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{SECT 0 {PARA 209 "" 0 "" {TEXT 250 38 " Worksheet 3, Math 152     Spr
ing 2006" }}{PARA 210 "" 0 "" {TEXT -1 0 "" }}{PARA 211 "" 0 "" {TEXT 
251 22 "Integration with Maple" }}{PARA 212 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 212 "> " 0 "" {MPLTEXT 1 252 8 "restart;" }}}{SECT 1 
{PARA 213 "" 0 "" {TEXT 253 37 "Introduction to Integration Technique
" }}{SECT 1 {PARA 214 "" 0 "" {TEXT 254 32 "Discovering Integration Fo
rmulas" }}{PARA 212 "" 0 "" {TEXT 255 142 "One of the ways we can make
 use of Maple's integration capabilities is to use them to discover (e
xperimentally) certain integration formulas. " }}{EXCHG {PARA 212 "" 
0 "" {TEXT 255 110 "Let's start with an integral whose value we know (
we obtained the answer in class using integration by parts):" }}{PARA 
212 "" 0 "" {TEXT 255 51 "                                            \+
       " }{XPPEDIT 18 0 "int(ln(x),x) = x ln(x) -x + C" "6#/-%$intG6$-
%#lnG6#%\"xGF*,(*&F*\"\"\"-F(6#F*F-F-F*!\"\"%\"CGF-" }{TEXT 255 2 "  \+
" }}{PARA 212 "> " 0 "" {MPLTEXT 1 252 13 "int(ln(x),x);" }}}{EXCHG 
{PARA 212 "" 0 "" {TEXT 255 71 "Note that Maple does not bother to giv
e us the constant of integration." }}{PARA 212 "" 0 "" {TEXT 255 14 "N
ow let's try " }{XPPEDIT 18 0 "int(x ln(x),x)" "6#-%$intG6$*&%\"xG\"\"
\"-%#lnG6#F'F(F'" }{TEXT 255 1 " " }}{PARA 212 "> " 0 "" {MPLTEXT 1 
252 15 "int(x*ln(x),x);" }}}{EXCHG {PARA 212 "" 0 "" {TEXT 255 4 "and \+
" }{XPPEDIT 18 0 "int(x^2*ln(x),x)" "6#-%$intG6$*&%\"xG\"\"#-%#lnG6#F'
\"\"\"F'" }{TEXT 255 1 " " }}{PARA 212 "> " 0 "" {MPLTEXT 1 252 17 "in
t(x^2*ln(x),x);" }}}{EXCHG {PARA 212 "" 0 "" {TEXT 255 19 "and maybe o
ne more " }}{PARA 212 "> " 0 "" {MPLTEXT 1 252 17 "int(x^3*ln(x),x);" 
}}}{EXCHG {PARA 212 "" 0 "" {TEXT 255 42 "I guess you might see a patt
ern emerging. " }}{PARA 212 "" 0 "" {TEXT 255 17 "Could it be that " }
{XPPEDIT 18 0 "int(x^4*ln(x),x) = x^5*ln(x)/5 -x^5/25" "6#/-%$intG6$*&
%\"xG\"\"%-%#lnG6#F(\"\"\"F(,&*(F(\"\"&-F+6#F(F-F0!\"\"F-*&F(F0\"#DF3F
3" }{TEXT 255 3 "  ?" }}{PARA 212 "> " 0 "" {MPLTEXT 1 252 17 "int(x^4
*ln(x),x);" }}}{EXCHG {PARA 212 "" 0 "" {TEXT 255 36 "Yes, so the gene
ral formula must be " }}{PARA 212 "" 0 "" {TEXT 255 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-6#F)F+,&F*F+F+F+!\"\"F+*&)F),&F*F+
F+F+F+*$,&F*F+F+F+\"\"#F6F6" }{TEXT 255 1 " " }}{PARA 212 "> " 0 "" 
{MPLTEXT 1 252 17 "int(x^n*ln(x),x);" }}}{EXCHG {PARA 212 "" 0 "" 
{TEXT 255 171 "This does not exactly match our guess, but trying to te
ll Maple to display the answer in exactly the form we want it, can be \+
rather involved, and is not worth our effort. " }}{PARA 212 "" 0 "" 
{TEXT 255 33 "One thing we can try, though, is " }{HYPERLNK 17 "simpli
fy" 2 "simplify" "" }{TEXT 255 22 " with the exp option: " }}{PARA 
212 "" 0 "" {TEXT 255 53 "(Note that this step may be required on Mapl
e 6 only)" }}{PARA 212 "> " 0 "" {MPLTEXT 1 252 17 "simplify(%, exp);
" }}}}{SECT 1 {PARA 214 "" 0 "" {TEXT 254 27 "Using the intparts Funct
ion" }}{EXCHG {PARA 212 "" 0 "" {TEXT 255 79 "We can also use Maple to
 see how the general formula found above evolves using " }{TEXT 257 
20 "integration by parts" }{TEXT 255 2 ". " }}{PARA 212 "" 0 "" {TEXT 
255 23 "To this end we use the " }{HYPERLNK 17 "intparts" 2 "intparts
" "" }{TEXT 255 19 " function from the " }{HYPERLNK 17 "student" 2 "st
udent" "" }{TEXT 257 1 " " }{TEXT 255 8 "package." }}{PARA 212 "> " 0 
"" {MPLTEXT 1 252 14 "with(student):" }}}{EXCHG {PARA 212 "" 0 "" 
{TEXT 255 4 "The " }{HYPERLNK 17 "intparts" 2 "intparts" "" }{TEXT 
255 29 " command takes two arguments:" }}{PARA 212 "" 0 "" {TEXT 255 
48 "the first is an (inert) integral specified with " }{HYPERLNK 17 "I
nt" 2 "Int" "" }{TEXT 255 4 "(); " }}{PARA 212 "" 0 "" {TEXT 255 48 "t
he second argument says what we want to use as " }{XPPEDIT 18 0 "u" "6
#%\"uG" }{TEXT 255 16 "  in the formula" }}{PARA 212 "" 0 "" {TEXT 
255 15 "               " }{XPPEDIT 18 0 "int(u,v) = u*v - int(v,u)" "6
#/-%$intG6$%\"uG%\"vG,&*&F'\"\"\"F(F+F+-F%6$F(F'!\"\"" }{TEXT 255 1 " \+
" }}{PARA 212 "" 0 "" {TEXT 255 42 "This is what needs to be done in o
ur case." }}{PARA 212 "> " 0 "" {MPLTEXT 1 252 35 "intparts(Int(x^n*ln
(x), x), ln(x));" }}}{EXCHG {PARA 212 "" 0 "" {TEXT 255 47 "This formu
la can be cleaned up a bit using the " }{HYPERLNK 17 "applyop" 2 "appl
yop" "" }{TEXT 255 69 " command, which allows us to apply a certain op
eration (in this case " }{HYPERLNK 17 "simplify" 2 "simplify" "" }
{TEXT 255 43 ") to only a specific part of an expression." }}{PARA 
212 "" 0 "" {TEXT 255 8 "Here we " }{HYPERLNK 17 "simplify" 2 "simplif
y" "" }{TEXT 255 22 " the second part (the " }{TEXT 257 1 "2" }{TEXT 
255 32 ")  of the previous formula (the " }{HYPERLNK 17 "%" 2 "percent
" "" }{TEXT 255 4 ").  " }}{PARA 212 "> " 0 "" {MPLTEXT 1 252 22 "appl
yop(simplify,2,%);" }}}{EXCHG {PARA 212 "" 0 "" {TEXT 255 56 "To obtai
n the same answer we had earlier we ask for the " }{HYPERLNK 17 "value
" 2 "value" "" }{TEXT 255 69 " of the integral (which you have to alwa
ys do with the inert command " }{HYPERLNK 17 "Int" 2 "Int" "" }{TEXT 
255 9 ") and get" }}{PARA 212 "> " 0 "" {MPLTEXT 1 252 9 "value(%);" }
}}{EXCHG {PARA 212 "" 0 "" {TEXT 255 55 "We could have used the evalua
ted form of the integral, " }{HYPERLNK 17 "int" 2 "int" "" }{TEXT 255 
9 "(), with " }{HYPERLNK 17 "intparts" 2 "intparts" "" }{TEXT 255 57 "
 also - but the pedagogical effect would not be the same:" }}{PARA 
212 "> " 0 "" {MPLTEXT 1 252 35 "intparts(int(x^n*ln(x), x), ln(x));" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 214 "" 0 "" 
{TEXT 254 28 "Partial Fractions with Maple" }}{EXCHG {PARA 212 "" 0 "
" {TEXT 255 52 "A third useful tool is the partial fractions option " 
}{HYPERLNK 17 "parfrac" 2 "parfrac" "" }{TEXT 257 1 " " }{TEXT 255 7 "
of the " }{HYPERLNK 17 "convert" 2 "convert" "" }{TEXT 257 1 " " }
{TEXT 255 9 "command, " }}{PARA 212 "" 0 "" {TEXT 255 106 "e.g., on a \+
future homework you will have to find the partial fractions decomposit
ion for the integrand of " }}{PARA 212 "" 0 "" {TEXT 255 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*!\"\"\"\"$F*F1F+" }{TEXT 255 3 "  ." }}{PARA 212 "
" 0 "" {TEXT 255 31 "This is how it's done in Maple:" }}{PARA 212 "> \+
" 0 "" {MPLTEXT 1 252 47 "convert((2*s+2)/((s^2+1)*(s-1)^3),'parfrac',
s);" }}}}{EXCHG }}{SECT 1 {PARA 214 "" 0 "" {TEXT 254 18 "Exercise Que
stions" }}{SECT 1 {PARA 214 "" 0 "" {TEXT 254 5 "Ex.1:" }}{PARA 212 "
" 0 "" {TEXT 255 26 "a) Evaluate the integrals " }{XPPEDIT 18 0 "int(l
n(x)/x^2,x), int(ln(x)/x^3,x), int(ln(x)/x^4,x)" "6%-%$intG6$*&-%#lnG6
#%\"xG\"\"\"*$F*\"\"#!\"\"F*-F$6$*&-F(6#F*F+*$F*\"\"$F.F*-F$6$*&-F(6#F
*F+*$F*\"\"%F.F*" }{TEXT 255 2 " ." }}{PARA 212 "" 0 "" {TEXT -1 0 "" 
}}{PARA 212 "" 0 "" {TEXT 255 44 "b) Can you see a pattern? What do yo
u think " }{XPPEDIT 18 0 "int(ln(x)/x^5,x)" "6#-%$intG6$*&-%#lnG6#%\"x
G\"\"\"*$F*\"\"&!\"\"F*" }{TEXT 255 5 "  is?" }}{PARA 212 "" 0 "" 
{TEXT 255 35 "c) What is the general formula for " }{XPPEDIT 18 0 "int
(ln(x)/x^n,x), n>=2" "6$-%$intG6$*&-%#lnG6#%\"xG\"\"\")F*%\"nG!\"\"F*1
\"\"#F-" }{TEXT 255 2 " ?" }}{PARA 212 "" 0 "" {TEXT 255 82 "Write out
 your own formula, and compare it to Maple's answer. Are they equivale
nt?" }}{PARA 212 "" 0 "" {TEXT -1 0 "" }}{PARA 212 "" 0 "" {TEXT 255 
31 "d) Derive the formula with the " }{HYPERLNK 17 "intparts" 2 "intpa
rts" "" }{TEXT 255 9 " command." }}{PARA 212 "" 0 "" {TEXT -1 0 "" }}}
{SECT 1 {PARA 214 "" 0 "" {TEXT 254 5 "Ex.2:" }}{PARA 212 "" 0 "" 
{TEXT 255 60 "According to the integral table in the back of our textb
ook " }}{PARA 212 "" 0 "" {TEXT 255 49 "                              \+
                   " }{XPPEDIT 18 0 "int(sqrt(x^2+1),x) = x*sqrt(x^2+1
)/2 + ln(x+sqrt(x^2+1))/2" "6#/-%$intG6$-%%sqrtG6#,&*$%\"xG\"\"#\"\"\"
F.F.F,,&*(F,F.-F(6#,&*$F,F-F.F.F.F.F-!\"\"F.*&-%#lnG6#,&F,F.-F(6#,&*$F
,F-F.F.F.F.F.F-F5F." }{TEXT 255 4 "  . " }}{PARA 212 "" 0 "" {TEXT 
255 65 "a) What answer does Maple give you for the value of the integr
al?" }}{PARA 212 "" 0 "" {TEXT -1 0 "" }}{PARA 212 "" 0 "" {TEXT 255 
60 "b)  Analytically show that the two formulas are equivalent. " }}}
{SECT 1 {PARA 214 "" 0 "" {TEXT 254 5 "Ex.3:" }}{PARA 212 "" 0 "" 
{TEXT 255 72 "(a) Use Maple to find the partial fraction decomposition
 of the function" }}{PARA 212 "" 0 "" {XPPEDIT 18 0 "f(x) = (4*x^3-27*
x^2+5*x-32)/(30*x^5-13*x^4+50*x^3-286*x^2-299*x-70);" "6#/-%\"fG6#%\"x
G*&,**&\"\"%\"\"\"*$F'\"\"$F,F,*&\"#FF,*$F'\"\"#F,!\"\"*&\"\"&F,F'F,F,
\"#KF3F,,.*&\"#IF,*$F'F5F,F,*&\"#8F,*$F'F+F,F3*&\"#]F,*$F'F.F,F,*&\"$'
GF,*$F'F2F,F3*&\"$*HF,F'F,F3\"#qF3F3" }{TEXT 255 1 " " }}{PARA 215 "" 
0 "" {TEXT -1 0 "" }}{PARA 212 "" 0 "" {TEXT 255 32 "      (b)  Use pa
rt (a) to find " }{XPPEDIT 18 0 "int(f(x),x);" "6#-%$intG6$-%\"fG6#%\"
xGF)" }{TEXT 255 2 " ." }}{PARA 212 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 
{PARA 214 "" 0 "" {TEXT 254 5 "Ex.4:" }}{PARA 212 "" 0 "" {TEXT 255 
15 "a) Integrate   " }{XPPEDIT 18 0 "int(x^2*(x^3+a)^3,x);" "6#-%$intG
6$*&%\"xG\"\"#,&*$F'\"\"$\"\"\"%\"aGF,F+F'" }{TEXT 255 31 "  by hand (
use a substitution)." }}{PARA 212 "" 0 "" {TEXT 255 61 "b) Use Maple t
o evaluate the integral in Part a) and get the " }{TEXT 269 12 "same r
esult." }}{PARA 212 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }{TEXT 255 27 "In Maple you will need the " }{HYPERLNK 17 "change
var" 2 "changevar" "" }{TEXT 255 18 " command from the " }{HYPERLNK 
17 "student" 2 "student" "" }{TEXT 255 51 " package (which does integr
ation by substitution). " }}{PARA 212 "" 0 "" {TEXT 255 15 "The syntax
 for " }{HYPERLNK 17 "changevar" 2 "changevar" "" }{TEXT 255 4 " is " 
}{TEXT 257 103 "changevar(expression defining the substitution, integr
al in which you want to substitute, new variable)" }{TEXT 255 2 ". " }
}{PARA 212 "" 0 "" {TEXT 255 29 "e.g., in order to substitute " }
{XPPEDIT 18 0 "u=cos(x) " "6#/%\"uG-%$cosG6#%\"xG" }{TEXT 255 18 "  in
 the integral " }{XPPEDIT 18 0 "int(sin(x)/(cos(x))^2, x)" "6#-%$intG6
$*&-%$sinG6#%\"xG\"\"\"*$-%$cosG6#F*\"\"#!\"\"F*" }{TEXT 255 4 "  do" 
}}{EXCHG {PARA 212 "> " 0 "" {MPLTEXT 1 252 52 "changevar(cos(x) = u, \+
Int(sin(x)/(cos(x))^2, x), u);" }}}{PARA 212 "" 0 "" {TEXT 255 44 "Or,
 if you want to evaluate at the same time" }}{EXCHG {PARA 212 "> " 0 "
" {MPLTEXT 1 252 52 "changevar(cos(x) = u, int(sin(x)/(cos(x))^2, x), \+
u);" }}}{PARA 212 "" 0 "" {TEXT 255 100 "After you have done the subst
itution and intergration you may want to change the variable back with
 " }{HYPERLNK 17 "subs" 2 "subs" "" }{TEXT 257 17 "(u=x,newintegral)" 
}{TEXT 255 27 " and show the final result." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "If it is not easy to tell whet
her or not the results from part(a) and part(b) are identical, you can
 use Maple to check it." }}{PARA 0 "" 0 "" {TEXT -1 91 "For example, g
iven a=(x+2)^5, b=x^5+10*x^4+40*x^3+80*x^2+80*x+32, you can do the fol
lowing:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "a:=(x+2)^5;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "b:=x^5+10*x^4+40*x^3+80*x^2+
80*x+32;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "'a-b'=simplify(a-b);" }}}}}{EXCHG }{PARA 220 "" 0 "" 
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