{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 259 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 264 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 266 "Courier" 0 24 255 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "Courier" 1 14 255 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" 18 270 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 276 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 277 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 278 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 280 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 281 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 282 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 283 "" 0 1 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 255 0 0 1 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 285 "" 0 1 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "" 0 1 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "" 0 1 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 288 "" 0 1 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 289 "" 0 1 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 290 "" 0 1 255 0 255 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 291 "" 1 14 0 0 0 0 1 2 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 292 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 293 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 294 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 295 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 296 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 297 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 298 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "" -1 299 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" 19 300 "" 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 301 "Courier" 0 24 255 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 302 "C ourier" 0 14 255 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 303 "Courier " 0 14 255 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 304 "" 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 305 "Courier" 0 14 255 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 306 "Courier" 0 1 255 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 307 "" 0 1 255 0 0 1 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 308 "" 0 1 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 309 "" 0 1 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 310 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 311 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 312 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 313 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 315 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 316 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "T imes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 6 6 1 0 1 0 2 2 0 1 }{PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 "Courier" 1 10 255 0 255 1 2 2 2 2 2 1 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Title " -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 } 3 1 0 0 12 12 1 0 1 0 2 2 19 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 256 24 "Calculus Single Variable " }{TEXT 298 3 " \n" }{TEXT 291 35 "Brian E. Blank and Steven G. Kran tz" }{TEXT 292 51 "\n\nSection 4.8\nAntidifferentiation and Applicatio ns\n" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 8 "1. The " }{TEXT 266 3 "i nt" }{TEXT -1 9 " Command" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 26 30 "Maple's Indefinite Integrator:" }}{PARA 0 "" 0 "" {TEXT -1 56 " The basic command for finding indefinite integrals is " }{TEXT 267 3 "int" }{TEXT -1 73 ". When used for this purpose it is a function t hat takes two arguments.\n" }}{PARA 0 "" 0 "usage" {TEXT 26 17 "Callin g Sequence:" }{TEXT -1 4 "\n " }{TEXT 283 15 "int( expr , x )" } {TEXT -1 3 " \n " }}{PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" }{TEXT -1 4 "\n " }{TEXT 23 7 "expr - " }{TEXT -1 57 "an algebraic expressi on or a procedure, the integrand\n " }{TEXT 23 7 "x - " }{TEXT -1 7 "a name\n" }}{PARA 0 "" 0 "synopsis" {TEXT 26 12 "Description:" } }{PARA 15 "" 0 "" {TEXT -1 13 "The command " }{TEXT 286 3 "int" } {TEXT -1 54 " computes the indefinite integral of the expression " } {TEXT 284 4 "expr" }{TEXT -1 32 " with respect to the variable " } {TEXT 285 1 "x" }{TEXT -1 13 ". \nThe name " }{TEXT 287 10 "integrate " }{TEXT -1 19 " is a synonym for " }{TEXT 288 3 "int" }{TEXT -1 3 " . \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "int(D(f)(x),x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"fG6#%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "integrate(D(f)(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"fG6#%\"xG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 " " }}{PARA 15 "" 0 "" {TEXT 290 5 "Note :" }{TEXT -1 65 " No constant of integration appears in the result of a call to " }{TEXT 289 3 "int" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "int(x^n,x); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%\"xG,&%\"nG\"\"\"F(F(F(F&!\"\" " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 15 "" 0 "" {TEXT -1 102 "If Maple cannot find a closed form expr ession for the integral, the function call itself is returned. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "int(x^x,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$intG6$)%\"xGF'F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 11 "2. Examples" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "The first example shows the error message that results \+ when the variable of integration is not specified:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "int( sin(x) \+ );" }}{PARA 8 "" 1 "" {TEXT -1 52 "Error, (in int) wrong number (or ty pe) of arguments\n" }}}{PARA 0 "" 0 "" {TEXT -1 90 "\nTo avoid this er ror, we simply specify the variable of integration. Of course the answ er " }{TEXT 275 4 "does" }{TEXT -1 40 " depend on the variable of inte gration.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "int(sin(x), x) ; \nint(sin(x), z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$cosG6#% \"xG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%$sinG6#%\"xG\"\"\"%\" zGF(" }}}{PARA 0 "" 0 "" {TEXT -1 247 "\nThe indefinite integral of an elementary function is not necessarily an elementary function. Nevert heless, many such functions appear so frequently as integrands that th eir indefinite integrals have been named and studied. Here are some ex amples:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "int( 2*exp(-x^2)/sqrt(Pi), x );" "6#-%$intG6$*(\"\"#\" \"\"-%$expG6#,$*$%\"xGF'!\"\"F(-%%sqrtG6#%#PiGF/F." }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%$erfG6#%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "int( sin(x)/x, x );" "6#-%$intG6$*&-%$sinG6#%\"xG\"\"\"F*!\"\"F* " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#SiG6#%\"xG" }}}{EXCHG {PARA 0 " > " 0 "" {XPPEDIT 19 1 "int( cos(x)/x, x );" "6#-%$intG6$*&-%$cosG6#% \"xG\"\"\"F*!\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#CiG6#%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "int( sin(Pi*x^2/2), x );" " 6#-%$intG6$-%$sinG6#*(%#PiG\"\"\"*$%\"xG\"\"#F+F.!\"\"F-" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%)FresnelSG6#%\"xG" }}}{EXCHG {PARA 0 "> " 0 " " {XPPEDIT 19 1 "int( cos(Pi*x^2/2), x );" "6#-%$intG6$-%$cosG6#*(%#Pi G\"\"\"*$%\"xG\"\"#F+F.!\"\"F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%)F resnelCG6#%\"xG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 37 "3. Example (Constant of Integration)" }}{PARA 3 "" 0 "" {TEXT 268 5 "Let " }{TEXT 257 4 "F(x)" }{TEXT 269 30 " be the antide rivative of " }{XPPEDIT 270 0 "f(x) = x^3*(1+x^4)" "6#/-%\"fG6#%\"x G*&F'\"\"$,&\"\"\"F+*$F'\"\"%F+F+" }{TEXT 271 16 " for which " } {TEXT 258 7 " \n F" }{TEXT 272 3 "(0)" }{TEXT 260 3 " = " }{TEXT 273 13 "1. What is " }{TEXT 259 1 "F" }{TEXT 274 5 "(1) ?" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT 295 9 "Solution:" }{TEXT -1 1 "\n" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "F := unapply(int(x^3*(1+x^4), x) + C, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6#%\"xG6\"6$%)operatorG%&arr owGF(,(*&#\"\"\"\"\")F/*$)9$F0F/F/F/*&#F/\"\"%F/*$)F3F6F/F/F/%\"CGF/F( F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "F(x);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,(*&\"\")!\"\"%\"xGF%\"\"\"*&\"\"%F&F'F*F(%\"CGF (" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "C := solve(F(0) = 1, C );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "F(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6## \"#6\"\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 60 "4. Example (Comparing Two Antiderivatives of Same Functi on)" }}{PARA 3 "" 0 "" {TEXT 276 14 "Suppose that " }{TEXT 262 13 "F' (x) = G'(x)" }{TEXT 277 11 " for all " }{TEXT 261 1 "x" }{TEXT 278 7 ". If " }{TEXT 263 1 "F" }{TEXT 279 12 "(0) = 2, if " }{TEXT 264 1 "G" }{TEXT 280 41 "(0) = 5, and if F(4) = 6, \nthen what is " } {TEXT 265 1 "G" }{TEXT 281 4 "(4)?" }{TEXT -1 3 " \n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 296 9 "Solution:" }{TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "F := x -> G(x) + C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&-%\"GG6 #9$\"\"\"%\"CGF1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "e qn1 := F(0) = 2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/,&-%\"GG6 #\"\"!\"\"\"%\"CGF+\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eqn2 := subs(G(0) = 5, eqn1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%e qn2G/,&\"\"&\"\"\"%\"CGF(\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "C := solve(eqn2, C);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG !\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "eqn3 := F(4) = 6;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/,&-%\"GG6#\"\"%\"\"\"\"\"$! \"\"\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "G(4) = solve( \+ eqn3 , G(4) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"GG6#\"\"%\"\"* " }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 25 "5. The Inert Integrator " }{TEXT 301 3 "Int" }}{PARA 0 "" 0 " " {TEXT 26 37 "\nMaple's Inert Indefinite Integrator:" }}{PARA 0 "" 0 "" {TEXT -1 23 "Maple has a command, " }{TEXT 302 3 "Int" }{TEXT -1 118 ", that forces the Maple kernel to recognize an integral but to s uppress its evaluation. (Such a command is called an " }{TEXT 304 5 "i nert" }{TEXT -1 24 " command.) The command " }{TEXT 303 3 "Int" } {TEXT -1 361 " is often used when it is known that an indefinite inte gral cannot be evaluated. Using it therefore saves the time it would o therwise have taken Maple to unsuccessfully attempt an evaluation. It \+ is also often used - especially for pedagogical purposes - when an ear ly integration would conceal mathematical structure. \n\nWhen used as \+ an indefinite integrator, " }{TEXT 305 3 "Int" }{TEXT -1 41 " is a fu nction that takes two arguments.\n" }}{PARA 0 "" 0 "usage" {TEXT 26 17 "Calling Sequence:" }{TEXT -1 4 "\n " }{TEXT 307 15 "Int( expr , \+ x )" }{TEXT -1 3 " \n" }}{PARA 0 "" 0 "" {TEXT 26 11 "Parameters:" } {TEXT -1 4 "\n " }{TEXT 308 4 "expr" }{TEXT 23 3 " - " }{TEXT -1 57 "an algebraic expression or a procedure, the integrand\n " }{TEXT 309 1 "x" }{TEXT 23 6 " - " }{TEXT -1 6 "a name" }}{PARA 0 "" 0 "sy nopsis" {TEXT 26 12 "Description:" }}{PARA 15 "" 0 "" {TEXT -1 110 "Th e capitalized function name Int is the inert version of the int functi on, which simply returns unevaluated. " }}{PARA 15 "" 0 "" {TEXT -1 70 "To force the evaluation of an inert indefinite integral, the funct ion " }{TEXT 306 5 "value" }{TEXT -1 9 " is used." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Int(D(f)(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$--%\"DG6#%\"fG6#%\"xGF," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "J := Int(D(f)(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%$IntG6$--%\"DG6#%\"fG6#%\"xGF." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 11 "value( J );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\" fG6#%\"xG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "K := Int(x^n,x );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG-%$IntG6$)%\"xG%\"nGF)" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "value( K );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%\"xG,&%\"nG\"\"\"F(F(F(F&!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "L := Int(x^x,x);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"LG-%$IntG6$)%\"xGF)F)" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 11 "value( L );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$i ntG6$)%\"xGF'F'" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 61 "6. Application (Constant Acceleration, Exercise 26 Pag e 323) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 310 31 "A car travels at the speed of " }{XPPEDIT 18 0 "v[0];" "6#&%\"vG6 #\"\"!" }{TEXT 311 76 " ft/s. To pass another car, the driver acceler ates at a constant rate for " }{XPPEDIT 18 0 "tau;" "6#%$tauG" } {TEXT 315 54 " seconds. During this acceleration the car travels " }{XPPEDIT 18 0 "L;" "6#%\"LG" }{TEXT 312 17 " ft. What (in " } {XPPEDIT 260 1 "ft/(s^2);" "6#*&%#ftG\"\"\"*$%\"sG\"\"#!\"\"" }{TEXT 313 109 " ) is the value of the car's acceleration and what (in ft/s) is its speed at the end of the acceleration? " }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "For Exercise 26, page 323, set " }{XPPEDIT 18 0 "v[0] = 55*88/60;" "6#/&%\"vG6#\"\"!*(\"#b\"\"\"\"#))F*\"#g!\"\" " }{TEXT -1 32 " (since 60 mph = 88 ft/s), " }{XPPEDIT 19 1 "tau \+ = 60;" "6#/%$tauG\"#g" }{TEXT -1 11 ", and " }{XPPEDIT 18 0 "L = \+ 1.3(5280);" "6#/%\"LG--%&FloatG6$\"#8!\"\"6#\"%!G&" }{TEXT -1 48 ". Al l units will then involve feet and seconds.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 316 9 "Solution:" }{TEXT -1 1 "\n" }} {PARA 0 "" 0 "" {TEXT -1 53 "Suppose that the constant value of accele ration is " }{XPPEDIT 18 0 "alpha;" "6#,$%&alphaG\"\"\"" }{TEXT -1 12 " where " }{XPPEDIT 18 0 "0 < alpha;" "6#2\"\"!%&alphaG" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "s( t);" "6#-%\"sG6#%\"tG" }{TEXT -1 31 " be the position of the car " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 43 " seconds after accelerati on begins. \nLet " }{XPPEDIT 18 0 "v(t);" "6#-%\"vG6#%\"tG" }{TEXT -1 31 " be the velocity of the car " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 43 " seconds after acceleration begins. \n\nThen" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 66 "accelerationEqn := diff(s(t),t$2) = alpha; #Accele ration equation" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0accelerationEqnG /-%%diffG6$-%\"sG6#%\"tG-%\"$G6$F,\"\"#%&alphaG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "velocityEqn := Int( lhs(accelerationEqn), t) \n \+ = Int(rhs(accelerationEqn), t) + C1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,velocityEqnG/-%$IntG6$-%%diffG6$-%\"sG6#%\"tG-% \"$G6$F/\"\"#F/,&-F'6$%&alphaGF/\"\"\"%#C1GF8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "velocityEqn := in t( lhs(accelerationEqn), t) \n = int(rhs(accelerati onEqn), t) + C1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,velocityEqnG/-% %diffG6$-%\"sG6#%\"tGF,,&*&%&alphaG\"\"\"F,F0F0%#C1GF0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "C1 := s olve(v[0] = subs(t=0, rhs(velocityEqn)), C1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#C1G&%\"vG6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "velocityEqn;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%d iffG6$-%\"sG6#%\"tGF*,&*&%&alphaG\"\"\"F*F.F.&%\"vG6#\"\"!F." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "positionEqn := Int( lhs(velo cityEqn), t) \n = Int(rhs(velocityEqn), t) + C2;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,positionEqnG/-%$IntG6$-%%diffG6$-% \"sG6#%\"tGF/F/,&-F'6$,&*&%&alphaG\"\"\"F/F6F6&%\"vG6#\"\"!F6F/F6%#C2G F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "positionEqn := int( l hs(velocityEqn), t) \n = int(rhs(velocityEqn), t) + C2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,positionEqnG/-%\"sG6#%\"tG, (*(\"\"#!\"\"%&alphaG\"\"\"F)F,F/*&&%\"vG6#\"\"!F/F)F/F/%#C2GF/" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "C2 := solve(0 = subs(t=0, rh s(positionEqn)), C2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#C2G\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "positionEqn;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"sG6#%\"tG,&*(\"\"#!\"\"%&alphaG\"\"\"F'F*F -*&&%\"vG6#\"\"!F-F'F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "positionEqn := subs(t=tau, positionEqn); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,positionEqnG/-%\"sG6#%$tauG,&*(\" \"#!\"\"%&alphaG\"\"\"F)F,F/*&&%\"vG6#\"\"!F/F)F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "positionEqn := subs(s(tau) = L, positionE qn); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,positionEqnG/%\"LG,&*(\"\" #!\"\"%&alphaG\"\"\"%$tauGF)F,*&&%\"vG6#\"\"!F,F-F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "acceleration := alpha = solve(posit ionEqn, alpha);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-accelerationG/%& alphaG,$*(\"\"#\"\"\",&%\"LG!\"\"*&&%\"vG6#\"\"!F*%$tauGF*F*F*F3!\"#F- " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "velocityEqn;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"sG6#%\"tGF*,&*&%&alphaG\"\"\" F*F.F.&%\"vG6#\"\"!F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "ve locityEqn := v(t) = rhs(velocityEqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,velocityEqnG/-%\"vG6#%\"tG,&*&%&alphaG\"\"\"F)F-F-&F'6#\"\"!F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "velocityEqn := subs( \{t= tau, acceleration\} , velocityEqn ); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,velocityEqnG/-%\"vG6#%$tauG,&*(\"\"#\"\"\",&%\"LG!\"\"*&&F'6#\" \"!F-F)F-F-F-F)F0F0F2F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 61 "7. Application (Consta nt Deceleration, Exercise 27 Page 323) " }}{PARA 0 "" 0 "" {TEXT 282 32 "\nA car travels at the speed of " }{XPPEDIT 18 0 "v[0];" "6#&%\"v G6#\"\"!" }{TEXT 293 60 " feet/second. It must come to a halt within \+ a distance of " }{XPPEDIT 18 0 "L;" "6#%\"LG" }{TEXT 294 67 " feet. I f the car decelerates at a constant rate, what value ( in " } {XPPEDIT 300 1 "ft/(s^2);" "6#*&%#ftG\"\"\"*$%\"sG\"\"#!\"\"" }{TEXT 299 35 " ) should that rate not exceed? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "(For Exercise 27, page 323, set " }{XPPEDIT 18 0 "v[0] = 50*88/60;" "6#/&%\"vG6#\"\"!*(\"#]\"\"\"\" #))F*\"#g!\"\"" }{TEXT -1 40 " (since 60 mph = 88 ft/s) and \+ " }{XPPEDIT 18 0 "L = 1200;" "6#/%\"LG\"%+7" }{TEXT -1 2 ".)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 297 9 "Solution:" } {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 53 "Suppose that the consta nt value of deceleration is " }{XPPEDIT 18 0 "-delta;" "6#,$%&deltaG !\"\"" }{TEXT -1 12 " where " }{XPPEDIT 18 0 "0 < delta;" "6#2\" \"!%&deltaG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " } {XPPEDIT 18 0 "s(t);" "6#-%\"sG6#%\"tG" }{TEXT -1 31 " be the positio n of the car " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 44 " seconds after deceleration begins. \n\nThen" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "decelera tionEqn := diff(s(t),t$2) = - delta; #Deceleration equation" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%0decelerationEqnG/-%%diffG6$-%\"sG6#%\"tG- %\"$G6$F,\"\"#,$%&deltaG!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "velocityEqn := Int( lhs(dec elerationEqn), t) \n = Int(rhs(decelerationEqn), t) + C1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,velocityEqnG/-%$IntG6$-%% diffG6$-%\"sG6#%\"tG-%\"$G6$F/\"\"#F/,&-F'6$,$%&deltaG!\"\"F/\"\"\"%#C 1GF:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "velocityEqn := int( lhs(decelerationEqn), t) \n \+ = int(rhs(decelerationEqn), t) + C1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,velocityEqnG/-%%diffG6$-%\"sG6#%\"tGF,,&*&%&deltaG\" \"\"F,F0!\"\"%#C1GF0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "C1 := solve(v[0] = subs(t=0, rhs(velocity Eqn)), C1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#C1G&%\"vG6#\"\"!" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "velocityEqn;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"sG6#%\"tGF*,&*&%&deltaG\"\"\"F*F. !\"\"&%\"vG6#\"\"!F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "pos itionEqn := Int( lhs(velocityEqn), t) \n = Int(rhs( velocityEqn), t) + C2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,positionE qnG/-%$IntG6$-%%diffG6$-%\"sG6#%\"tGF/F/,&-F'6$,&*&%&deltaG\"\"\"F/F6! \"\"&%\"vG6#\"\"!F6F/F6%#C2GF6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "positionEqn := int( lhs(velocityEqn), t) \n \+ = int(rhs(velocityEqn), t) + C2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %,positionEqnG/-%\"sG6#%\"tG,(*(\"\"#!\"\"%&deltaG\"\"\"F)F,F-*&&%\"vG 6#\"\"!F/F)F/F/%#C2GF/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "C 2 := solve(0 = subs(t=0, rhs(positionEqn)), C2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#C2G\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "positionEqn;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"sG6#%\"tG,&*( \"\"#!\"\"%&deltaG\"\"\"F'F*F+*&&%\"vG6#\"\"!F-F'F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "T := solve(rhs(velocityEqn) = 0, t); # Durat ion of deceleration" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG*&&%\"vG6 #\"\"!\"\"\"%&deltaG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "stoppedPositionEqn := subs(t = T, positionEqn); \n \+ #Position car comes to rest " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%3stoppedPositionEqnG/-%\"sG6#*&&%\"vG6#\"\"!\"\" \"%&deltaG!\"\",$*&#F.\"\"#F.*&F/F0F*F4F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "eqnForMinimumDeceleration := \ndelta = solve(rhs( stoppedPositionEqn) = L, delta); #Minimum deceleration" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%:eqnForMinimumDecelerationG/%&deltaG,$*&#\"\"\" \"\"#F**&&%\"vG6#\"\"!F+%\"LG!\"\"F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 32 "Copyright and Author Information" }}{PARA 0 "" 0 "" {TEXT -1 101 "\nWorksheet Title: BlankKrantz-04_8-R8.mws A Maple Release 8 worksheet.\n\nAuthor: Brian E. Blank " }}{PARA 0 "" 0 "" {TEXT -1 42 "Date Created: 26 January 2000 (MapleV R4)" }}{PARA 0 "" 0 "" {TEXT -1 485 "Date Last Revised: 29 August 2006\n\nThis document may n ot be distributed by any medium,\nincluding print, disk, and electroni c transfer, without\nprior written permission of the author.\n\nFor mo re information, please contact the author:\n \n Department of \+ Mathematics, \n Washington University in St. Louis\n St. Louis , MO 63130\n \n Telephone: (314) 935-6763\n e-m ail: brian@math.wustl.edu\n\nCopyright: \251 2000-2007 Brian E. Bl ank, All Rights Reserved.\n" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}}} {MARK "8 3 0" 446 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }