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{SECT 0 {PARA 206 "" 0 "" {TEXT 224 22 "Integration with Maple" }}
{EXCHG {PARA 207 "> " 0 "" {MPLTEXT 1 225 8 "restart;" }}}{SECT 0 
{PARA 208 "" 0 "" {TEXT 226 12 "Introduction" }}{SECT 0 {PARA 209 "" 0
 "" {TEXT 227 32 "Discovering Integration Formulas" }}{PARA 207 "" 0 "
" {TEXT 228 142 "One of the ways we can make use of Maple's integratio
n capabilities is to use them to discover (experimentally) certain int
egration formulas. " }}{EXCHG {PARA 207 "" 0 "" {TEXT 228 110 "Let's s
tart with an integral whose value we know (we obtained the answer in c
lass using integration by parts):" }}{PARA 207 "" 0 "" {TEXT 228 51 " \+
                                                  " }{XPPEDIT 18 0 "in
t(ln(x),x) = x ln(x) -x + C" "6#/-%$intG6$-%#lnG6#%\"xGF*,(*&F*\"\"\"F
'F-F-F*!\"\"%\"CGF-" }{TEXT 229 1 " " }{TEXT 229 1 " " }{TEXT 228 1 " \+
" }}{PARA 207 "> " 0 "" {MPLTEXT 1 225 13 "int(ln(x),x);" }}}{EXCHG 
{PARA 207 "" 0 "" {TEXT 228 71 "Note that Maple does not bother to giv
e us the constant of integration." }}{PARA 207 "" 0 "" {TEXT 228 14 "N
ow let's try " }{XPPEDIT 18 0 "int(x ln(x),x)" "6#-%$intG6$*&%\"xG\"\"
\"-%#lnG6#F'F(F'" }{TEXT 229 1 " " }{TEXT 229 1 " " }}{PARA 207 "> " 0
 "" {MPLTEXT 1 225 15 "int(x*ln(x),x);" }}{PARA 210 "" 1 "" {XPPMATH 
20 "6#,&*&I\"xG6\"\"\"#-I#lnG6$I*protectedGF+I(_syslibGF&6#F%\"\"\"#F.
F'*$F%F'#!\"\"\"\"%" }{TEXT 230 1 " " }}}{EXCHG {PARA 207 "" 0 "" 
{TEXT 228 4 "and " }{XPPEDIT 18 0 "int(x^2*ln(x),x)" "6#-%$intG6$*&%\"
xG\"\"#-%#lnG6#F'\"\"\"F'" }{TEXT 229 1 " " }{TEXT 229 1 " " }}{PARA 
207 "" 0 "" }{PARA 207 "> " 0 "" {MPLTEXT 1 225 17 "int(x^2*ln(x),x);"
 }}{PARA 210 "" 1 "" {XPPMATH 20 "6#,&*&I\"xG6\"\"\"$-I#lnG6$I*protect
edGF+I(_syslibGF&6#F%\"\"\"#F.F'*$F%F'#!\"\"\"\"*" }{TEXT 230 1 " " }}
}{EXCHG {PARA 207 "" 0 "" {TEXT 228 19 "and maybe one more " }}{PARA 
207 "> " 0 "" {MPLTEXT 1 225 17 "int(x^3*ln(x),x);" }}}{EXCHG {PARA 
207 "" 0 "" {TEXT 228 42 "I guess you might see a pattern emerging. " 
}}{PARA 207 "" 0 "" {TEXT 228 17 "Could it be that " }{XPPEDIT 18 0 "i
nt(x^4*ln(x),x) = x^5*ln(x)/5 -x^5/25" "6#/-%$intG6$*&%\"xG\"\"%-%#lnG
6#F(\"\"\"F(,&*(F(\"\"&F*F-F0!\"\"F-*&F(F0\"#DF1F1" }{TEXT 229 1 " " }
{TEXT 229 1 " " }{TEXT 228 2 " ?" }}{PARA 207 "> " 0 "" {MPLTEXT 1 
225 17 "int(x^4*ln(x),x);" }}}{EXCHG {PARA 207 "" 0 "" {TEXT 228 36 "Y
es, so the general formula must be " }}{PARA 207 "" 0 "" {TEXT 228 46 
"                                              " }{XPPEDIT 18 0 "int(x
^n*ln(x),x) = x^(n+1)*ln(x)/(n+1) - x^(n+1)/(n+1)^2" "6#/-%$intG6$*&)%
\"xG%\"nG\"\"\"-%#lnG6#F)F+F),&*()F),&F*F+F+F+F+F,F+F2!\"\"F+*&F1F+*$F
2\"\"#F3F3" }{TEXT 229 1 " " }{TEXT 229 1 " " }}{PARA 207 "> " 0 "" 
{MPLTEXT 1 225 17 "int(x^n*ln(x),x);" }}{PARA 210 "" 1 "" {XPPMATH 20 
"6#,&**,&\"\"\"F&I\"nG6\"F&!\"\"I\"xGF(F&-I#lnG6$I*protectedGF.I(_sysl
ibGF(6#F*F&-I$expGF-6#*&F'F&F+F&F&F&*(,(F&F&F'\"\"#*$F'F7F&F)F*F&F1F&F
)" }{TEXT 230 1 " " }}}{EXCHG {PARA 207 "" 0 "" {TEXT 228 171 "This do
es not exactly match our guess, but trying to tell Maple to display th
e answer in exactly the form we want it, can be rather involved, and i
s not worth our effort. " }}{PARA 207 "" 0 "" {TEXT 228 33 "One thing \+
we can try, though, is " }{HYPERLNK 231 "simplify" 2 "simplify" "" }
{TEXT 228 22 " with the exp option: " }}{PARA 207 "" 0 "" {TEXT 228 
53 "(Note that this step may be required on Maple 6 only)" }}{PARA 
207 "> " 0 "" {MPLTEXT 1 225 17 "simplify(%, exp);" }}{PARA 210 "" 1 "
" {XPPMATH 20 "6#,&**,&\"\"\"F&I\"nG6\"F&!\"\"I\"xGF(F&-I#lnG6$I*prote
ctedGF.I(_syslibGF(6#F*F&)F*F'F&F&*(,(F&F&F'\"\"#*$F'F4F&F)F*F&F1F&F)"
 }{TEXT 230 1 " " }}}}{SECT 0 {PARA 209 "" 0 "" {TEXT 227 27 "Using th
e intparts Function" }}{EXCHG {PARA 207 "" 0 "" {TEXT 228 79 "We can a
lso use Maple to see how the general formula found above evolves using
 " }{TEXT 232 20 "integration by parts" }{TEXT 228 2 ". " }}{PARA 207 
"" 0 "" {TEXT 228 23 "To this end we use the " }{HYPERLNK 231 "intpart
s" 2 "intparts" "" }{TEXT 228 19 " function from the " }{HYPERLNK 231 
"student" 2 "student" "" }{TEXT 232 1 " " }{TEXT 228 8 "package." }}
{PARA 207 "> " 0 "" {MPLTEXT 1 225 14 "with(student);" }}}{EXCHG 
{PARA 207 "" 0 "" {TEXT 228 4 "The " }{HYPERLNK 231 "intparts" 2 "intp
arts" "" }{TEXT 228 29 " command takes two arguments:" }}{PARA 207 "" 
0 "" {TEXT 228 48 "the first is an (inert) integral specified with " }
{HYPERLNK 231 "Int" 2 "Int" "" }{TEXT 228 4 "(); " }}{PARA 207 "" 0 ""
 {TEXT 228 48 "the second argument says what we want to use as " }
{XPPEDIT 18 0 "u" "6#%\"uG" }{TEXT 229 1 " " }{TEXT 229 1 " " }{TEXT 
228 15 " in the formula" }}{PARA 207 "" 0 "" {TEXT 228 15 "           \+
    " }{XPPEDIT 18 0 "int(u,v) = u*v - int(v,u)" "6#/-%$intG6$%\"uG%\"
vG,&*&F'\"\"\"F(F+F+-F%6$F(F'!\"\"" }{TEXT 229 1 " " }{TEXT 229 1 " " 
}}{PARA 207 "" 0 "" {TEXT 228 42 "This is what needs to be done in our
 case." }}{PARA 207 "> " 0 "" {MPLTEXT 1 225 35 "intparts(Int(x^n*ln(x
), x), ln(x));" }}}{EXCHG {PARA 207 "" 0 "" {TEXT 228 47 "This formula
 can be cleaned up a bit using the " }{HYPERLNK 231 "applyop" 2 "apply
op" "" }{TEXT 228 69 " command, which allows us to apply a certain ope
ration (in this case " }{HYPERLNK 231 "simplify" 2 "simplify" "" }
{TEXT 228 43 ") to only a specific part of an expression." }}{PARA 
207 "" 0 "" {TEXT 228 8 "Here we " }{HYPERLNK 231 "simplify" 2 "simpli
fy" "" }{TEXT 228 22 " the second part (the " }{TEXT 232 1 "2" }{TEXT 
228 32 ")  of the previous formula (the " }{HYPERLNK 231 "%" 2 "percen
t" "" }{TEXT 228 4 ").  " }}{PARA 207 "> " 0 "" {MPLTEXT 1 225 22 "app
lyop(simplify,2,%);" }}}{EXCHG {PARA 207 "" 0 "" {TEXT 228 56 "To obta
in the same answer we had earlier we ask for the " }{HYPERLNK 231 "val
ue" 2 "value" "" }{TEXT 228 69 " of the integral (which you have to al
ways do with the inert command " }{HYPERLNK 231 "Int" 2 "Int" "" }
{TEXT 228 9 ") and get" }}{PARA 207 "> " 0 "" {MPLTEXT 1 225 9 "value(
%);" }}}{EXCHG {PARA 207 "" 0 "" {TEXT 228 55 "We could have used the \+
evaluated form of the integral, " }{HYPERLNK 231 "int" 2 "int" "" }
{TEXT 228 9 "(), with " }{HYPERLNK 231 "intparts" 2 "intparts" "" }
{TEXT 228 57 " also - but the pedagogical effect would not be the same
:" }}{PARA 207 "> " 0 "" {MPLTEXT 1 225 35 "intparts(int(x^n*ln(x), x)
, ln(x));" }}}}{SECT 1 {PARA 209 "" 0 "" {TEXT 227 28 "Partial Fractio
ns with Maple" }}{EXCHG {PARA 207 "" 0 "" {TEXT 228 52 "A third useful
 tool is the partial fractions option " }{HYPERLNK 231 "parfrac" 2 "pa
rfrac" "" }{TEXT 232 1 " " }{TEXT 228 7 "of the " }{HYPERLNK 231 "conv
ert" 2 "convert" "" }{TEXT 232 1 " " }{TEXT 228 9 "command, " }}{PARA 
207 "" 0 "" {TEXT 228 106 "e.g., on a future homework you will have to
 find the partial fractions decomposition for the integrand of " }}
{PARA 207 "" 0 "" {TEXT 228 44 "                                      \+
      " }{XPPEDIT 18 0 "int((2*s+2)/((s^2+1)*(s-1)^3), s)" "6#-%$intG6
$*&,&*&\"\"#\"\"\"%\"sGF*F*F)F*F**&,&*$F+F)F*F*F*F**$,&F+F*F*!\"\"\"\"
$F*F1F+" }{TEXT 229 1 " " }{TEXT 229 1 " " }{TEXT 228 2 " ." }}{PARA 
207 "" 0 "" {TEXT 228 31 "This is how it's done in Maple:" }}{PARA 
207 "> " 0 "" {MPLTEXT 1 225 47 "convert((2*s+2)/((s^2+1)*(s-1)^3),'pa
rfrac',s);" }}}{EXCHG {PARA 207 "> " 0 "" }}}}{SECT 0 {PARA 208 "" 0 "
" {TEXT 226 12 "Assignment 3" }}{SECT 0 {PARA 209 "" 0 "" {TEXT 227 5 
"Ex.1:" }}{PARA 207 "" 0 "" {TEXT 228 26 "a) Evaluate the integrals " 
}{XPPEDIT 18 0 "int(ln(x)/x^2,x), int(ln(x)/x^3,x), int(ln(x)/x^4,x)" 
"6%-%$intG6$*&-%#lnG6#%\"xG\"\"\"*$F*\"\"#!\"\"F*-F$6$*&F'F+*$F*\"\"$F
.F*-F$6$*&F'F+*$F*\"\"%F.F*" }{TEXT 229 1 " " }{TEXT 229 1 " " }{TEXT 
228 1 "." }}{PARA 207 "" 0 "" {TEXT 228 44 "b) Can you see a pattern? \+
What do you think " }{XPPEDIT 18 0 "int(ln(x)/x^5,x)" "6#-%$intG6$*&-%
#lnG6#%\"xG\"\"\"*$F*\"\"&!\"\"F*" }{TEXT 229 1 " " }{TEXT 229 1 " " }
{TEXT 228 4 " is?" }}{PARA 207 "" 0 "" {TEXT 228 35 "c) What is the ge
neral formula for " }{XPPEDIT 18 0 "int(ln(x)/x^n,x), n>=2" "6$-%$intG
6$*&-%#lnG6#%\"xG\"\"\")F*%\"nG!\"\"F*1\"\"#F-" }{TEXT 229 1 " " }
{TEXT 229 1 " " }{TEXT 228 1 "?" }}{PARA 207 "" 0 "" {TEXT 228 82 "Wri
te out your own formula, and compare it to Maple's answer. Are they eq
uivalent?" }}{PARA 207 "" 0 "" {TEXT 228 31 "d) Derive the formula wit
h the " }{HYPERLNK 231 "intparts" 2 "intparts" "" }{TEXT 228 9 " comma
nd." }}}{SECT 0 {PARA 209 "" 0 "" {TEXT 227 5 "Ex.2:" }}{PARA 207 "" 0
 "" {TEXT 228 8 "Use the " }{HYPERLNK 231 "intparts" 2 "intparts" "" }
{TEXT 228 23 " command to derive the " }{TEXT 232 17 "reduction formul
a" }}{PARA 207 "" 0 "" {TEXT 228 42 "                                 \+
         " }{XPPEDIT 18 0 "int(x^n*(ln(a*x))^m,x) = (x^(n+1)*(ln(a*x))
^m)/(n+1) - (m/(n+1))*int(x^n*(ln(a*x))^(m-1),x)" "6#/-%$intG6$*&)%\"x
G%\"nG\"\"\")-%#lnG6#*&%\"aGF+F)F+%\"mGF+F),&*()F),&F*F+F+F+F+F,F+F6!
\"\"F+*(F2F+F6F7-F%6$*&F(F+)F-,&F2F+F+F7F+F)F+F7" }{TEXT 229 1 " " }
{TEXT 229 1 " " }{TEXT 228 1 "." }}{PARA 207 "" 0 "" }}{SECT 0 {PARA 
209 "" 0 "" {TEXT 227 5 "Ex.3:" }}{PARA 207 "" 0 "" {TEXT 228 60 "Acco
rding to the integral table in the back of our textbook " }}{PARA 207 
"" 0 "" {TEXT 228 49 "                                                \+
 " }{XPPEDIT 18 0 "int(sqrt(x^2+1),x) = x*sqrt(x^2+1)/2 + ln(x+sqrt(x^
2+1))/2" "6#/-%$intG6$-%%sqrtG6#,&*$%\"xG\"\"#\"\"\"F.F.F,,&*(F,F.F'F.
F-!\"\"F.*&-%#lnG6#,&F,F.F'F.F.F-F1F." }{TEXT 229 1 " " }{TEXT 229 1 "
 " }{TEXT 228 3 " . " }}{PARA 207 "" 0 "" {TEXT 228 65 "a) What answer
 does Maple give you for the value of the integral?" }}{PARA 207 "" 0 
"" {TEXT 228 86 "b)  Show that the two formulas are equivalent. A few \+
things you might want to try are:" }}{PARA 207 "" 0 "" {TEXT 228 112 "
Graph the two different right-hand side functions (the one from the te
xtbook, and the one from Maple) on [-1,1]." }}{PARA 207 "" 0 "" {TEXT 
228 41 "Compute the derivatives and compare them." }}}{SECT 0 {PARA 
209 "" 0 "" {TEXT 227 5 "Ex.4:" }}{PARA 207 "" 0 "" {TEXT 228 14 "The \+
functions " }{XPPEDIT 2 0 "sin(nx), cos(mx)" "6#6$-I$sinG6$I*protected
GF'I(_syslibG6\"6#I#nxGF)-I$cosGF&6#I#mxGF)" }{TEXT 229 1 " " }{TEXT 
229 1 " " }{TEXT 228 55 "   are an othogonal set of functions.  This m
eans that " }{XPPEDIT 18 0 "Int(sin(nx)*cos(mx),x = -pi .. pi);" "6#-%
$IntG6$*&-%$sinG6#%#nxG\"\"\"-%$cosG6#%#mxGF+/%\"xG;,$%#piG!\"\"F4" }
{TEXT 229 1 " " }{TEXT 229 1 " " }{TEXT 228 15 " = 0 for n <> m" }}
{PARA 207 "" 0 "" }{PARA 207 "" 0 "" {XPPEDIT 18 0 "Int(sin(nx)^2,x = \+
-pi .. pi);" "6#-%$IntG6$*$-%$sinG6#%#nxG\"\"#/%\"xG;,$%#piG!\"\"F0" }
{TEXT 229 1 " " }{TEXT 229 1 " " }{TEXT 228 3 " = " }{XPPEDIT 18 0 "pi
;" "6#%#piG" }{TEXT 229 1 " " }{TEXT 229 1 " " }{TEXT 228 10 " for all
 n" }}{PARA 207 "" 0 "" {XPPEDIT 18 0 "Int(cos(nx)^2,x = -pi .. pi);" 
"6#-%$IntG6$*$-%$cosG6#%#nxG\"\"#/%\"xG;,$%#piG!\"\"F0" }{TEXT 229 1 "
 " }{TEXT 229 1 " " }{TEXT 228 3 " = " }{XPPEDIT 18 0 "pi;" "6#%#piG" 
}{TEXT 229 1 " " }{TEXT 229 1 " " }}{PARA 207 "" 0 "" }{PARA 207 "" 0 
"" {TEXT 228 174 "Verify these relationships for various values of n,m
 and combinations of the functions, including the constant function 1.
  Does Maple have any limitation on the values of n?" }}{PARA 207 "" 0
 "" {TEXT 228 1 "." }}{PARA 207 "" 0 "" {TEXT 228 50 "What is \"specia
l\" about the constant function 1?. " }}}}{EXCHG {PARA 207 "> " 0 "" 
{MPLTEXT 1 225 3 "   " }}}{PARA 211 "" 0 "" }{PARA 212 "" 0 "" }}
{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }