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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 58 "Integration of ``polynomi
als'' in a logarithmic extension." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 43 "First the simplest case. Our base-field i
s " }{TEXT 257 5 "K = C" }{TEXT -1 55 "(x), the differentiation is x'=
1 and c'=0 for all c in " }{TEXT 258 1 "C" }{TEXT -1 17 ", and theta'=
1/x." }}{PARA 0 "" 0 "" {TEXT -1 55 "We've seen that then theta must b
e transcendental over " }{TEXT 259 1 "K" }{TEXT -1 160 ", which is equ
ivalent to saying for every positive integer n and for every a0..an in
 K, one has that a0*theta^0+...+an*theta^n is zero only when a0=a1=...
=an=0." }}{PARA 0 "" 0 "" {TEXT -1 21 "Now consider the set " }{TEXT 
260 1 "K" }{TEXT -1 265 "[theta] of all polynomials in theta with coef
fs in K. That theta is transcendental over K means that only the zero-
polynomial in theta is zero (so theta is not like a function like sqrt
(x) for which a non-zero polynomial sqrt(x)^2 - x can have a zero-valu
e anyway)." }}{PARA 0 "" 0 "" {TEXT -1 29 "How to integrate elements o
f " }{TEXT 261 1 "K" }{TEXT -1 85 "[theta]? To make it easier, we will
 assume that we are able to integrate elements of " }{TEXT 262 1 "K" }
{TEXT -1 31 ", which is true in the example " }{TEXT 263 1 "K" }{TEXT 
-1 1 "=" }{TEXT 264 1 "C" }{TEXT -1 78 "(x). So what we'll do is just \+
use Maple's int, but use it only on elements of " }{TEXT 265 1 "K" }
{TEXT -1 21 ", not on elements of " }{TEXT 266 1 "K" }{TEXT -1 83 "[th
eta]. The question now is, under these conditions, how to integrate el
ements of " }{TEXT 267 1 "K" }{TEXT -1 8 "[theta]?" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 14 "theta:=log(x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&thetaG-%#lnG6#%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 72 "f:=-86*x^2*ln(x)+23*x-84/x^3*ln(x)+19/x^2*ln(x)^3-50*
x^2*ln(x)^3+88/x^5;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"fG,.*&)%\"xG\"\"#\"\"\"-%#lnG6#F(\"\"\"!#')F(\"#B
*&F+F**$)F(\"\"$F*!\"\"!#%)*&*$)F+\"\"$F*F**$)F(\"\"#F*F5\"#>*&F'F*F9F
*!#]*&F*F**$)F(\"\"&F*F5\"#))" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 27 "f:=collect(f,theta,normal);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%\"fG,(*&*&,&!#>\"\"\"*$)%\"xG\"\"%\"\"\"\"#]F*)-%#lnG6#F-\"\"$F/F/*$)
F-\"\"#F/!\"\"!\"\"*&*&,&*$)F-\"\"&F/\"#V\"#UF*F*F2F*F/*$)F-\"\"$F/F9!
\"#*&,&*$)F-\"\"'F/\"#B\"#))F*F/*$)F-\"\"&F/F9F*" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 167 "collect(f,theta) writes f as an*theta^n+...+a0*th
eta^0, so if you have something like a*ln^d+...+b*ln^d, then the a*ln^
d+b*ln^d are being put into one term (a+b)*ln^d." }}{PARA 0 "" 0 "" 
{TEXT -1 125 "collect(f,theta,normal) will also bring f into the form \+
an*theta^n+...+a0*theta^0, but it will also apply normal on a0,..,an.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "d:=degree(f,theta);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG\"\"$" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 39 "lcoeff(f,theta);   # = coeff(f,theta,d)" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,$*&,&!#>\"\"\"*$)%\"xG\"\"%\"\"\"\"#]F,*$)F
*\"\"#F,!\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "A1:=i
nt(lcoeff(f,theta),x) * theta^d;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
#A1G*&,&*$)%\"xG\"\"$\"\"\"#!#]F**&F+F+F)!\"\"!#>\"\"\")-%#lnG6#F)F*F+
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "a1:=diff(%,x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#a1G,&*&,&*&\"\"\"F)*$)%\"xG\"\"#F)!
\"\"\"#>*$)F,\"\"#F)!#]\"\"\")-%#lnG6#F,\"\"$F)F4*&*&,&*$)F,F9F)#F3F9*
&F)F)F,F.!#>F4)F6F2F)F)F,F.F9" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 131 
"As you can see, a1 has the same degree and same lcoeff as f. Therefor
e, f-a1 will have degree less than f as a polynomial in theta." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "new_f := collect( f-a1 ,thet
a,normal);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&new_fG,(*&*&,&*$)%\"x
G\"\"%\"\"\"\"#]\"#d\"\"\"F0)-%#lnG6#F+\"\"#F-F-*$)F+\"\"#F-!\"\"F0*&*
&,&*$)F+\"\"&F-\"#V\"#UF0F0F2F0F-*$)F+\"\"$F-F9!\"#*&,&*$)F+\"\"'F-\"#
B\"#))F0F-*$)F+\"\"&F-F9F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
35 "Int('f',x) = Int('new_f',x) + 'A1';" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%$IntG6$%\"fG%\"xG,&-F%6$%&new_fGF(\"\"\"%#A1GF-" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "%;" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#/-%$IntG6$,(*&*&,&!#>\"\"\"*$)%\"xG\"\"%\"\"\"\"#]F,)-%#lnG6#F/\"\"$
F1F1*$)F/\"\"#F1!\"\"!\"\"*&*&,&*$)F/\"\"&F1\"#V\"#UF,F,F4F,F1*$)F/\"
\"$F1F;!\"#*&,&*$)F/\"\"'F1\"#B\"#))F,F1*$)F/\"\"&F1F;F,F/,&-F%6$,(*&*
&,&F-F2\"#dF,F,)F4\"\"#F1F1*$)F/\"\"#F1F;F,F=FHFIF,F/F,*&,&*$)F/F7F1#!
#]F7*&F1F1F/F;F+F,F3F1F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 
"normal(diff(lhs(%)-rhs(%),x)); # check previous equation" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "N
ow lets try to reduce the degree of new_f." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 23 "d:=degree(new_f,theta);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"dG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
20 "lcoeff(new_f,theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*$)%\"
xG\"\"%\"\"\"\"#]\"#d\"\"\"F)*$)F'\"\"#F)!\"\"" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 9 "int(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&
*$)%\"xG\"\"$\"\"\"#\"#]F'*&F(F(F&!\"\"!#d" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 14 "A2:=%*theta^d;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%#A2G*&,&*$)%\"xG\"\"$\"\"\"#\"#]F**&F+F+F)!\"\"!#d\"\"\")-%#lnG6#F)\"
\"#F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "new_f:=collect(new
_f - diff(A2,x),theta,normal);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&n
ew_fG,&*&*&,(*$)%\"xG\"\"&\"\"\"\"$z\"F+!$r\"\"$E\"\"\"\"F1-%#lnG6#F+F
1F-*$)F+\"\"$F-!\"\"#!\"#\"\"$*&,&*$)F+\"\"'F-\"#B\"#))F1F-*$)F+\"\"&F
-F8F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "d:=degree(new_f,th
eta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG\"\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "int(lcoeff(new_f,theta),x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"$\"\"\"#!$e$\"\"**&F(F(*
$)F&\"\"#F(!\"\"\"#U*&F(F(F&F0!$9\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "A3:=%*theta^d;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#
A3G*&,(*$)%\"xG\"\"$\"\"\"#!$e$\"\"**&F+F+*$)F)\"\"#F+!\"\"\"#U*&F+F+F
)F3!$9\"\"\"\"-%#lnG6#F)F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
48 "new_f:=collect(new_f - diff(A3,x),theta,normal);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%&new_fG,$*&,,*$)%\"xG\"\"'\"\"\"\"$2#\"$#z\"\"\"*$
)F*\"\"(F,\"$e$*$)F*\"\"#F,!$y$*$)F*\"\"$F,\"%E5F,*$)F*\"\"&F,!\"\"#F/
\"\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "degree(new_f,theta
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 30 "degree 0, so this is now in K." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 33 "Int(f,x) = A1+A2+A3+int(new_f,x);" }}{PARA 12 "
" 1 "" {XPPMATH 20 "6#/-%$IntG6$,(*&*&,&!#>\"\"\"*$)%\"xG\"\"%\"\"\"\"
#]F,)-%#lnG6#F/\"\"$F1F1*$)F/\"\"#F1!\"\"!\"\"*&*&,&*$)F/\"\"&F1\"#V\"
#UF,F,F4F,F1*$)F/\"\"$F1F;!\"#*&,&*$)F/\"\"'F1\"#B\"#))F,F1*$)F/\"\"&F
1F;F,F/,2*&,&*$)F/F7F1#!#]F7*&F1F1F/F;F+F,F3F1F,*&,&FV#F2F7FZ!#dF,)F4
\"\"#F1F,*&,(FV#!$e$\"\"**&F1F1*$)F/\"\"#F1F;FDFZ!$9\"F,F4F1F,FV#\"$e$
\"#F*$)F/FjnF1#FNFjn*&F1F1*$)F/\"\"%F1F;!#AF`o\"#@FZFdo" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 6 "Check:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "normal(diff(lhs(%)-rhs(%),x));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "OK." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 248 "This was
 a little too easy. We had to integrate lcoeff(f,theta) and later lcoe
ff(f_new,theta), and in the example the result was a rational function
 every time. But what if the result included a logarithm like theta? C
onsider the following example:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 39 "f:=(1+x+x^2)/x^2*theta^2+(x+3/x)*theta;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"fG,&*&*&,(\"\"\"F)%\"xGF)*$)F*\"\"#\"\"\"F)F))-%#ln
G6#F*F-F.F.*$)F*\"\"#F.!\"\"F)*&,&F*F)*&F.F.F*F6\"\"$F)F0F)F)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "d:=degree(f,theta);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG\"\"#" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 16 "lcoeff(f,theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#*&,(\"\"\"F%%\"xGF%*$)F&\"\"#\"\"\"F%F**$)F&\"\"#F*!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "int(%,x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,(%\"xG\"\"\"*&\"\"\"F'F$!\"\"!\"\"-%#lnG6#F$F%" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "A1:=%*theta^d;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#A1G*&,(%\"xG\"\"\"*&\"\"\"F*F'!\"\"!\"\"-%#ln
G6#F'F(F()F-\"\"#F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "f:=c
ollect(f-diff(A1,x),theta,normal);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%\"fG,&*&*$)-%#lnG6#%\"xG\"\"#\"\"\"F.F,!\"\"!\"#*&*&,**$)F,\"\"$F.\"
\"\"F,F6*$)F,F-F.F0F-F7F7F)F7F.*$)F,\"\"#F.F/F7" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 19 "d:=degree(f,theta);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"dG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
35 "A2:=int(lcoeff(f,theta),x)*theta^d;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#A2G,$*$)-%#lnG6#%\"xG\"\"$\"\"\"!\"#" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 38 "f:=collect(f-diff(A2,x),theta,normal);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"fG,&*&*$)-%#lnG6#%\"xG\"\"#\"\"\"F.F,!\"
\"\"\"%*&*&,**$)F,\"\"$F.\"\"\"F,F6*$)F,F-F.!\"#F-F7F7F)F7F.*$)F,\"\"#
F.F/F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "d:=degree(f,theta
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG\"\"#" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 35 "A3:=int(lcoeff(f,theta),x)*theta^d;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A3G,$*$)-%#lnG6#%\"xG\"\"$\"\"\"\"
\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "f:=collect(f-diff(A3
,x),theta,normal);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,&*&*$)-%#
lnG6#%\"xG\"\"#\"\"\"F.F,!\"\"!\")*&*&,**$)F,\"\"$F.\"\"\"F,F6*$)F,F-F
.!\"#F-F7F7F)F7F.*$)F,\"\"#F.F/F7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
26 "The process does not stop." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 35 "When we have:  c * theta' * theta^d" }}
{PARA 0 "" 0 "" {TEXT -1 90 "where c is a constant and d is an integer
, if we try to integrate in the above way we get:" }}{PARA 0 "" 0 "" 
{TEXT -1 32 "int(c*theta')  which is c*theta." }}{PARA 0 "" 0 "" 
{TEXT -1 21 "Then A=c*theta^(d+1)." }}{PARA 0 "" 0 "" {TEXT -1 36 "The
n A'=c * (d+1) * theta' * theta^d" }}{PARA 0 "" 0 "" {TEXT -1 101 "whi
ch is a factor (d+1) too much. If we correct for that factor, by divid
ing by (d+1), then it works." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 39 "f:=(1+x+x^2)/x^2*theta^2+(x+3/x)*theta;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"fG,&*&*&,(\"\"\"F)%\"xGF)*$)F*\"\"#\"\"\"F)F))-%#ln
G6#F*F-F.F.*$)F*\"\"#F.!\"\"F)*&,&F*F)*&F.F.F*F6\"\"$F)F0F)F)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "d:=degree(f,theta);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG\"\"#" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 16 "lcoeff(f,theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#*&,(\"\"\"F%%\"xGF%*$)F&\"\"#\"\"\"F%F**$)F&\"\"#F*!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "int(%,x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,(%\"xG\"\"\"*&\"\"\"F'F$!\"\"!\"\"-%#lnG6#F$F%" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "subs(theta=theta/(d+1),%);  \+
# correct the factor (d+1) we have too much" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,(%\"xG\"\"\"*&\"\"\"F'F$!\"\"!\"\"-%#lnG6#F$#F%\"\"$" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "A1:=%*theta^d;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#A1G*&,(%\"xG\"\"\"*&\"\"\"F*F'!\"\"!\"\"-
%#lnG6#F'#F(\"\"$F()F-\"\"#F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 42 "new_f:=collect(f-diff(A1,x),theta,normal);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&new_fG*&*&,**$)%\"xG\"\"$\"\"\"\"\"\"F*F+*$)F*\"\"#F
,!\"#F0F-F--%#lnG6#F*F-F,*$)F*\"\"#F,!\"\"" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 57 "d:=degree(new_f,theta); # this way the degree did g
o down" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG\"\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "lcoeff(new_f,theta);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#*&,**$)%\"xG\"\"$\"\"\"\"\"\"F'F(*$)F'\"\"#F)!\"
#F-F*F)*$)F'\"\"#F)!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "
int(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**$)%\"xG\"\"#\"\"\"#\"
\"\"F'F&!\"#*&F(F(F&!\"\"F+-%#lnG6#F&\"\"$" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 26 "subs(theta=theta/(d+1),%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,**$)%\"xG\"\"#\"\"\"#\"\"\"F'F&!\"#*&F(F(F&!\"\"F+-%#l
nG6#F&#\"\"$F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "A2:=%*the
ta^d;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A2G*&,**$)%\"xG\"\"#\"\"\"
#\"\"\"F*F)!\"#*&F+F+F)!\"\"F.-%#lnG6#F)#\"\"$F*F-F1F-" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "new_f:=collect(new_f-diff(A2,x),the
ta,normal);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&new_fG,$*&,(*$)%\"xG
\"\"$\"\"\"\"\"\"*$)F*\"\"#F,!\"%F1F-F,*$)F*\"\"#F,!\"\"#!\"\"F0" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "d:=degree(new_f,theta);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG\"\"!" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 5 "Done." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Int(
'f',x)='A1'+'A2'+Int('new_f',x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-
%$IntG6$%\"fG%\"xG,(%#A1G\"\"\"%#A2GF+-F%6$%&new_fGF(F+" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "rhs(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,(*&,(%\"xG\"\"\"*&\"\"\"F)F&!\"\"!\"\"-%#lnG6#F&#F'\"
\"$F')F,\"\"#F)F'*&,**$)F&F2F)#F'F2F&!\"#F(F8F,#F0F2F'F,F'F'-%$IntG6$,
$*&,(*$)F&F0F)F'F5!\"%FBF'F)*$)F&\"\"#F)F*#F+F2F&F'" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#,,*&,(%\"xG\"\"\"*&\"\"\"F)F&!\"\"!\"\"-%#lnG6#F&#F'\"\"$F')F,\"\"#
F)F'*&,**$)F&F2F)#F'F2F&!\"#F(F8F,#F0F2F'F,F'F'F5#F+\"\"%F&F2F(F8" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "normal(diff(%,x)-f);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 6 "0, OK." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 67 "But what happens when, if we integrate lcoeff(f,theta) wh
ich is in " }{TEXT 268 1 "K" }{TEXT -1 84 ", if we need a logarithmic \+
extension other than theta? This can happen, for example:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "f:=1/(x-1) * theta^3 + x*theta;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,&*&*$)-%#lnG6#%\"xG\"\"$\"\"\"F
.,&F,\"\"\"!\"\"F0!\"\"F0*&F,F0F)F0F0" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "d:=degree(f,theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%\"dG\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "lcoeff(f,t
heta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$,&%\"xG\"\"\"!\"\"
F'!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "int(%,x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#,&%\"xG\"\"\"!\"\"F(" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "A new logarithm, which is not an e
lement of " }{TEXT 269 1 "K" }{TEXT -1 41 "(theta). Let's see if we ca
n do anything:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "A1:=%*the
ta^d;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A1G*&-%#lnG6#,&%\"xG\"\"\"
!\"\"F+F+)-F'6#F*\"\"$\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 38 "f:=collect(f-diff(A1,x),theta,normal);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"fG,&*&%\"xG\"\"\"-%#lnG6#F'F(F(*&*&-F*6#,&F'F(!\"\"
F(F()F)\"\"#\"\"\"F4F'!\"\"!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 19 "d:=degree(f,theta);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG
\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "lcoeff(f,theta);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%#lnG6#,&%\"xG\"\"\"!\"\"F*\"\"
\"F)!\"\"!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "int(%,x);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%&dilogG6#%\"xG!\"$*&-%#lnG6#,&F'
\"\"\"!\"\"F.F.-F+F&F.F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "The d
ilog function is not an elementary function." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 201 "Liouvilles principle, which we
 will see next time, will tell us that using just elementary functions
 it is not possible to integrate the last example f, that is, int(f,x)
 is not an elementary function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 270 88 "Homework: integrate the following functi
on. Use Maple's int only on rational functions. " }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 117 "f := ln(x)^4+2*(x+x^2+1)/x*ln(x)^3-(2+2*x+x^
3+4*x^2+x^4)/x^2*ln(x)^2-(-6*x^3-3*x^4+\n2*x^6+2+3*x+x^2+2*x^5)/x^3*ln
(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,**$)-%#lnG6#%\"xG\"\"%
\"\"\"\"\"\"*&*&,(F.F.F+F.*$)F+\"\"#F-F.F.)F(\"\"$F-F-F+!\"\"F4*&*&,,F
4F.F+F4*$)F+F6F-F.F2F,*$)F+F,F-F.F.)F(F4F-F-*$)F+\"\"#F-F7!\"\"*&*&,0F
;!\"'F=!\"$*$)F+\"\"'F-F4F4F.F+F6F2F.*$)F+\"\"&F-F4F.F(F.F-*$)F+\"\"$F
-F7FC" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "75 0 0
" 0 }{VIEWOPTS 1 1 0 1 1 1803 }