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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 257 10 "Math 1322 " }}{PARA 18 "
" 0 "" {TEXT 256 49 "Techniques of Integration - III\nPartial Fraction
s" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 15 "1. Introduction" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "A function of the
 form\n\n" }{TEXT 267 29 "                             " }{XPPEDIT 
268 1 "P(x)/Q(x);" "6#*&-%\"PG6#%\"xG\"\"\"-%\"QG6#F'!\"\"" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 13 "is called a  " }{TEXT 385 17 "rational function" }
{TEXT -1 12 "  if both   " }{XPPEDIT 381 1 "P(x);" "6#-%\"PG6#%\"xG" }
{TEXT 382 1 " " }{TEXT -1 5 " and " }{TEXT 379 1 " " }{XPPEDIT 383 1 "
Q(x);" "6#-%\"QG6#%\"xG" }{TEXT 384 1 " " }{TEXT -1 133 " are polynomi
als.  If the degree of the numerator is strictly less than that of the
 denominator we will refer to the function as a   " }{TEXT 386 24 "pro
per rational function" }{TEXT -1 434 ".  By division we can write any \+
rational function as the sum of a polynomial and a proper rational fun
ction.\n\nThis worksheet concerns a method, the Method of Partial Frac
tions, that converts a proper rational function into a sum of several \+
simpler rational functions. The method is purely algebraic - there is \+
no calculus involved. But once converted into its partial fraction dec
omposition, a rational function is easy to integrate. " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 46 "2. Partial \+
Fractions - Distinct Linear Factors" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 13 "Suppose that " }{XPPEDIT 387 1 "Q(x) = (
x-r[1])*(x-r[2])*`...`*(x-r[N]);" "6#/-%\"QG6#%\"xG**,&F'\"\"\"&%\"rG6
#\"\"\"!\"\"F*,&F'F*&F,6#\"\"#F/F*%$...GF*,&F'F*&F,6#%\"NGF/F*" }
{TEXT 388 2 "  " }{TEXT -1 15 "where the roots" }{TEXT 390 1 " " }
{XPPEDIT 391 1 "r[1],r[2],`...`,r[N];" "6&&%\"rG6#\"\"\"&F$6#\"\"#%$..
.G&F$6#%\"NG" }{TEXT -1 5 "  of " }{TEXT 392 1 " " }{XPPEDIT 393 1 "Q(
x);" "6#-%\"QG6#%\"xG" }{TEXT -1 18 " are distinct. If " }{XPPEDIT 
394 1 "deg(P(x)) < deg(Q(x));" "6#2-%$degG6#-%\"PG6#%\"xG-F%6#-%\"QG6#
F*" }{TEXT 395 1 " " }{TEXT -1 66 "then a theorem of algebra asserts t
hat there are unique constants " }{XPPEDIT 396 1 "A[1],A[2],`...`,A[N]
;" "6&&%\"AG6#\"\"\"&F$6#\"\"#%$...G&F$6#%\"NG" }{TEXT 397 2 "  " }
{TEXT -1 9 "such that" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 7 "       " }{TEXT 398 13 "             " }{XPPEDIT 399 1 
"P(x)/Q(x) = A[1]/(x-r[1])+A[2]/(x-r[2])+`...`+A[N]/(x-r[N]);" "6#/*&-
%\"PG6#%\"xG\"\"\"-%\"QG6#F(!\"\",**&&%\"AG6#\"\"\"F),&F(F)&%\"rG6#\"
\"\"F-F-F)*&&F16#\"\"#F),&F(F)&F66#\"\"#F-F-F)%$...GF)*&&F16#%\"NGF),&
F(F)&F66#FEF-F-F)" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 402 8 "Exa
mple:" }{TEXT -1 17 "  Evaluate\n\n     " }{TEXT 400 13 "             \+
" }{XPPEDIT 401 1 "Int((2*x^5-x^4-33*x^3-65*x^2-32*x-9)/(x^3-2*x^2-13*
x-10),x);" "6#-%$IntG6$*&,.*&\"\"#\"\"\"*$)%\"xG\"\"&F*F*F**$)F-\"\"%F
*!\"\"*&\"#LF**$)F-\"\"$F*F*F2*&\"#lF**$)F-\"\"#F*F*F2*&\"#KF*F-F*F2\"
\"*F2F*,**$)F-\"\"$F*F**&\"\"#F**$)F-\"\"#F*F*F2*&\"#8F*F-F*F2\"#5F2F2
F-" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 403 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT -1 
178 "The integrand is a rational function. It is not, however, proper.
 We must first divide to reduce it to a proper rational function. The \+
Maple function to use for this purpose is  " }{HYPERLNK 17 "quo" 2 "qu
o" "" }{TEXT -1 2 ".\n" }}{PARA 3 "" 0 "" {TEXT -1 3 "quo" }{TEXT 30 
26 " - quotient of polynomials" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 4 "" 0 "usage" {TEXT -1 17 "Calling Sequence:" }}{PARA 0 "" 0 "
" {TEXT -1 17 "     quo(a, b, x)" }}{PARA 0 "" 0 "" {TEXT -1 22 "     \+
quo(a, b, x, 'r')" }}{PARA 4 "" 0 "" {TEXT -1 11 "Parameters:" }}
{PARA 0 "" 0 "" {TEXT -1 5 "     " }{TEXT 23 8 "a, b  - " }{TEXT -1 
17 "polynomials in x " }}{PARA 0 "" 0 "" {TEXT -1 5 "     " }{TEXT 23 
8 "x     - " }{TEXT -1 7 "a name " }}{PARA 0 "" 0 "" {TEXT -1 3 "   " 
}{TEXT 23 9 "'r'    - " }{TEXT -1 28 "(optional) unevaluated names" }}
{PARA 4 "" 0 "" {TEXT -1 13 "\nDescription:" }}{PARA 15 "" 0 "" {TEXT 
-1 144 "The quo function returns the quotient q of a divided by b. The
 remainder r and quotient q satisfy: a = b*q + r where degree(r,x) < d
egree(b,x). " }}{PARA 15 "" 0 "" {TEXT -1 75 "If a fourth argument is \+
passed to quo it will be assigned the remainder r. " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 65 "J := Int((2*x^5-x^4-33*x^3-65*x^2-32*x-9)/(x^3-2
*x^2-13*x-10),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%$IntG6$*&
,.*$)%\"xG\"\"&\"\"\"\"\"#*$)F,\"\"%F.!\"\"*$)F,\"\"$F.!#L*$)F,F/F.!#l
F,!#K!\"*\"\"\"F.,*F4F=F8!\"#F,!#8!#5F=!\"\"F," }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 27 "f := student[integrand](J);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"fG*&,.*$)%\"xG\"\"&\"\"\"\"\"#*$)F)\"\"%F+!\"\"*$
)F)\"\"$F+!#L*$)F)F,F+!#lF)!#K!\"*\"\"\"F+,*F1F:F5!\"#F)!#8!#5F:!\"\"
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "p := numer(f):\nq := de
nom(f):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "g := quo(p,q,x,'
r');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG,(*$)%\"xG\"\"#\"\"\"F)F
(\"\"$!\"\"\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 12 "We now have " }{TEXT 405 1 " " }{XPPEDIT 406 1 "f = g+r/q
;" "6#/%\"fG,&%\"gG\"\"\"*&%\"rGF'%\"qG!\"\"F'" }{TEXT -1 8 "  where \+
" }{TEXT 407 1 " " }{XPPEDIT 408 1 "r/q;" "6#*&%\"rG\"\"\"%\"qG!\"\"" 
}{TEXT -1 63 "   is a proper rational function. Let us verify this sta
tement." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "eqn := f = g + r/q;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/*&,.*$)%\"xG\"\"&\"\"\"\"\"#*$
)F*\"\"%F,!\"\"*$)F*\"\"$F,!#L*$)F*F-F,!#lF*!#K!\"*\"\"\"F,,*F2F;F6!\"
#F*!#8!#5F;!\"\",*F6F-F*F4F1F;*&,(!#>F;F6!\")F*!#:F,F<F@F;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "testeq(eqn);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 76 "The next step is to factor the denominator of the pr
oper rational function.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 
"factor(q);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,&%\"xG\"\"\"F&F&F&,&
F%F&\"\"#F&F&,&F%F&!\"&F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 42 "According to the quoted theorem on partial" }
{TEXT 409 1 " " }{TEXT 410 24 "fraction decompositions," }{TEXT -1 22 
"  there are constants " }{TEXT 413 1 "A" }{TEXT -1 2 ", " }{TEXT 414 
2 " B" }{TEXT -1 6 ", and " }{TEXT 422 3 " C " }{TEXT -1 9 "such that
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "     \+
            " }{TEXT 411 6 "      " }{XPPEDIT 412 1 "r/q = A/(x+1)+B/(
x+2)+C/(x-5);" "6#/*&%\"rG\"\"\"%\"qG!\"\",(*&%\"AGF&,&%\"xGF&\"\"\"F&
F(F&*&%\"BGF&,&F-F&\"\"#F&F(F&*&%\"CGF&,&F-F&\"\"&F(F(F&" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "for every" }{TEXT 
423 1 " " }{XPPEDIT 424 1 "x;" "6#%\"xG" }{TEXT -1 42 ".  Let us solve
 for these three constants." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
57 "solve(identity(r/q = A/(x+1)+B/(x+2)+C/(x-5),x),\{A,B,C\});" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%\"BG!\"$/%\"AG\"\"#/%\"CG!\"(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "eqn2 := subs(\{B = -3, A = 2
, C = -7\},r/q = A/(x+1)+B/(x+2)+C/(x-5));" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/*&,(!#>
\"\"\"*$)%\"xG\"\"#\"\"\"!\")F,!#:F.,**$)F,\"\"$F.F)F*!\"#F,!#8!#5F)!
\"\",(*&F.F.,&F,F)F)F)F8F-*&F.F.,&F,F)F-F)F8!\"$*&F.F.,&F,F)!\"&F)F8!
\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "testeq(eqn2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "Fi
nally, we piece together all summands of the original integrand:" }}
{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 23 "J = Int(g+rhs(eqn2),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$I
ntG6$*&,.*$)%\"xG\"\"&\"\"\"\"\"#*$)F+\"\"%F-!\"\"*$)F+\"\"$F-!#L*$)F+
F.F-!#lF+!#K!\"*\"\"\"F-,*F3F<F7!\"#F+!#8!#5F<!\"\"F+-F%6$,.F7F.F+F5F2
F<*&F-F-,&F+F<F<F<FAF.*&F-F-,&F+F<F.F<FA!\"$*&F-F-,&F+F<!\"&F<FA!\"(F+
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "The \+
resulting integral is elementary! We evaluate it with the " }
{HYPERLNK 17 "value" 2 "value" "" }{TEXT -1 10 "  command." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 23 "lhs(%) = value(rhs(%));" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#/-%$IntG6$*&,.*$)%\"xG\"\"&\"\"\"\"\"#*$)F+\"\"%F-!
\"\"*$)F+\"\"$F-!#L*$)F+F.F-!#lF+!#K!\"*\"\"\"F-,*F3F<F7!\"#F+!#8!#5F<
!\"\"F+,.F3#F.F5F7#F5F.F+F2-%#lnG6#,&F+F<F<F<F.-FF6#,&F+F<F.F<!\"$-FF6
#,&F+F<!\"&F<!\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 120 "As a check we will ask Maple to differentiate the right \+
side of this equation and compare the result with the integrand:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
29 "testeq( f = diff(rhs(%),x) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%
%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" 
{TEXT -1 46 "3. Partial Fractions - Repeated Linear Factors" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Suppose that" }
}{PARA 0 "" 0 "" {TEXT -1 7 "       " }{TEXT 451 7 "       " }
{XPPEDIT 256 1 "Q(x) = (x-r[1])^m[1]*(x-r[2])^m[2]*`...`*(x-r[N])^m[N]
;" "6#/-%\"QG6#%\"xG**),&F'\"\"\"&%\"rG6#\"\"\"!\"\"&%\"mG6#\"\"\"F+),
&F'F+&F-6#\"\"#F0&F26#\"\"#F+%$...GF+),&F'F+&F-6#%\"NGF0&F26#FBF+" }
{TEXT 428 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 15 "where the roots" }{TEXT 429 1 " " }{XPPEDIT 259 1 "r[1],r
[2],`...`,r[N];" "6&&%\"rG6#\"\"\"&F$6#\"\"#%$...G&F$6#%\"NG" }{TEXT 
-1 28 "  are the distinct roots of " }{TEXT 431 1 " " }{XPPEDIT 261 1 
"Q(x);" "6#-%\"QG6#%\"xG" }{TEXT -1 6 ". If  " }{XPPEDIT 262 1 "deg(P(
x)) < deg(Q(x));" "6#2-%$degG6#-%\"PG6#%\"xG-F%6#-%\"QG6#F*" }{TEXT 
434 2 "  " }{TEXT -1 65 "then a theorem of algebra asserts that there \+
are unique constants" }{TEXT 452 1 " " }{XPPEDIT 264 1 "R[1],R[2],`...
`,R[N];" "6&&%\"RG6#\"\"\"&F$6#\"\"#%$...G&F$6#%\"NG" }{TEXT 436 2 "  \+
" }{TEXT -1 9 "such that" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 7 "       " }{TEXT 437 8 "        " }{TEXT 453 5 "    \+
 " }{XPPEDIT 267 1 "P(x)/Q(x) = R[1]+R[2]+`...`+R[N];" "6#/*&-%\"PG6#%
\"xG\"\"\"-%\"QG6#F(!\"\",*&%\"RG6#\"\"\"F)&F06#\"\"#F)%$...GF)&F06#%
\"NGF)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "
where" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 " \+
  " }{TEXT 454 2 "  " }{XPPEDIT 455 1 "R[j] = A[j,1]/(x-r[j])+A[j,2]/(
(x-r[j])^2)+`...`+A[j,m]/((x-r[j])^m[j]);" "6#/&%\"RG6#%\"jG,**&&%\"AG
6$F'\"\"\"\"\"\",&%\"xGF.&%\"rG6#F'!\"\"F4F.*&&F+6$F'\"\"#F.*$),&F0F.&
F26#F'F4\"\"#F.F4F.%$...GF.*&&%\"AG6$%\"jG%\"mG\"\"\"),&%\"xGFF&%\"rG6
#FD!\"\"&FE6#FDFM\"\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 456 8 "Example:" }{TEXT -1 
23 "  Evaluate\n\n           " }{XPPEDIT 19 1 "Int((3*x^7+24*x^6+71*x^
5+83*x^4-99*x^2-76*x-8*x^3-19)/(x^5+8*x^4+25*x^3+38*x^2+28*x+8),x);" "
6#-%$IntG6$*&,2*&\"\"$\"\"\"*$)%\"xG\"\"(F*F*F**&\"#CF**$)F-\"\"'F*F*F
**&\"#rF**$)F-\"\"&F*F*F**&\"#$)F**$)F-\"\"%F*F*F**&\"#**F**$)F-\"\"#F
*F*!\"\"*&\"#wF*F-F*FC*&\"\")F**$)F-\"\"$F*F*FC\"#>FCF*,.*$)F-\"\"&F*F
**&\"\")F**$)F-\"\"%F*F*F**&\"#DF**$)F-\"\"$F*F*F**&\"#QF**$)F-\"\"#F*
F*F**&\"#GF*F-F*F*\"\")F*FCF-" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 457 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 95 "J := Int((3*x^7+24*x^6+71*x^5+83*x^4-99*x^2-76*x-8*x^3-19)/(x^
5+8*x^4+25*x^3+38*x^2+28*x+8),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%\"JG-%$IntG6$*&,2!#>\"\"\"%\"xG!#w*$)F,\"\"#\"\"\"!#***$)F,\"\"$F1!\"
)*$)F,\"\"%F1\"#$)*$)F,\"\"&F1\"#r*$)F,\"\"'F1\"#C*$)F,\"\"(F1F5F1,.F;
F+F7\"\")F3\"#DF.\"#QF,\"#GFGF+!\"\"F," }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "f := student[integrand](J);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"fG*&,2!#>\"\"\"%\"xG!#w*$)F)\"\"#\"\"\"!#***$)F)\"
\"$F.!\")*$)F)\"\"%F.\"#$)*$)F)\"\"&F.\"#r*$)F)\"\"'F.\"#C*$)F)\"\"(F.
F2F.,.F8F(F4\"\")F0\"#DF+\"#QF)\"#GFDF(!\"\"" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 29 "p := numer(f):\nq := denom(f):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 20 "g := quo(p,q,x,'r');" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"gG,&*$)%\"xG\"\"#\"\"\"\"\"$!\"%\"\"\"" }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "We now have " }
{TEXT 458 1 " " }{XPPEDIT 257 1 "f = g+r/q;" "6#/%\"fG,&%\"gG\"\"\"*&%
\"rGF'%\"qG!\"\"F'" }{TEXT -1 8 "  where " }{TEXT 460 1 " " }{XPPEDIT 
259 1 "r/q;" "6#*&%\"rG\"\"\"%\"qG!\"\"" }{TEXT -1 63 "   is a proper \+
rational function. Let us verify this statement." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 19 "eqn := f = g + r/q;" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#>%$eqnG/*&,2!#>\"\"\"%\"xG!#w*$)F*\"\"#\"\"\"!#***$)F*\"\"$F/!\"
)*$)F*\"\"%F/\"#$)*$)F*\"\"&F/\"#r*$)F*\"\"'F/\"#C*$)F*\"\"(F/F3F/,.F9
F)F5\"\")F1\"#DF,\"#QF*\"#GFEF)!\"\",(F,F3!\"%F)*&,,\"#8F)F*\"#OF,\"#H
F1FEF5F)F/FDFIF)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "testeq(
eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "The next step is to facto
r the denominator of the proper rational function.\n" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 10 "factor(q);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#*&),&%\"xG\"\"\"F'F'\"\"#\"\"\"),&F&F'F(F'\"\"$F)" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "According to the q
uoted theorem on partial" }{TEXT 462 1 " " }{TEXT 463 24 "fraction dec
ompositions," }{TEXT -1 22 "  there are constants " }{TEXT 466 14 "A, \+
B, C, d, E " }{TEXT -1 9 "such that" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 17 "                 " }{TEXT 464 6 "      \+
" }{XPPEDIT 263 1 "r/q = A/(x+1)+B/((x+1)^2)+C/(x+2)+d/((x+2)^2)+E/((x
+2)^3);" "6#/*&%\"rG\"\"\"%\"qG!\"\",,*&%\"AG\"\"\",&%\"xGF,\"\"\"F,!
\"\"F,*&%\"BGF,*$),&F.F,\"\"\"F,\"\"#F,F0F,*&%\"CGF,,&F.F,\"\"#F,F0F,*
&%\"dG\"\"\"*$),&F.F,\"\"#F,\"\"#F,F0F,*&%\"EGF,*$),&F.F,\"\"#F,\"\"$F
,F0F," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "f
or every" }{TEXT 469 1 " " }{XPPEDIT 268 1 "x;" "6#%\"xG" }{TEXT -1 
43 ".  Let us solve for these five constants.\n\n" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 67 "ident := r/q = A/(x+1)+B/((x+1)^2)+C/(x+2)+d/
((x+2)^2)+E/((x+2)^3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&identG/*&
,,\"#8\"\"\"%\"xG\"#O*$)F*\"\"#\"\"\"\"#H*$)F*\"\"$F/\"\")*$)F*\"\"%F/
F)F/,.*$)F*\"\"&F/F)F5F4F1\"#DF,\"#QF*\"#GF4F)!\"\",,*&%\"AGF/,&F*F)F)
F)F?F)*&%\"BGF/*$)FC\"\"#F/F?F)*&%\"CGF/,&F*F)F.F)F?F)*&%\"dGF/*$)FK\"
\"#F/F?F)*&%\"EGF/*$)FK\"\"$F/F?F)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "solve(identity(ident,x),\{A,B,C,d,E\});" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#<'/%\"AG\"\"\"/%\"BG!\"\"/%\"CG\"\"!/%\"dG\"\"#/
%\"EG\"\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "eqn2 := subs(
\{A = 1, B = -1, C = 0, d = 2, E = 9\},ident);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%eqn2G/*&,,\"#8\"\"\"%\"xG\"#O*$)F*\"\"#\"\"\"\"#H*$)
F*\"\"$F/\"\")*$)F*\"\"%F/F)F/,.*$)F*\"\"&F/F)F5F4F1\"#DF,\"#QF*\"#GF4
F)!\"\",**&F/F/,&F*F)F)F)F?F)*&F/F/*$)FB\"\"#F/F?!\"\"*&F/F/*$),&F*F)F
.F)\"\"#F/F?F.*&F/F/*$)FK\"\"$F/F?\"\"*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "testeq(eqn2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%tr
ueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 66 "Finally, we piece together all summands
 of the original integrand:" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "J = Int(g+rhs(eqn2),x);" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&,2!#>\"\"\"%\"xG!#w*$)F+\"
\"#\"\"\"!#***$)F+\"\"$F0!\")*$)F+\"\"%F0\"#$)*$)F+\"\"&F0\"#r*$)F+\"
\"'F0\"#C*$)F+\"\"(F0F4F0,.F:F*F6\"\")F2\"#DF-\"#QF+\"#GFFF*!\"\"F+-F%
6$,.F-F4!\"%F**&F0F0,&F+F*F*F*FJF**&F0F0*$)FP\"\"#F0FJ!\"\"*&F0F0*$),&
F+F*F/F*\"\"#F0FJF/*&F0F0*$)FY\"\"$F0FJ\"\"*F+" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "The resulting integral is
 elementary! We evaluate it with the " }{HYPERLNK 17 "value" 2 "value
" "" }{TEXT -1 10 "  command." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
23 "lhs(%) = value(rhs(%));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%$Int
G6$*&,2!#>\"\"\"%\"xG!#w*$)F+\"\"#\"\"\"!#***$)F+\"\"$F0!\")*$)F+\"\"%
F0\"#$)*$)F+\"\"&F0\"#r*$)F+\"\"'F0\"#C*$)F+\"\"(F0F4F0,.F:F*F6\"\")F2
\"#DF-\"#QF+\"#GFFF*!\"\"F+,.F2F*F+!\"%-%#lnG6#,&F+F*F*F*F**&F0F0FPFJF
**&F0F0,&F+F*F/F*FJ!\"#*&F0F0*$)FS\"\"#F0FJ#!\"*F/" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 120 "As a check we will ask
 Maple to differentiate the right side of this equation and compare th
e result with the integrand:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "testeq( f = diff(rhs(%),x) )
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 49 "4. Partial Fract
ions - Distinct Quadratic Factors" }}{PARA 0 "" 0 "" {TEXT -1 180 "Whe
n an irreducible quadratic polynomial is a factor of the denominator a
 linear term  is called for in the numerator of the partial fraction s
ummand.  The example will illustrate." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT 477 8 "Example:" }{TEXT -1 51 "  Evaluate\n
\n                                       " }{XPPEDIT 19 1 "Int((7*x^3+
6*x^2+22*x+10)/(x^4+2*x^3+6*x^2+8*x+8),x);" "6#-%$IntG6$*&,**&\"\"(\"
\"\"*$)%\"xG\"\"$F*F*F**&\"\"'F**$)F-\"\"#F*F*F**&\"#AF*F-F*F*\"#5F*\"
\"\",,*$)%\"xG\"\"%\"\"\"F=*&\"\"#F=*$)F;\"\"$F=F=F=*&\"\"'F=*$)F;\"\"
#F=F=F=*&\"\")F=F;F=F=\"\")F=!\"\"%\"xG" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT 478 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 58 "J := Int((7*x^3+6*x^2+22*x+10)/(x^4+2*x^3+6*x^2+8*x+8
),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%$IntG6$*&,**$)%\"xG\"
\"$\"\"\"\"\"(F,\"#A*$)F,\"\"#F.\"\"'\"#5\"\"\"F.,,*$)F,\"\"%F.F6F1F4F
*F3F,\"\")F;F6!\"\"F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "f \+
:= student[integrand](J);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*&,
**$)%\"xG\"\"$\"\"\"\"\"(F)\"#A*$)F)\"\"#F+\"\"'\"#5\"\"\"F+,,*$)F)\"
\"%F+F3F.F1F'F0F)\"\")F8F3!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 29 "p := numer(f):\nq := denom(f):" }}}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 140 "The integrand is already a proper r
ational function. We can proceed directly to factoring the denominator
 of this proper rational function.\n" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "factor(q);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,(*$)
%\"xG\"\"#\"\"\"\"\"\"F'F(F(F*F*,&F%F*\"\"%F*F*" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "According to the quoted t
heorem on partial" }{TEXT 481 1 " " }{TEXT 482 24 "fraction decomposit
ions," }{TEXT -1 22 "  there are constants " }{TEXT 484 14 "A, B, C, d
, E " }{TEXT -1 9 "such that" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 13 "             " }{TEXT 486 4 "    " }{TEXT 483 
6 "      " }{XPPEDIT 258 1 "p/q = (A*x+B)/(x^2+2*x+2)+(C*x+d)/(x^2+4);
" "6#/*&%\"pG\"\"\"%\"qG!\"\",&*&,&*&%\"AGF&%\"xGF&F&%\"BGF&F&,(*$)F.
\"\"#F&F&*&\"\"#F&F.F&F&\"\"#F&F(F&*&,&*&%\"CGF&F.F&F&%\"dGF&F&,&*$)F.
\"\"#F&F&\"\"%F&F(F&" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 9 "for every" }{TEXT 485 1 " " }{XPPEDIT 259 1 "x;" "6#%\"x
G" }{TEXT -1 43 ".  Let us solve for these five constants.\n\n" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "ident := p/q = (A*x+B)/(x^2+
2*x+2)+(C*x+d)/(x^2+4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&identG/*
&,**$)%\"xG\"\"$\"\"\"\"\"(F*\"#A*$)F*\"\"#F,\"\"'\"#5\"\"\"F,,,*$)F*
\"\"%F,F4F/F2F(F1F*\"\")F9F4!\"\",&*&,&*&%\"AGF4F*F4F4%\"BGF4F,,(F/F4F
*F1F1F4F:F4*&,&*&%\"CGF4F*F,F4%\"dGF4F,,&F/F4F8F4F:F4" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "solve(identity(ident,x),\{A,B,C,d\}
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<&/%\"AG\"\"&/%\"BG\"\"$/%\"dG!
\"\"/%\"CG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "eqn2 := \+
subs(\{A = 5, B = 3, d = -1, C = 2\},ident);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%eqn2G/*&,**$)%\"xG\"\"$\"\"\"\"\"(F*\"#A*$)F*\"\"#F,
\"\"'\"#5\"\"\"F,,,*$)F*\"\"%F,F4F/F2F(F1F*\"\")F9F4!\"\",&*&,&F*\"\"&
F+F4F,,(F/F4F*F1F1F4F:F4*&,&F*F1!\"\"F4F,,&F/F4F8F4F:F4" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "testeq(eqn2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "Finally, we piece to
gether all summands of the original integrand:" }}{PARA 0 "" 0 "" 
{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "J = Int(rh
s(eqn2),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&,**$)%\"xG
\"\"$\"\"\"\"\"(F+\"#A*$)F+\"\"#F-\"\"'\"#5\"\"\"F-,,*$)F+\"\"%F-F5F0F
3F)F2F+\"\")F:F5!\"\"F+-F%6$,&*&,&F+\"\"&F,F5F-,(F0F5F+F2F2F5F;F5*&,&F
+F2!\"\"F5F-,&F0F5F9F5F;F5F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 256 "The resulting integrals on the right are
 not difficult but they do require some work if evaluated by hand. Sin
ce the focus of this worksheet is partial fraction decompositions, we \+
will bypass the aditional steps and call for the final answer by using
 the  " }{HYPERLNK 17 "value" 2 "value" "" }{TEXT -1 10 "  command." }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "lhs(%) = value(rhs(%));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&,**$)%\"xG\"\"$\"\"\"\"\"(
F+\"#A*$)F+\"\"#F-\"\"'\"#5\"\"\"F-,,*$)F+\"\"%F-F5F0F3F)F2F+\"\")F:F5
!\"\"F+,*-%#lnG6#,(F0F5F+F2F2F5#\"\"&F2-%'arctanG6#,&F+F5F5F5!\"#-F>6#
,&F0F5F9F5F5-FD6#,$F+#F5F2#!\"\"F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 120 "As a check we will ask Maple to differe
ntiate the right side of this equation and compare the result with the
 integrand:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 29 "testeq( f = diff(rhs(%),x) );" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 49 "5. Partial Fractions - Repeated Quadratic
 Factors" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
46 "When an irreducible quadratic polynomial is a " }{TEXT 502 8 "repe
ated" }{TEXT -1 123 " factor of the denominator we must use a combinat
ion of the previously stated principles. The next example will illustr
ate." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 494 8 "E
xample:" }{TEXT -1 51 "  Evaluate\n\n                                 \+
      " }{XPPEDIT 19 1 "Int((2*x^7+7*x^6+3*x^5+25*x^4+2*x^3+30*x^2-x+7
)/((x^2+1)^3*x^2),x);" "6#-%$IntG6$*&,2*&\"\"#\"\"\"*$)%\"xG\"\"(F*F*F
**&\"\"(F**$)F-\"\"'F*F*F**&\"\"$F**$)F-\"\"&F*F*F**&\"#DF**$)F-\"\"%F
*F*F**&\"\"#F**$)F-\"\"$F*F*F**&\"#IF**$)F-\"\"#F*F*F*F-!\"\"\"\"(F*F*
*&),&*$)F-\"\"#F*F*\"\"\"F*\"\"$F*)F-\"\"#F*FH%\"xG" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 495 9 "Solution:" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 72 "J := Int((2*x^7+7*x^6+3*x^5+25*x^4+2*x^3+30
*x^2-x+7)/((x^2+1)^3*x^2),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"J
G-%$IntG6$*&,2*$)%\"xG\"\"(\"\"\"\"\"#*$)F,\"\"'F.F-*$)F,\"\"&F.\"\"$*
$)F,\"\"%F.\"#D*$)F,F6F.F/*$)F,F/F.\"#IF,!\"\"F-\"\"\"F.*&),&F=FAFAFA
\"\"$F.)F,\"\"#F.!\"\"F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 
"f := student[integrand](J);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG
*&,2*$)%\"xG\"\"(\"\"\"\"\"#*$)F)\"\"'F+F**$)F)\"\"&F+\"\"$*$)F)\"\"%F
+\"#D*$)F)F3F+F,*$)F)F,F+\"#IF)!\"\"F*\"\"\"F+*&),&F:F>F>F>\"\"$F+)F)
\"\"#F+!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "p := numer(
f):\nq := denom(f):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eval
b(degree(p) < degree(q));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" 
}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 140 "The i
ntegrand is already a proper rational function. We can proceed directl
y to factoring the denominator of this proper rational function.\n" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(q);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#*&),&*$)%\"xG\"\"#\"\"\"\"\"\"F+F+\"\"$F*F'F*" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "It is imp
ortant to understand that  " }{XPPEDIT 19 1 "x^2;" "6#*$)%\"xG\"\"#\"
\"\"" }{TEXT -1 61 "   is a repeated linear term - not an irreducible \+
quadratic: " }{XPPEDIT 19 1 "x^2 = (x-0)*(x-0);" "6#/*$)%\"xG\"\"#\"\"
\"*&,&F&F(\"\"!!\"\"F(,&F&F(F+F,F(" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "
" {TEXT -1 26 "There are eight constants " }{TEXT 499 23 "A, B, C, d, \+
E, F, G, H " }{TEXT -1 9 "such that" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 13 "             " }{TEXT 501 4 "    " }
{TEXT 498 6 "      " }{XPPEDIT 256 1 "p/q = (A*x+B)/(x^2+1)+(C*x+d)/((
x^2+1)^2)+(E*x+F)/((x^2+1)^3)+G/x+H/(x^2);" "6#/*&%\"pG\"\"\"%\"qG!\"
\",,*&,&*&%\"AG\"\"\"%\"xGF.F.%\"BGF.F.,&*$)F/\"\"#F.F.\"\"\"F.!\"\"F.
*&,&*&%\"CGF.F/F.F.%\"dGF.F.*$),&*$)F/\"\"#F.F.\"\"\"F.\"\"#F.F6F.*&,&
*&%\"EGF.F/F.F.%\"FGF.F.*$),&*$)F/\"\"#F.F.\"\"\"F.\"\"$F.F6F.*&%\"GGF
.F/F6F.*&%\"HGF.*$)F/\"\"#F.F6F." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 9 "for every" }{TEXT 500 1 " " }{XPPEDIT 257 
1 "x;" "6#%\"xG" }{TEXT -1 44 ".  Let us solve for these eight constan
ts.\n\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "ident :=  p/q = (
A*x+B)/(x^2+1)+(C*x+d)/((x^2+1)^2)+(E*x+F)/((x^2+1)^3)+G/x+H/(x^2);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&identG/*&,2*$)%\"xG\"\"(\"\"\"\"\"
#*$)F*\"\"'F,F+*$)F*\"\"&F,\"\"$*$)F*\"\"%F,\"#D*$)F*F4F,F-*$)F*F-F,\"
#IF*!\"\"F+\"\"\"F,*&),&F;F?F?F?\"\"$F,)F*\"\"#F,!\"\",,*&,&*&%\"AGF?F
*F?F?%\"BGF?F,FBFFF?*&,&*&%\"CGF?F*F,F?%\"dGF?F,*$)FB\"\"#F,FFF?*&,&*&
%\"EGF?F*F,F?%\"FGF?F,*$)FB\"\"$F,FFF?*&%\"GGF,F*FFF?*&%\"HGF,*$)F*\"
\"#F,FFF?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "solve(identity
(ident,x),\{A,B,C,d,E,F,G,H\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<*/
%\"GG!\"\"/%\"HG\"\"(/%\"BG\"\"!/%\"AG\"\"$/%\"CGF,/%\"EG\"\"#/%\"dG\"
\"%/%\"FG\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "eqn2 := s
ubs(\{G = -1, H = 7, B = 0, A = 3, C = 0, E = 2, d = 4, F = 5\},ident)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/*&,2*$)%\"xG\"\"(\"\"\"
\"\"#*$)F*\"\"'F,F+*$)F*\"\"&F,\"\"$*$)F*\"\"%F,\"#D*$)F*F4F,F-*$)F*F-
F,\"#IF*!\"\"F+\"\"\"F,*&),&F;F?F?F?\"\"$F,)F*\"\"#F,!\"\",,*&F*F,FBFF
F4*&F,F,*$)FB\"\"#F,FFF7*&,&F*F-F3F?F,*$)FB\"\"$F,FFF?*&F,F,F*FFF>*&F,
F,*$)F*\"\"#F,FFF+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "teste
q(eqn2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 66 "Finally, we piece together all summands of the original i
ntegrand:" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 21 "J = Int(rhs(eqn2),x);" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#/-%$IntG6$*&,2*$)%\"xG\"\"(\"\"\"\"\"#*$)F+\"\"'F-F,*$)F+\"\"&F-
\"\"$*$)F+\"\"%F-\"#D*$)F+F5F-F.*$)F+F.F-\"#IF+!\"\"F,\"\"\"F-*&),&F<F
@F@F@\"\"$F-)F+\"\"#F-!\"\"F+-F%6$,,*&F+F-FCFGF5*&F-F-*$)FC\"\"#F-FGF8
*&,&F+F.F4F@F-*$)FC\"\"$F-FGF@*&F-F-F+FGF?*&F-F-*$)F+\"\"#F-FGF,F+" }}
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 256 "The res
ulting integrals on the right are not difficult but they do require so
me work if evaluated by hand. Since the focus of this worksheet is par
tial fraction decompositions, we will bypass the aditional steps and c
all for the final answer by using the  " }{HYPERLNK 17 "value" 2 "valu
e" "" }{TEXT -1 10 "  command." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
23 "lhs(%) = value(rhs(%));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%$Int
G6$*&,2*$)%\"xG\"\"(\"\"\"\"\"#*$)F+\"\"'F-F,*$)F+\"\"&F-\"\"$*$)F+\"
\"%F-\"#D*$)F+F5F-F.*$)F+F.F-\"#IF+!\"\"F,\"\"\"F-*&),&F<F@F@F@\"\"$F-
)F+\"\"#F-!\"\"F+,.-%#lnG6#FC#F5F.*&F+F-FCFG#\"#J\"\")-%'arctanG6#F+FN
*&,&F+\"#5!\"%F@F-*$)FC\"\"#F-FG#F@FP-FJFSF?*&F-F-F+FG!\"(" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 120 "As a check we \+
will ask Maple to differentiate the right side of this equation and co
mpare the result with the integrand:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "testeq( f = diff(rhs(%),x)
 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 34 "6. Converting to
 Partial Fractions" }}{PARA 0 "" 0 "" {TEXT -1 157 "Maple has a single
 command with which to obtain a partial fraction decomposition. It eve
n does the division to obtain a proper rational function. It is the  \+
" }{HYPERLNK 17 "convert/parfrac" 2 "convert,parfrac" "" }{TEXT -1 10 
"  command." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "usage" 
{TEXT -1 17 "Calling Sequence:" }}{PARA 0 "" 0 "" {TEXT -1 27 "     co
nvert(f, parfrac, x)" }}{PARA 4 "" 0 "" {TEXT -1 11 "Parameters:" }}
{PARA 0 "" 0 "" {TEXT -1 5 "     " }{TEXT 23 11 "f        - " }{TEXT 
-1 22 "rational function in x" }}{PARA 0 "" 0 "" {TEXT -1 5 "     " }
{TEXT 23 11 "x        - " }{TEXT -1 19 "main variable name " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "info" {TEXT -1 12 "Description
:" }}{PARA 15 "" 0 "" {TEXT -1 107 "Convert to parfrac performs a part
ial fraction decomposition of the rational function f in the variable \+
x. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 505 8 "Ex
ample:" }{TEXT -1 44 "  Find the partial fraction decomposition of" }}
{PARA 0 "" 0 "" {TEXT -1 9 "         " }}{PARA 0 "" 0 "" {TEXT -1 15 "
               " }{XPPEDIT 19 1 "(7*x^13+76*x^12+367*x^11+1025*x^10+18
40*x^9+2275*x^8+2063*x^7+1442*x^6+957*x^5+995*x^4+1062*x^3+673*x^2+304
*x+178)/(x^9+11*x^8+54*x^7+154*x^6+284*x^5+364*x^4+346*x^3+246*x^2+115
*x+25);" "6#*&,>*&\"\"(\"\"\"*$)%\"xG\"#8F'F'F'*&\"#wF'*$)F*\"#7F'F'F'
*&\"$n$F'*$)F*\"#6F'F'F'*&\"%D5F'*$)F*\"#5F'F'F'*&\"%S=F'*$)F*\"\"*F'F
'F'*&\"%vAF'*$)F*\"\")F'F'F'*&\"%j?F'*$)F*\"\"(F'F'F'*&\"%U9F'*$)F*\"
\"'F'F'F'*&\"$d*F'*$)F*\"\"&F'F'F'*&\"$&**F'*$)F*\"\"%F'F'F'*&\"%i5F'*
$)F*\"\"$F'F'F'*&\"$t'F'*$)F*\"\"#F'F'F'*&\"$/$F'F*F'F'\"$y\"F'F',6*$)
F*\"\"*F'F'*&\"#6F'*$)F*\"\")F'F'F'*&\"#aF'*$)F*\"\"(F'F'F'*&\"$a\"F'*
$)F*\"\"'F'F'F'*&\"$%GF'*$)F*\"\"&F'F'F'*&\"$k$F'*$)F*\"\"%F'F'F'*&\"$
Y$F'*$)F*\"\"$F'F'F'*&\"$Y#F'*$)F*\"\"#F'F'F'*&\"$:\"F'F*F'F'\"#DF'!\"
\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 506 9 "Sol
ution:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 
"What a mess!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 189 "f := (7*x^13+76*x^12+367*x^11+1025*x^10+1840*x^
9+2275*x^8+2063*x^7+1442*x^6+957*x^5+995*x^4+1062*x^3+673*x^2+304*x+17
8)/(x^9+11*x^8+54*x^7+154*x^6+284*x^5+364*x^4+346*x^3+246*x^2+115*x+25
)\"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"fG*&,>*$)%\"xG\"#8\"\"\"\"
\"(*$)F)\"#7F+\"#w*$)F)\"#6F+\"$n$*$)F)\"#5F+\"%D5*$)F)\"\"*F+\"%S=*$)
F)\"\")F+\"%vA*$)F)F,F+\"%j?*$)F)\"\"'F+\"%U9*$)F)\"\"&F+\"$d**$)F)\"
\"%F+\"$&***$)F)\"\"$F+\"%i5*$)F)\"\"#F+\"$t'F)\"$/$\"$y\"\"\"\"F+,6F9
FZF=F3FA\"#aFD\"$a\"FH\"$%GFL\"$k$FP\"$Y$FT\"$Y#F)\"$:\"\"#DFZ!\"\"" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "convert(f, parfrac, x);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#,4*$)%\"xG\"\"%\"\"\"\"\"(*$)F&\"\"$F
(!\"\"F&\"\"\"!\"&F.*&,&F&\"\"#F.F.F(,(*$)F&F2F(F.F&F'\"\"&F.!\"\"F.*&
,&F&F.!\"#F.F(*$)F3\"\"#F(F7F.*&F&F(,&F4F.F.F.F7\"\"'*&F(F(,&F&F.F.F.F
7F'*&F(F(*$)FB\"\"$F(F7\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 11 "Not so bad!" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 140 "It is worth pointing out that Map
le will use the Method of Partial Fractionsin computing integrals with
out specific prompting from the user:" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 10 "int(f, x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,6*$)%\"xG\"\"&
\"\"\"#\"\"(F'*$)F&\"\"%F(#!\"\"F-*$)F&\"\"#F(#\"\"\"F2F&!\"&*&F(F(*$)
,&F&F4F4F4\"\"#F(!\"\"!\"%-%#lnG6#F9F--F>6#,&F0F4F4F4\"\"$-F>6#,(F0F4F
&F-F'F4F4-%'arctanG6#,&F&F4F2F4F5*&,&F&!\")!#=F4F(FFF;#F4F-" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 33 " Copyr
ight and Author Information" }}{PARA 0 "" 0 "" {TEXT -1 571 "Copyright
 and Author Information\nM132L6R5.mws     A MapleV Release 5 worksheet
.\n\nAuthor:  Brian E. Blank  (20 February 2000)\n\nThis document may \+
not be distributed by any medium,\nincluding print, disk, and electron
ic transfer, without\nprior written permission of the author.\n\nFor m
ore information,  please contact the author:\n    \n     Department of
 Mathematics, \n     Washington University in St. Louis\n     St. Loui
s, MO   63130\n   \n     Telephone:    (314) 935-6763\n             e-
mail:    brian@math.wustl.edu\n\nCopyright:  \251 2000 Brian E. Blank,
  All Rights Reserved.\n" }}}}{MARK "7 0 0" 1 }{VIEWOPTS 1 1 0 1 1 
1803 }