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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 60 "Maple and Integral Tables
, and Partial Fraction Integration " }}{PARA 0 "" 0 "" {TEXT -1 385 "M
ost functions do not have elementary antiderivatives.  The ones that d
o are so few in number that they can almost all be looked up in tables
 of integrals.  Some students feel that since most of the integrals th
at have antiderivatives can be looked up in tables that they need not \+
study how to find integrals by their own devices.  However,  the probl
ems encountered while using tables" }}{PARA 0 "" 0 "" {TEXT -1 818 "in
clude some rather sophisticated algebraic techniques such as long divi
sion of polynomials,  completing the square,  and converting rational \+
functions to partial fractions.  The student must also develop an abil
ity to recognize the general class of the function that is being integ
rated. Indeed since tables of integrals are developed by humans,  ther
e are errors in the tables.  You also need to develop skills in verify
ing that the results you get are are correct Maple V is an excellent t
ool for finding antiderivatives of functions. For this reason its use \+
can virtually replace the need for using integral tables.  Remember th
at there are bugs in any computer program,  including Maple V.  Hence,
  even when using Maple V, you need to develop skills in verifying tha
t the results that you get are are correct.   " }}{PARA 0 "" 0 "" 
{TEXT -1 138 "You  can probably use Maple V to integrate most any inte
gral that has an elementary integral or that you could use a table to \+
integrate.  " }}{PARA 0 "" 0 "" {TEXT -1 34 "We illustrate with a few \+
examples." }}{PARA 0 "" 0 "" {TEXT 259 9 "Example 1" }{TEXT -1 27 " Ev
aluate  the integral of " }}{PARA 0 "" 0 "" {TEXT -1 61 "             \+
                           sin(12*x)* sin (7*x)." }}{PARA 0 "" 0 "" 
{TEXT 260 9 "Solution:" }{TEXT -1 202 "  Without Maple V this would mi
ght be a rather challenging integration by parts problem.  Can you wor
k it that way?   You could also use a table of integrals.  The problem
 is easily worked with Maple V." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Int(sin(12*x)*sin(7*x),x) : % = val
ue(%)+C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&-%$sinG6#,$%\"
xG\"#7\"\"\"-F)6#,$F,\"\"(F.F,,(-F)6#,$F,\"\"&#F.\"#5-F)6#,$F,\"#>#!\"
\"\"#Q%\"CGF." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Now that  was ea
sy,  " }{TEXT 258 44 "but how do we know that the answer is correc" }
{TEXT -1 216 "t?  This is an example in which verifying the correctnes
s of the answer is more difficult than the actual integration.  In ord
er to show the integral is correct differentiate the right-hand side o
f the last equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 15 "diff(rhs(%),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#,&-%$cosG6#,$%\"xG\"\"&#\"\"\"\"\"#-F%6#,$F(\"#>#!\"\"F," }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 190 " Does this look like the integran
d in the original problem?  You can verify that it is by using trig id
entities.   To see that the integrand is equal to the preceding Maple \+
V output, use the" }}{PARA 0 "" 0 "" {TEXT -1 41 "\"combine\" command \+
with the \"trig\" option:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 33 "combine(sin(12*x)*sin(7*x),trig);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,&-%$cosG6#,$%\"xG\"\"&#\"\"\"\"\"#-F%6#,$F(
\"#>#!\"\"F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "This verifies tha
t the value of the integral is correct." }}{PARA 0 "" 0 "" {TEXT 256 
9 "Example 2" }{TEXT -1 16 ".  Evaluate the " }{XPPEDIT 18 0 "int(x^10
*cos(7*x),x)" "6#-%$intG6$*&%\"xG\"#5-%$cosG6#*&\"\"(\"\"\"F'F.F.F'" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 9 "Solution:" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Int(x^10*cos(7
*x),x): %=value(%)+C;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "This tim
e despite the complicated answer the problem is answer is" }}{PARA 0 "
" 0 "" {TEXT -1 15 "easily checked." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "diff(rhs(%),x);" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 9 "Example 3" }
{TEXT -1 12 ".  Evaluate " }{XPPEDIT 18 0 "int((x^2+3*x-2)/(x^2+6*x+14
),x" "6#-%$intG6$*&,(*$%\"xG\"\"#\"\"\"*&\"\"$F+F)F+F+\"\"#!\"\"F+,(*$
F)\"\"#F+*&\"\"'F+F)F+F+\"#9F+F/F)" }}{PARA 0 "" 0 "" {TEXT -1 190 "So
lution:  This is the kind of problem that you would have to use partia
l fractions to work by hand.  We can use Maple V to evaluate this this
 integral immediately,  but first we will expand" }}{PARA 0 "" 0 "" 
{TEXT -1 142 "the integrand into partial fractions and then integrate.
  Then we will use Maple V to get the same answer by integrating the p
roblem directly." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 50 "f:= x -> (x^2+3*x-2)/((x+1)*(x+2)^2*(x^2+6*x+14));" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%\"xG6\"6$%)operatorG%&arrow
GF(**,(*$9$\"\"#\"\"\"F/\"\"$!\"#F1F1,&F/F1F1F1!\"\",&F/F1F0F1F3,(F.F1
F/\"\"'\"#9F1F5F(F(6\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "Use the
 \"convert\" command with the \"parfrac\" option to expand the express
ion by partial fractions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 29 "g := convert(f(x),parfrac,x);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"gG,**$,&%\"xG\"\"\"F)F)!\"\"#!\"%\"\"**$,&F(
F)\"\"#F)!\"##F0\"\"$*$F/F*#\"#6\"#=*&,&\"#;F)F(F3F),(*$F(F0F)F(\"\"'
\"#9F)F*#F*F7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 293 "The idea of par
tial fractions is to reduce the rather complicated rational function i
nto a sum of fractions that are easily integrated.  Can you integrate \+
each expression in the preceding sum?  Remember that you should be abl
e to evaluate each of the integrals in the above expression by hand." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "I
1 := int(g,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#I1G,,-%#lnG6#,&%
\"xG\"\"\"F+F+#!\"%\"\"**$,&F*F+\"\"#F+!\"\"#!\"#\"\"$-F'6#F0#\"#6\"#=
-F'6#,(*$F*F1F+F*\"\"'\"#9F+#F2\"#7*&\"\"&#F+F1-%'arctanG6#,$*&,&F*F1F
?F+F+FDFE#F+\"#5F+#!\"(\"#!*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "N
ow check the result." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 11 "diff(I1,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,
,*$,&%\"xG\"\"\"F'F'!\"\"#!\"%\"\"**$,&F&F'\"\"#F'!\"##F.\"\"$*$F-F(#
\"#6\"#=*&,&F&F.\"\"'F'F',(*$F&F.F'F&F8\"#9F'F(#F(\"#7*$,&F'F'*$F7F.#F
'\"#?F(#!\"(\"#!*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Upon simplif
ying by using \"normal\"  we arrive at the integrand." }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "normal(%);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#**,(*$%\"xG\"\"#\"\"\"F&\"\"$!\"#F(F(,
&F&F(F(F(!\"\",&F&F(F'F(F*,(F%F(F&\"\"'\"#9F(F," }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 92 "If all one wants is the answer then one can work the
 problem in a single step using Maple V." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Int(f(x),x):% = value(%) + C
;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$**,(*$%\"xG\"\"#\"\"\"F
*\"\"$!\"#F,F,,&F*F,F,F,!\"\",&F*F,F+F,F.,(F)F,F*\"\"'\"#9F,F0F*,.-%#l
nG6#F/#!\"%\"\"**$F1F0#F.F--F76#F1#\"#6\"#=-F76#F2#F0\"#7*&\"\"&#F,F+-
%'arctanG6#,$*&,&F*F+F3F,F,FHFI#F,\"#5F,#!\"(\"#!*%\"CGF," }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 109 "Sometimes Maple V gives an answer that i
sn't acceptable,  and you can remedy this with the \"assume\"  command
." }}{PARA 0 "" 0 "" {TEXT -1 74 "Example 4.  Verify the following for
mula which comes from integral tables." }}{PARA 0 "" 0 "" {XPPEDIT 18 
0 "int(1/sqrt(a^2-x^2),x)" "6#-%$intG6$*&\"\"\"\"\"\"-%%sqrtG6#,&*$%\"
aG\"\"#F(*$%\"xG\"\"#!\"\"F3F1" }{TEXT -1 1 " " }{TEXT 257 19 "=  arcs
in (x/a) + C" }{TEXT -1 10 ",   where " }{XPPEDIT 18 0 "a<>0 " "6#0%\"
aG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 105 "Solution:  In
 this case, if you do the obvious,  you get a rather strange answer,  \+
which is unacceptable." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Int(1/sqr
t(a^2-x^2),x): %=value(%) +C;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 262 
"What went wrong here?  Maple V often gives answers that make little s
ense to us whenever the problem involves square roots of numbers which
 may  or may not be negative such as,  in this case,  a^2 - x^2.  If y
ou imform Maple V that you want to assume that  a>0, " }}{PARA 0 "" 0 
"" {TEXT -1 31 "then use the \"assume\"  command." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "assume(a>0);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Now try the same command as before
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
38 "Int(1/sqrt(a^2-x^2),x): %=value(%) +C;" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 75 "This time value of the integral is the same one as the on
e given in tables." }}}}{MARK "13 1 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }