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{SECT 0 {PARA 18 "" 0 "" {TEXT -1 33 "Integration With Student in Mapl
e" }}{PARA 257 "" 0 "" {TEXT -1 18 "Jim Herod, Retired" }}{PARA 258 "
" 0 "" {TEXT -1 21 "School of Mathematics" }}{PARA 259 "" 0 "" {TEXT
-1 12 "Georgia Tech" }}{PARA 260 "" 0 "" {TEXT -1 21 "herod@math.gatec
h.edu" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 8 "restart;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256
360 " The techniques of integration contain more that methods for \+
evaluating integrals. For example, there are many places where it is u
seful to know how to find the partial fraction decomposition of the qu
otient of polynomials. Since this process seems so automated, it is no
surprise that performing a partial fraction decomposition is automate
d within Maple." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "convert(1
/((x+1)*(x+2)^2),parfrac,x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 257 45 "There is a beautiful package in Maple ca
lled " }{TEXT 258 7 "Student" }{TEXT 259 57 " that needs exploration. \+
Look first at all that is there." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student);" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 138 "We will explore t
his package in this section. First, I point out that Maple can do inte
gral that many good students of mathematics cannot." }}{EXCHG {PARA 0
"> " 0 "" {MPLTEXT 1 0 40 "int(sin(x^2),x=0..sqrt(Pi/2)); evalf(%);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "plot(sin(x^2),x=0..sqrt(Pi
/2));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 260 1 " " }{TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT 262 228 "We explore the evaluation of the in
tegral more as a human would. (One should not be discouraged that Mapl
e knows how to evaluate so many integrals!) Either from recalling the \+
previous graph or from considering the derivative of " }{XPPEDIT 18 0
"sin(x^2);" "6#-%$sinG6#*$%\"xG\"\"#" }{TEXT -1 0 "" }{TEXT 261 8 " on
[0, " }{XPPEDIT 18 0 "sqrt(Pi);" "6#-%%sqrtG6#%#PiG" }{TEXT -1 1 " "
}{TEXT 269 162 "], we see that the graph is increasing on that interva
l. Thus the value of the integral lies between the \"left-sums\" and t
he \"right-sums.\" We evaluate both these." }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 90 "leftbox(sin(x^2),x=0..sqrt(Pi/2),10); \n leftsum(s
in(x^2),x=0..sqrt(Pi/2),10);\n evalf(%);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 92 "rightbox(sin(x^2),x=0..sqrt(Pi/2),10); \n rightsum
(sin(x^2),x=0..sqrt(Pi/2),10);\n evalf(%);" }}}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 71 "For better behaved functions, \+
we might expect some use of special sums." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 32 "nsum:=leftsum(x^2+x+1,x=0..1,n);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 30 "evalf(limit(nsum,n=infinity));" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 70 "We examine the ch
ange-of-variable property next with two illustrations" }}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 28 "Int((cos(x)+1)^3*sin(x), x);" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "changevar(cos(x)+1=u, Int((cos(x)+1
)^3*sin(x), x), u);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Int(
sqrt(1-x^2), x=a...b);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "c
hangevar(x=sin(u), Int(sqrt(1-x^2), x=a...b), u);" }}}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 163 "In doing integration by
parts, the problem is to make the appropriate choice for u and dv. Co
nsider the following integral which Maple already knows how to handle.
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Int(x*exp(x),x) = int(x*
exp(x),x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 ""
{TEXT 266 144 "Humans, upon considering this integral the first time m
ight wonder which to take as u and which to take as dv. We watch Maple
take both choices." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "intpa
rts(Int(x*exp(x),x),x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "
intparts(Int(x*exp(x),x),exp(x));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 256 "" 0 "" {TEXT 267 50 "Here is another integral with a number
of choices." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "intparts(Int
(x*exp(x)/(x+1)^2,x),exp(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 43 "intparts(Int(x*exp(x)/(x+1)^2,x),x*exp(x));" }}}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 268 87 "Be aware, that Maple k
new how to do that integral without any coaxing by with(student)." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "int(x*exp(x)/(x+1)^2,x);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT 270 9 "Exercise:" }{TEXT -1 48 " Const
ruct left and right sums for the integral " }{XPPEDIT 18 0 "int(sin(Pi
*x),x = 0 .. 1);" "6#-%$intG6$-%$sinG6#*&%#PiG\"\"\"%\"xGF+/F,;\"\"!F+
" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "1 0" 0 }
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