"#\"\"\"\"\"\"F2F4F2F3!\"\"F1F4!
\"#-F'6#,&F,F4F5F4\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "
new_f:=normal(new_f-diff(A2,x)); # we know how to integrate this, so l
ets just use Maple's int" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&new_fG*
&,**&-%#lnG6#,&*$)%\"xG\"\"#\"\"\"\"\"\"F/F1F1F.F1\"\"%F/F1F,F1F.F/F0F
+!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "int(new_f,x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)-%#lnG6#,&*$)%\"xG\"\"#\"\"\"\"\"
\"F-F/F-F.F/F,F/F&F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Int
('f',x)=A1+A2+%;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$%\"fG%\"
xG,.*&,&*$)F(\"\"#\"\"\"\"\"\"*&-%#lnG6#,&F,F0F.F0F0F(F0F0F/,&*$)F2F.F
/F0F(!\"\"!\"\"F0-F36#,&F7F9F(F0!\"#-F36#,&F2F0F9F0\"\"$F7F0F(F0F2F0" 
}}}{EXCHG {PARA 12 "" 1 "" {TEXT -1 10 "Now check:" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 73 "normal(diff(rhs(%),x) - f); # must be 0, ot
herwise answer would be wrong." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"
!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 132 "When we calculated the resi
dues, the result turned out to be constants. This, however, does not a
lways need to be true, for example:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "f:=(x-3)/(theta+2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%\"fG*&,&%\"xG\"\"\"!\"$F(\"\"\",&-%#lnG6#,&*$)F'\"\"#F*F(F2F(F(F2F
(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "a,b:=numer(f),den
om(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6$%\"aG%\"bG6$,&%\"xG\"\"\"
!\"$F*,&-%#lnG6#,&*$)F)\"\"#\"\"\"F*F3F*F*F3F*" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 19 "b_prime:=diff(b,x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%(b_primeG,$*&%\"xG\"\"\",&*$)F'\"\"#F(\"\"\"F,F-!\"\"
F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "a,b,b_prime := op(sub
s(theta=Theta,[a,b,b_prime]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>6%%
\"aG%\"bG%(b_primeG6%,&%\"xG\"\"\"!\"$F+,&%&ThetaGF+\"\"#F+,$*&F*\"\"
\",&*$)F*F/F2F+F/F+!\"\"F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
31 "resultant(a-z*b_prime,b,Theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#*&,,*$)%\"xG\"\"$\"\"\"\"\"\"F'\"\"#*$)F'F+F)!\"$!\"'F**&%\"zGF*F'F*!
\"#F),&F,F*F+F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "res
idues:=\{solve(%,z)\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)residuesG
<#,$*&*&,&%\"xG\"\"\"!\"$F+F+,&*$)F*\"\"#\"\"\"F+F0F+F+F1F*!\"\"#F+F0
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "A1:=add(r*log(gcd(a-b_p
rime*r,b)),r=residues);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A1G,$*&*
(,&%\"xG\"\"\"!\"$F*F*,&*$)F)\"\"#\"\"\"F*F/F*F*-%#lnG6#,&%&ThetaGF*F/
F*F*F0F)!\"\"#F*F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "norma
l(f-diff(A1,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,2*$)%\"xG\"\"
$\"\"\"!\"#*$)F(\"\"#F*\"\"'*(-%#lnG6#,&%&ThetaG\"\"\"F.F6F6F'F*-F26#,
&F,F6F.F6F6F.*&F1F*F'F*\"\"%*(F1F*F-F*F7F*!\"$*&F1F*F-F*!\"'*&F1F*F7F*
F/F1\"#7F**&,&F7F6F.F6\"\"\")F(\"\"#F*!\"\"#!\"\"F." }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 79 "As you can see, we still have theta+2 in the de
nominator. We can't integrate f." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "int(f,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$intG6$
*&,&%\"xG\"\"\"!\"$F)\"\"\",&-%#lnG6#,&*$)F(\"\"#F+F)F3F)F)F3F)!\"\"F(
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "There exists no elementary fu
nction F such that diff(F,x)=f. We can prove this using " }{TEXT 257 
21 "Liouvilles principle." }}{PARA 0 "" 0 "" {TEXT -1 132 "It will fol
low from Liouvilles principle that the residues we compute must be con
stants, if not, then no elementary integral exists." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 9 "Homework:" }{TEXT -1 
51 " integrate the following function in the above way:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "f:=2*ln(x^2+3)*x*(ln(x^2+3)^3-3*ln(
x^2+3)+2*ln(x^2+3)^2-2)/(ln(x^2+3)^2-1)^2/(x^2+3);" }}}}{MARK "52 0 0
" 10 }{VIEWOPTS 1 1 0 1 1 1803 }