{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 Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
284 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 287 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 288 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
289 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 290 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 291 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 292 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }
{CSTYLE "" -1 293 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
294 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 295 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 296 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 297 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 298 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
299 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 300 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim
es" 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 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 
4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 
0 8 2 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 "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 
0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" -1 13 1 
{CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 
0 1 0 1 0 2 2 0 1 }{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 }
{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 
2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 
257 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 
0 0 8 2 1 0 1 0 2 2 0 1 }}
{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 10 "C\341lculo II" }}{PARA 
257 "" 0 "" {TEXT -1 31 "Li\347\343o 13a: Integrais Impr\363prias" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "At\351 ent\343o, em nosso estudo d
e integra\347\343o, temos considerado " }{XPPEDIT 18 0 "Int(f(x),x = a
 .. b);" "6#-%$IntG6$-%\"fG6#%\"xG/F);%\"aG%\"bG" }{TEXT -1 9 " , send
o " }{TEXT 268 1 "f" }{TEXT -1 36 " \351 uma fun\347\343o no intervalo
 limitado " }{XPPEDIT 18 0 "[a, b];" "6#7$%\"aG%\"bG" }{TEXT -1 57 ". \+
 Deseja-se agora analisar o que acontece quando ambos, " }{TEXT 269 1 
"f" }{TEXT -1 15 " ou o intervalo" }{XPPEDIT 18 0 "[a, b];" "6#7$%\"aG
%\"bG" }{TEXT -1 56 " , tornam-se ilimitados.  Em ambos os casos, tem-
se uma " }{TEXT 270 18 "integral impr\363pria" }{TEXT -1 49 " (as inte
grais estudadas at\351 agora s\343o integrais " }{TEXT 271 8 "pr\363pr
ias" }{TEXT -1 157 ").  Como veremos mais adiante, uma integral impr
\363pria n\343o \351 definida diretamente em termos de parti\347\365es
 e somas, mas sim como o limite de integrais pr\363prias." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 28 "Tipo 1: I
ntervalos Infinitos" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Iniciaremos
 com o caso onde o intervalo de integra\347\343o, [" }{XPPEDIT 18 0 "a
,infinity;" "6$%\"aG%)infinityG" }{TEXT -1 16 "), \351 ilimitado \340
" }{TEXT 272 1 " " }{TEXT -1 32 "esquerda.  Neste caso, definimos" }}}
{EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "Int(f(x),x = a \+
.. infinity) := limit(Int(f(x),x = a .. b),b = infinity);" "6#>-%$IntG
6$-%\"fG6#%\"xG/F*;%\"aG%)infinityG-%&limitG6$-F%6$-F(6#F*/F*;F-%\"bG/
F8F." }{TEXT -1 2 " ," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "contanto
 que o limite exista.  Se " }{TEXT 273 1 "f" }{TEXT -1 99 " \351 uma f
un\347\343o positiva, este limite pode ser interpretado como sendo a \+
\341rea total sob o gr\341fico de " }{TEXT 274 1 "f" }{TEXT -1 15 ", \+
\340 direita de " }{TEXT 275 1 "a" }{TEXT -1 72 "; em geral, tem-se as
 mesmas interpreta\347\365es que para a integral pr\363pria." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "O " }{TEXT 276 
5 "Maple" }{TEXT -1 67 " \351 capaz de resolver muitas integrais impr
\363prias como, por exemplo," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "Int(1/(2*x - 5)^3, x=4..infi
nity) = int(1/(2*x - 5)^3, x=4..infinity);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(*$),&*&\"\"#F(%\"xGF(F(\"\"&!\"\"\"
\"$F(F0/F.;\"\"%%)infinityG#F(\"#O" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "f := x->1/(2*x - 5)^3;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&\"\"\"F-*$),&*&\"\"#F
-9$F-F-\"\"&!\"\"\"\"$F-F5F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 124 "p1 := plot(f(x), x=3.5..8,thickness=2):\np2 := plott
ools[line]([4,0],[4,f(4)],color=blue,thickness=2):\nplots[display](p1,
p2);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVE
SG6%7Y7$$\"3++++++++N!#<$\"3+++++++]7!#=7$$\"38+vV)z@X_$F*$\"35M\"3?Qv
B;\"F-7$$\"3E+](ofV!\\NF*$\"3%onGO,`F3\"F-7$$\"3S+DJ&RlNd$F*$\"3r$f<0'
)R-,\"F-7$$\"33++v$>(3)f$F*$\"32=g#*4')fS%*!#>7$$\"3+]i:l(f2k$F*$\"3I'
fb!RtH?%)FB7$$\"3!**\\ilLKMo$F*$\"3j%QWGs\")=a(FB7$$\"3.]P41@UJPF*$\"3
))3GDbM0%p'FB7$$\"39+]iv=TzPF*$\"3#Qi!)f.)oofFB7$$\"3=++]_(>x#QF*$\"3#
Hl0Qw31M&FB7$$\"3y**\\PHw-wQF*$\"31mX_r)\\wz%FB7$$\"3'**\\7`=%=sRF*$\"
3w**RWr8j<RFB7$$\"3W+Dc@OLhSF*$\"3/.P'z=\\TG$FB7$$\"3%**\\i!*pUO:%F*$
\"3wSSvk()HkFFB7$$\"3!)*\\i!z)3\"\\UF*$\"31Wv#p9GfL#FB7$$\"3Z*\\7y*)oU
M%F*$\"3;xy%[tyE*>FB7$$\"3*)***\\Pn_@W%F*$\"3El\"4Hw<jq\"FB7$$\"3S+](o
vo$GXF*$\"3I-y*HG`y\\\"FB7$$\"3O++DYwUDYF*$\"3Tr9FX-)=I\"FB7$$\"39++vo
])Gs%F*$\"3\\Z9(e#Q/Q6FB7$$\"38++v`L!o\"[F*$\"3JbP&f)o<05FB7$$\"3$**\\
iS:!4-\\F*$\"3*R%ed_=m=!*!#?7$$\"3Z++v3W].]F*$\"3#**[vKx^k'zFbr7$$\"3C
+++&e:%*3&F*$\"31!)3Bc)R&*>(Fbr7$$\"3=+Dc12N*=&F*$\"3.^$Q93*REkFbr7$$
\"3U****\\A-\"yF&F*$\"3m&3W(\\czJeFbr7$$\"3k+DcJV'[P&F*$\"3%z&z)3fu3E&
Fbr7$$\"3o+vo%z#GnaF*$\"3[36krHY%y%Fbr7$$\"3O+]il<rjbF*$\"3-EkO%QmnM%F
br7$$\"3g*\\7GmjAl&F*$\"3yXDyh'Q1*RFbr7$$\"3u**\\(o%)yxu&F*$\"3Q:r*450
)[OFbr7$$\"3_*\\i!zA*p%eF*$\"3UTeQi'[QL$Fbr7$$\"3;+vVjyNLfF*$\"3#p(G#*
**o`)3$Fbr7$$\"3()**\\ig]jEgF*$\"3%=bdtJ#*)\\GFbr7$$\"3t****\\K&**H7'F
*$\"3S:eQRI[GEFbr7$$\"3U**\\7oLF<iF*$\"3\"QE;D#*GNV#Fbr7$$\"3)**\\i::)
[3jF*$\"3[\"z&*yMJGE#Fbr7$$\"3F+](ohm(4kF*$\"3[tNqu#)\\\"4#Fbr7$$\"3o*
***\\A)p2]'F*$\"3#[#eBkx*>&>Fbr7$$\"3I++vo^$zf'F*$\"3S#fo]n8k\"=Fbr7$$
\"3y*\\iST\")fo'F*$\"3zw'QWW!>/<Fbr7$$\"3#)***\\i@RAy'F*$\"3)pfbG.K=f
\"Fbr7$$\"3C*\\ilI5G(oF*$\"3e<[*y#\\&\\\\\"Fbr7$$\"3%)*\\7G>$[npF*$\"3
4YD`UN\">S\"Fbr7$$\"3/++vVK/gqF*$\"3(>voxBm#=8Fbr7$$\"3M*\\i!R]%p:(F*$
\"3eE(yC;vwB\"Fbr7$$\"3]+++&)HF]sF*$\"3=a!H2r\\h;\"Fbr7$$\"3g**\\P*G9d
M(F*$\"3E</$zU#f)4\"Fbr7$$\"3E+Dc\"Hl.W(F*$\"3f*o!o6;lO5Fbr7$$\"3x****
\\K(Rt_(F*$\"3.Qw4'HRx$)*!#@7$$\"3p**\\(oDAqi(F*$\"35\"Hy))Q5]F*F_[l7$
$\"3W+++&\\zhr(F*$\"3J&4:l]&\\2))F_[l7$$\"3m*\\ilqR7\"yF*$\"3>M=]%>,IM
)F_[l7$$\"3))*\\P%eWA-zF*$\"3Gm@#pA?&GzF_[l7$$\"\")\"\"!$\"3-yd,4![J^(
F_[l-%'COLOURG6&%$RGBG$\"#5!\"\"$Fg\\lFg\\lFa]l-%*THICKNESSG6#\"\"#-F$
6%7$7$$\"\"%Fg\\lFa]l7$Fj]l$\"+/Pq.P!#6-F[]l6&F]]lFa]lFa]l$\"*++++\"!
\")Fb]l-%+AXESLABELSG6$Q\"x6\"Q!Fi^l-%%VIEWG6$;$\"#NF`]lFe\\l%(DEFAULT
G" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "
Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 197 "Contudo, o uso deste
s comando nos oferecem pouca clareza.  Desta forma, resolveremos a mes
ma integral passo a passo.  Inicialmente, \351 preciso resolver a inte
gral de 4 a um limite superior vari\341vel " }{TEXT 277 1 "b" }{TEXT 
-1 117 ".  Manualmente, \351 preciso encontrar uma anti-derivada para \+
a fun\347\343o e aplicar ent\343o o Teorema Fundamental do C\341lculo:
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "ad := int(f(x), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%#adG,$*&\"\"\"F'*&\"\"%F'),&*&\"\"#F'%\"xGF'F'\"\"&!\"\"F-F'F0F0" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "inttob := subs(x=b, ad) -
 subs(x=4, ad);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'inttobG,&*&\"\"
\"F'*&\"\"%F'),&*&\"\"#F'%\"bGF'F'\"\"&!\"\"F-F'F0F0#F'\"#OF'" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Pode-se obter o mesmo resuldado di
retamente com o " }{TEXT 278 5 "Maple" }{TEXT -1 34 ", no c\341lculo d
a integral de 4 at\351 " }{TEXT 279 1 "b" }{TEXT -1 78 ".  (A resposta
 aparece de forma diferente.  Por\351m, verifica-se que \351 a mesma.)
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "int(f(x), x=4..b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#,$*(\"\"*!\"\",(\"\"%\"\"\"*$)%\"bG\"\"#F)F)*&\"\"&F)F,F)F&F),&*&F-F)
F,F)F)F/F&!\"#F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "simplif
y(inttob - int(f(x), x=4..b));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"
!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Tendo encontrado a integral \+
at\351 " }{TEXT 280 1 "b" }{TEXT -1 66 ", de uma forma ou de outra, po
de-se ent\343o tormar o limite quando  " }{XPPEDIT 18 0 "proc (b) opti
ons operator, arrow; infinity end;" "6#f*6#%\"bG7\"6$%)operatorG%&arro
wG6\"%)infinityGF*F*F*" }{TEXT -1 2 " :" }{TEXT 281 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "Int(1/(2*x - 5)^3, x=4..infinity) =
 limit(inttob, b=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$Int
G6$*&\"\"\"F(*$),&*&\"\"#F(%\"xGF(F(\"\"&!\"\"\"\"$F(F0/F.;\"\"%%)infi
nityG#F(\"#O" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Esta resposta enc
ontra-se em acordo com a resposta original do " }{TEXT 282 5 "Maple" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 79 "Se o intervalo for ilimitado \340 esquerda, procedemos de
 forma similar, definindo" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 256 "" 0 "" {XPPEDIT 18 0 "Int(f(x),x = -infinity .. b) := limit
(Int(f(x),x = a .. b),a = -infinity);" "6#>-%$IntG6$-%\"fG6#%\"xG/F*;,
$%)infinityG!\"\"%\"bG-%&limitG6$-F%6$-F(6#F*/F*;%\"aGF0/F:,$F.F/" }
{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "Int(exp(3
*x), x=-infinity..2) = int(exp(3*x), x=-infinity..2);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/-%$IntG6$-%$expG6#,$*&\"\"$\"\"\"%\"xGF-F-/F.;,$%
)infinityG!\"\"\"\"#,$*&#F-F,F--F(6#\"\"'F-F-" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 32 "inttoa := int(exp(3*x), x=a..2);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%'inttoaG,&*&#\"\"\"\"\"$F(-%$expG6#\"\"'F(F(*&#F
(F)F(-F+6#,$*&F)F(%\"aGF(F(F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "limit(inttoa, a=-infinity);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$*&#\"\"\"\"\"$F&-%$expG6#\"\"'F&F&" }}}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 41 "Finalmente, se a integral \351 ilimitada de " }
{TEXT 283 5 "ambos" }{TEXT -1 44 " os lados, escolhe-se um ponto conve
niente, " }{TEXT 284 2 "c " }{TEXT -1 11 "(em geral, " }{TEXT 285 1 "c
" }{TEXT -1 17 " = 0) e define-se" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 256 "" 0 "" {XPPEDIT 18 0 "Int(f(x),x = -infini
ty .. infinity) := Int(f(x),x = -infinity .. c)+Int(f(x),x = c .. infi
nity);" "6#>-%$IntG6$-%\"fG6#%\"xG/F*;,$%)infinityG!\"\"F.,&-F%6$-F(6#
F*/F*;,$F.F/%\"cG\"\"\"-F%6$-F(6#F*/F*;F8F.F9" }{TEXT -1 2 " ," }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 286 55 "contanto que ambas
 as integrais do lado direito existam" }{TEXT -1 66 ".  Esta condi\347
\343o \351 crucial.  Consideremos, por exemplo, a integral" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "f := x->(1 + (1+x^2)*sin(x))/(1 + x
^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operator
G%&arrowGF(*&,&\"\"\"F.*&,&*$)9$\"\"#F.F.F.F.F.-%$sinG6#F3F.F.F.F0!\"
\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Int(f(x), x=-in
finity..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&,&\"
\"\"F(*&,&*$)%\"xG\"\"#F(F(F(F(F(-%$sinG6#F-F(F(F(F*!\"\"/F-;,$%)infin
ityGF2F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "plot(f(x), x=-5
0..50, thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 354 354 354 
{PLOTDATA 2 "6&-%'CURVESG6$7aem7$$!#]\"\"!$\"3*=.zw$puFE!#=7$$!3CLekyn
P')\\!#;$\"3q'zCf7\"*R\"RF-7$$!3[m;HdNvs\\F1$\"31P)4;m\"yF^F-7$$!3s*\\
PfLI\"f\\F1$\"35HXHtniYiF-7$$!3'HL$e9r]X\\F1$\"3SN!p>3%z\\sF-7$$!3X**
\\(=ng#=\\F1$\"3$e5')o5Gs$))F-7$$!3kmm;HU,\"*[F1$\"333y1?#oHx*F-7$$!3#
He*[=E?%)[F1$\"3)yve*G4%e*)*F-7$$!3))*\\7y+\"Rx[F1$\"3Ar$y&3h$G(**F-7$
$!3'oTNrRz0([F1$\"3,%414nf.+\"!#<7$$!3%QLekynP'[F1$\"3wIYA<,)z)**F-7$$
!33nT5lX9][F1$\"3BaiJe+7=)*F-7$$!3K++vV8_O[F1$\"3+/$48(RSm%*F-7$$!3!om
T5!\\F4[F1$\"37\"**y=bCnC)F-7$$!3GLLLe%G?y%F1$\"3a!G]'o=$*=kF-7$$!3/Ug
Fm\\<qZF1$\"37T$p;9=nY&F-7$$!33](=UZ@$eZF1$\"3>6CNYe%yV%F-7$$!37e9;#)z
YYZF1$\"3mx;erXvYLF-7$$!3(o;/,\\9Yt%F1$\"3,%yE\\4d(3AF-7$$!3ivo/)*4wAZ
F1$\"3%Gnm=[C)R5F-7$$!3m$e*)f]24r%F1$!3$Q!fbM0QO9!#>7$$!3q\"HKR,a!*p%F
1$!3:NjN!Q@]K\"F-7$$!3Y+](=_+so%F1$!31&e;baXx[#F-7$$!3?4x\")HqMvYF1$!3
,r%4uX#\\:OF-7$$!3C</wPN\\jYF1$!3mNnt&*[V#p%F-7$$!3GDJqX+k^YF1$!3qrq$>
jeMq&F-7$$!3/Mek`lyRYF1$!398(3NauVj'F-7$$!37]7`p&zgh%F1$!3ThJDV4$\\?)F
-7$$!3!pm;aesBf%F1$!3[s%ymXXiJ*F-7$$!3g3-)Q4U!zXF1$!3)4*\\Mi2$\\r*F-7$
$!3e\\PM-;rlXF1$!380`#4Z]6%**F-7$$!33?bdcj/fXF1$!3eK\\PC$4#))**F-7$$!3
G\"H236\"Q_XF1$!3g610l+*3***F-7$$!3[i!R]'erXXF1$!3U]T7D2=\\**F-7$$!3'H
$3F>10RXF1$!33)>]'okEj)*F-7$$!3L;z>O'*Q7XF1$!3c(pP%zF4)3*F-7$$!3s**\\7
`'Gd[%F1$!33v%>#z'*RqwF-7$$!3ST&)eh\")RsWF1$!3!)*zow(=J]nF-7$$!35$3_+n
n!fWF1$!3.,*R$z#e.r&F-7$$!3yCc^yrtXWF1$!3q5*=T=\"**oXF-7$$!3Zm\"zpo1CV
%F1$!31\\&>\"eBYYLF-7$$!3;3FW&>w!>WF1$!3/-4H)ojW1#F-7$$!3%)\\i!RqXdS%F
1$!3ybviOGUduF^r7$$!3a\"zpB@:CR%F1$\"3ih;mr9.jeF^r7$$!3ALL$3s%3zVF1$\"
3y[\"f/[P!3>F-7$$!3-Le9memlVF1$\"3F)3YR&>Q/KF-7$$!3!GLe9,ZAN%F1$\"3`BD
,Dw?VWF-7$$!3HL3xc\"G)QVF1$\"3Sp1gN:C-cF-7$$!33LL3-$4aK%F1$\"3%\\roSNX
1m'F-7$$!3OL$3Ffr&)H%F1$\"3-SZ%[(zf,%)F-7$$!3%HLLL)QtrUF1$\"34ZlJGoUT&
*F-7$$!3ULekG]JeUF1$\"3[8=?b`be)*F-7$$!3@L$eR<'*[C%F1$\"3;6H]%zI&)***F
-7$$!3Y$e9mu'=QUF1$\"3<D4Fyt6+5Ffn7$$!3+L3F>tZJUF1$\"3W_$z.3P)e**F-7$$
!3C$3F>*ywCUF1$\"3M$)pj!G6<()*F-7$$!3\\LLek%e!=UF1$\"3\\RS#e_)=S(*F-7$
$!31L$3_v?7>%F1$\"3ETE#R8,\\y)F-7$$!3MLL$e/$QkTF1$\"3<D8K0w1,sF-7$$!3'
=z>'oz-^TF1$\"3%3'\\2q'HB@'F-7$$!3o\\iS\"*GnPTF1$\"37PG!e0eI6&F-7$$!3?
3F>9yJCTF1$\"3\"*R#>*32$G#RF-7$$!3sm\"zpti46%F1$\"3'Hl!oTa%Gm#F-7$$!3C
Dcwfwg(4%F1$\"3ym)fPTUbN\"F-7$$!3y$3_De_U3%F1$\"3qHa\\Y)[/U#!#?7$$!3eT
&Q`](*32%F1$!3nlY,>uX28F-7$$!35+]7GCadSF1$!3Vn5n;ms:EF-7$$!3je9\"4N(=W
SF1$!3YV5i8IIxQF-7$$!3W;zptA$3.%F1$!3Uxlmy!=(p]F-7$$!3'\\P%['>xu,%F1$!
3=ubM*)\\trhF-7$$!3[L3F>@7/SF1$!3Y@H4XosjrF-7$$!3a]P%['>TxRF1$!3IB<j^(
R#\\()F-7$$!3'om;/\"=q]RF1$!3P'f[b31Qr*F-7$$!3;L3FW*>$QRF1$!3)\\Q])**3
XH**F-7$$!3=+]7y!Qf#RF1$!3!yt3%[?'H***F-7$$!3>n\"z>@cN\"RF1$!300!)e`kO
.**F-7$$!3\\LL$eMu6!RF1$!3UwOBB`.i'*F-7$$!3#omTNh5k(QF1$!3nwt')p6@T()F
-7$$!37++D\")ok^QF1$!3CI!yTFlmG(F-7$$!3VmT5:]ERQF1$!3)>#f:(HTeQ'F-7$$!
3WL$e*[J)o#QF1$!3_!QOHQOrQ&F-7$$!3X+D\"GG,X\"QF1$!3fe583:%eI%F-7$$!3vm
mm;%>@!QF1$!3%**G#e<A^eJF-7$$!31L3_]vt*y$F1$!3#Gr8V8:F'>F-7$$!33+]P%ob
tx$F1$!3A-kr$))*entF^r7$$!33n\"H#=Q(\\w$F1$\"3fo(\\4<de+&F^r7$$!3QLL3_
>f_PF1$\"3<It=5TPI<F-7$$!34+](oCr(RPF1$\"3'*H&Hv&GrvHF-7$$!34mmmT0&ps$
F1$\"3iZ/,9WKsTF-7$$!3zK$ek$)HTr$F1$\"3F`G0fxc+`F-7$$!3]***\\7848q$F1$
\"3#*3-r8T#>M'F-7$$!3#HLL3sncn$F1$\"3W%4%G)R7t4)F-7$$!3KmmT5j-]OF1$\"3
B42'3O8PK*F-7$$!3.L$3_g0sj$F1$\"3)HN7Uvt>r*F-7$$!3t********[QCOF1$\"3s
JeEb5%4%**F-7$$!3WLeRZX(zh$F1$\"3?rt]^9T%***F-7$$!3Wm;z%>k:h$F1$\"3/2#
ybx&o+5Ffn7$$!3W*\\(=UQ:0OF1$\"3#ReGG&*G#y**F-7$$!39LLe*[V()f$F1$\"34;
hdeFk3**F-7$$!3cmm;z?5tNF1$\"3$)fLI*eP*G#*F-7$$!3)****\\(o1YZNF1$\"3#o
f\"GXzFYzF-7$$!3.LL3x9?MNF1$\"3u\"QlBM#osqF-7$$!3\"om;aGU4_$F1$\"3K'>!
>#zn]2'F-7$$!3e++v$4$o2NF1$\"3)yS`CO[4(\\F-7$$!3kLL3-RU%\\$F1$\"3a&*[!
H\"yqzPF-7$$!3smmT5Z;\"[$F1$\"3)QnbgJfA_#F-7$$!3y***\\(=b!zY$F1$\"3k)*
*R#H$y1A\"F-7$$!3%GL$3FjkaMF1$!3D7K$3P\\=-\"F^r7$$!3hmmTNrQTMF1$!3im+s
^m5B9F-7$$!3R++vVz7GMF1$!3`i9*R0(*)=FF-7$$!3XLL3_(o[T$F1$!3o%4xoa2o'RF
-7$$!3^mmTg&4;S$F1$!3,^4eX+$\\9&F-7$$!3e***\\(o.N)Q$F1$!3?\"o15d\"eKiF
-7$$!3UmmT&)>$=O$F1$!3!=Sq\\&f,i!)F-7$$!3DLL3-OJNLF1$!3e$4nay-sK*F-7$$
!37+D\"G$p4ALF1$!3j!ynCUUyr*F-7$$!3)pmTNE!))3LF1$!3A*GRuk=)Q**F-7$$!30
]i!*G>F-LF1$!3#fw2=GgV)**F-7$$!39L3F%fjcH$F1$!3K-')4zdF')**F-7$$!3#pTN
'f_0*G$F1$!3eoJ>:mbW**F-7$$!3++++DpW#G$F1$!3oalf[ZQf)*F-7$$!3-L$eke8gD
$F1$!3Yy+))3(3W4*F-7$$!3umm\"zC!eHKF1$!3O*>`(H*>qp(F-7$$!3gLekyNO;KF1$
!3#=r2iR^**y'F-7$$!3Y+]P4p9.KF1$!3%**H**e))yUw&F-7$$!3)p;/,CI**=$F1$!3
9THs39*yj%F-7$$!3[LL$3d8n<$F1$!3iW5+!pM/V$F-7$$!3++Dc,p\\jJF1$!371/B(
\\oH;#F-7$$!3]m;HK-G]JF1$!3cT=?AI*fd)F^r7$$!3OL3-jN1PJF1$\"3Tm(R/9`!HY
F^r7$$!3()***\\P*o%Q7$F1$\"3G4*>!)y8bx\"F-7$$!3l\"zp[#>D5JF1$\"3yRF%**
*RM$4$F-7$$!33$e*)f&pl'4$F1$\"3]@Gh%z\"GaVF-7$$!3)[P4r)>1$3$F1$\"3@9b2
(of]`&F-7$$!3mm\"H#=qYpIF1$\"375nNv%*)Qh'F-7$$!3))\\(o/3xA/$F1$\"3k?/@
!4E$)Q)F-7$$!34L$3F9(3:IF1$\"3!p\\MM=Hsa*F-7$$!3_C\"GQ<#\\,IF1$\"3quc[
4zGn)*F-7$$!3K;z%\\?(*y)HF1$\"3ah&ppHY0+\"Ffn7$$!3Q7y]?(*4\")HF1$\"3J[
Ec0:a+5Ffn7$$!353x1OAIuHF1$\"31Teh)Q1#f**F-7$$!3#QgF;v/v'HF1$\"3e3#yxj
]q')*F-7$$!3a*\\(=nsqgHF1$\"3L!>8Vpt$H(*F-7$$!35$3F%Ht^LHF1$\"3(o4@bNi
#R()F-7$$!3Kmmm\"RFj!HF1$\"3nHWo0J*y5(F-7$$!39$3xc$HN%*GF1$\"3ru=(Qjua
@'F-7$$!3i*\\(oz%yB)GF1$\"3A2Rd\"Q7UB&F-7$$!3W;zpBSSqGF1$\"3'RPf%o5;yT
F-7$$!3EL$3xcH%eGF1$\"3WG^*R`YC1$F-7$$!3s\\(=<6bk%GF1$\"3YK3,6\"\\I!>F
-7$$!3bm\"Hdl![MGF1$\"3CzwG:wulrF^r7$$!3P$eR(*>1D#GF1$!3Gj`Kuu#)*z%F^r
7$$!3$)***\\PuJ0\"GF1$!3#H%)[?A&[p;F-7$$!3I;/w(Gd&)z#F1$!3L!f,$)[&*[$G
F-7$$!39L3xJGe'y#F1$!3![R8=R@&fRF-7$$!3'*\\7yv$3Yx#F1$!3m\")3Vc[DF]F-7
$$!3Um;z>RjiFF1$!31f[%yd-G-'F-7$$!3r*\\7y+&oQFF1$!3)>AoexS:u(F-7$$!3+L
L$e4OZr#F1$!3DhW+VYh<!*F-7$$!3n\\7`%pb7q#F1$!3#[aO4kOE^*F-7$$!3Om\"HKH
vxo#F1$!3i6'Q)*40[$)*F-7$$!3qC\"yD4N5o#F1$!3=xzQ*)e6H**F-7$$!3/$3F>*[H
uEF1$!3O#*)*og=Fy**F-7$$!3QTgF\"pavm#F1$!3wfP\"zX\\?)**F-7$$!3s**\\i!
\\93m#F1$!3'H(R0+7VS**F-7$$!31L3-)o`Qj#F1$!3\"p[SA)3%oK*F-7$$!3UmmT&)G
*og#F1$!3)R'oe*GW$Q!)F-7$$!35$e9T[7Mf#F1$!3#RP/yoZ$orF-7$$!3y*\\7G3K*z
DF1$!3=b$=\"oB,ohF-7$$!3Y;/^\"o^kc#F1$!3%p7yf;'[b]F-7$$!39L$3-GrHb#F1$
!3!)H4(>`_4&QF-7$$!3\")\\i!*y3\\RDF1$!3%Qv/)4VEwDF-7$$!3\\mTgx/,EDF1$!
3C*HhPAZXD\"F-7$$!3;$3-j2ID^#F1$\"3W.AC\")H(>-*F[`l7$$!3[*****\\n\\!*
\\#F1$\"3oFzLt$ROV\"F-7$$!3s**\\(=#R^&[#F1$\"3%)[wmMMmcFF-7$$!3e***\\(
o\"y>Z#F1$\"32-Ke;kbHSF-7$$!3Y**\\i:CWeCF1$\"3#=d_]pN!H_F-7$$!3L****\\
im!\\W#F1$\"3gq0(*4<;LjF-7$$!3V***\\i:NyT#F1$\"3s-H&*esow\")F-7$$!3a**
****\\Ow!R#F1$\"37ExIs#yeU*F-7$$!3S**\\(o*yAxBF1$\"3oo=qmiF(z*F-7$$!3k
***\\P9#pjBF1$\"37V!o]Ms(*)**F-7$$!3e*\\(=nU#pN#F1$\"3Z%R\")oGq<+\"Ffn
7$$!3')**\\i!Rc,N#F1$\"3#*=StC)[)****F-7$$!3!)*\\iS^)QVBF1$\"3o1GXWCHO
**F-7$$!3t****\\P1iOBF1$\"3*RuJ'>_KF)*F-7$$!3$)***\\78\\&4BF1$\"35T?t.
0S]*)F-7$$!3%******\\ixCG#F1$\"3(p-GRd0HU(F-7$$!3u***\\(=MVpAF1$\"3;Gt
]jw)e['F-7$$!3!*****\\7#*QcAF1$\"3)yFt#)*4+RaF-7$$!3/++D1]MVAF1$\"3F[.
%*[r.+VF-7$$!3%)*******z+.B#F1$\"3'\\U-0C_$)3$F-7$$!3k***\\Pfcs@#F1$\"
3Y_#*et!QX#=F-7$$!3z****\\(Q7U?#F1$\"3jJ$Q[]A2I&F^r7$$!3%****\\7=o6>#F
1$!3*>`<[\\n/t(F^r7$$!3w*****\\(R7y@F1$!3#394ZMtE1#F-7$$!3c***\\(o(z];
#F1$!3](zya7\"*oJ$F-7$$!3q****\\ib._@F1$!3lvV<M_Q9XF-7$$!3')***\\iN\"*
*Q@F1$!3J(\\kW_/[j&F-7$$!3m******\\r%f7#F1$!3[Rg)fM2\"fmF-7$$!3g****\\
P(e)*4#F1$!3#fExQ#zl^$)F-7$$!3c*****\\KqP2#F1$!31H#3,pauZ*F-7$$!353x1O
\\#>1#F1$!3KNk\"3>&Qz(*F-7$$!3I;a8Z&z+0#F1$!3^^P/\\[#R%**F-7$$!3dq#pE&
o:W?F1$!3g\"y'zw\\(Q(**F-7$$!3[CJ?eTBQ?F1$!3)\\'p7WPwo**F-7$$!3vyptj9J
K?F1$!39o$G^d3'G**F-7$$!3.L3Fp()QE?F1$!3)pQD<y\\N&)*F-7$$!3w\\iS\"*zp-
?F1$!39Q1DGhM6#*F-7$$!3]m;a8s+z>F1$!3uTQ0Cj:`!)F-7$$!3A$3xcV;`&>F1$!3^
]3#oEYOW'F-7$$!3'**\\7ylD;$>F1$!3#RnXm!GpsWF-7$$!393-))o-y>>F1$!3CJ\\P
(*QW&Q$F-7$$!3L;z%*z[$z!>F1$!3Yca\"e'GO]AF-7$$!3)[i:5\\*3'*=F1$!3_\\>k
BHN$3\"F-7$$!31LL3-TC%)=F1$\"3!41SA#H`B**F[`l7$$!3e\"HK*)ye,(=F1$\"3c<
OMGrz-:F-7$$!3u\\7yvM2c=F1$\"3?(\\srinr(GF-7$$!3)y?IE;))>%=F1$\"3MT8(>
yL^>%F-7$$!3/m\"z%\\G!z#=F1$\"3.O%*4[**fIaF-7$$!3>C\"Gj`<Q\"=F1$\"3ECu
ZeZ5flF-7$$!3q#3xJAK(*z\"F1$\"3I+YYKcIevF-7$$!3ATg-5pk&y\"F1$\"3DK#z&=
8U3%)F-7$$!3Q**\\(ofh:x\"F1$\"3q;(3dqBE4*F-7$$!3_dRs$Gwuv\"F1$\"3_+yGU
6P(f*F-7$$!3/;Hdq4RV<F1$\"3'GAl9mwE\"**F-7$$!36&R(*RJ[jt\"F1$\"3?P-I_*
*=(***F-7$$!3bu=UdcIH<F1$\"3Yh&e*z0B.5Ffn7$$!3i`j%3+jAs\"F1$\"3W[*=!\\
^y,5Ffn7$$!3qK3FW.A:<F1$\"3-Nu%)*3+R&**F-7$$!3O\\(ozr\\qo\"F1$\"3$GwJY
)G88#*F-7$$!3/mmm\"4z)e;F1$\"3Mt#G^um)[xF-7$$!3ImT&Q)p%pk\"F1$\"3Pn_2E
'eo$pF-7$$!3?m;/w[,N;F1$\"3^%RJ'*yPn-'F-7$$!36m\"H#oF3B;F1$\"3^.'>Apa9
.&F-7$$!3PmmTg1:6;F1$\"3'R4Q?.r^'RF-7$$!3GmTg_&=#*f\"F1$\"3.`dAm$eI%GF
-7$$!3Om;zWkG(e\"F1$\"3co_SnB3\"o\"F-7$$!3Ym\"zpLa`d\"F1$\"3c&HAi(yvd
\\F^r7$$!3Omm;HAUj:F1$!3EO^6<k'*fpF^r7$$!3WmTN@,\\^:F1$!3)>:(eI!ys(=F-
7$$!3`m;a8!e&R:F1$!3I?bYh/EJIF-7$$!3Wm\"Hd!fiF:F1$!3dV7f,S_TTF-7$$!3_m
m\"zz$p::F1$!3_cWj(>qA>&F-7$$!3_m;H#eH=\\\"F1$!3UR(fu/gk0(F-7$$!3ommmm
`'zY\"F1$!372@iAT3=&)F-7$$!3A$3x1h0-W\"F1$!3m+Y@:]B.'*F-7$$!3$**\\(oae
W79F1$!3Iw>iksJ\\**F-7$$!3k;zp)4'o%Q\"F1$!3H@%RA')y'H&*F-7$$!3=L$3FMEp
N\"F1$!3-kK\\ppGw$)F-7$$!3WTNrkk/V8F1$!3_wZS!QR*\\vF-7$$!3q\\(=nem\"H8
F1$!3)RB-$o%zsd'F-7$$!39eRs3nG:8F1$!33GA-d>*pZ&F-7$$!3Tm\"H2$oS,8F1$!3
3V8N2c@qUF-7$$!3ouVt_p_(G\"F1$!3GOVN&fO,)HF-7$$!37$eRZ2ZOF\"F1$!3bNkN3
DaJ;F-7$$!3S\"zWn>n(f7F1$!3v)Q7(p`X.DF^r7$$!3m***\\(=t)eC\"F1$\"3N%=cf
'f\"p8\"F-7$$!3G\"zp[F,OB\"F1$\"3CN*)zpLa[BF-7$$!3!He*)4B:8A\"F1$\"36c
Ju'G(yDNF-7$$!3qu$4r=H!47F1$\"3KR)eRC@4l%F-7$$!3Mm\"HK9Vn>\"F1$\"3!>1a
@/0qq&F-7$$!3w\\(oa0r@<\"F1$\"3iX>c=&4(\\vF-7$$!3-L$3x'*)fZ6F1$\"3R&R#
zt\"*QV*)F-7$$!3E;z%*zo-B6F1$\"3%)p,I_2_/)*F-7$$!3o*\\(=#za%)4\"F1$\"3
Y+'>*=)e\"35Ffn7$$!37$3FWq#)Q2\"F1$\"3KS$*p2)y\"e(*F-7$$!3Ommm;1J\\5F1
$\"3K`6QD3(R&))F-7$$!3fuoHz3$e.\"F1$\"3!GMoDl)fH\")F-7$$!3*H3F>9^B-\"F
1$\"3EZd]v'*\\fsF-7$$!3R\"HdXSr)35F1$\"3OrZmm)3&fiF-7$$!31'*\\(=n;R&**
Ffn$\"3Iz(*)R))>y9&F-7$$!3Hyq<)H>\">)*Ffn$\"3U-Vx1^lWRF-7$$!3Gi\"zW#>K
%o*Ffn$\"3#3.iO())*=n#F-7$$!3IY7y]X_\\&*Ffn$\"3YpVnx0q_8F-7$$!3^GL3xrs
9%*Ffn$\"3<_U/8?[06F[`l7$$!3u5aQ.)H*z#*Ffn$!3y(Ri$ePjG8F-7$$!3u%\\(oHC
8X\"*Ffn$!3@\"fgN2$*>k#F-7$$!3ty&*)f0N.,*Ffn$!3^X-t)eD^!RF-7$$!3'4m\"H
#oPb())Ffn$!3C\\)o4oT]4&F-7$$!3=VPf3.uS()Ffn$!3+*)*3(=l2!>'F-7$$!3=Fe*
[$H%fg)Ffn$!3+*y)Q&)GGqrF-7$$!3=6z>hb9r%)Ffn$!3lM%z/#Gz<!)F-7$$!3S$***
\\(=[jL)Ffn$!3!GNvX^Trr)F-7$$!3%R*\\PM-jz!)Ffn$!3-vo+Vyh&f*F-7$$!3\\%*
*\\7G7H#yFfn$!3-ViqyoRM)*F-7$$!3/&*\\7GV>mvFfn$!3$)>Ut_21<%*F-7$$!3f&*
***\\Pw%4tFfn$!3q)oIoA4,P)F-7$$!3'e\\P%)R<6=(Ffn$!3Z8UO])p,j(F-7$$!39'
*\\(=UeF0(Ffn$!3ctO!3P47w'F-7$$!3S'\\7`W*RCpFfn$!3wZ[;JsQxdF-7$$!3o'**
\\(o//'z'Ffn$!3il?.#GWZp%F-7$$!3%p\\(=#\\\"onmFfn$!3F*=)\\Y\"Q4`$F-7$$
!3A(*\\i:DKRlFfn$!3'z&)f#3>&\\I#F-7$$!3](\\i!RN'4T'Ffn$!3*yZ\">14yO5F-
7$$!3y(***\\iXg#G'Ffn$\"3#=)pQDA(*GDF^r7$$!379z>Oan[hFfn$\"3!yuI6(pt)f
\"F-7$$!3[Ie*)4ju9gFfn$\"3k5Qu\"*)[7#HF-7$$!3#ou$f$=<3)eFfn$\"3-nz,T%4
q>%F-7$$!3?j;Hd!))ou&Ffn$\"3Mt$*\\,EX.aF-7$$!3az&*)4$*eHh&Ffn$\"3&>qr)
fhF>lF-7$$!3!f\\(o/)H!zaFfn$\"3,=2>Jc#[_(F-7$$!3E7aQy15X`Ffn$\"3*3#=2)
**[CS)F-7$$!3hGL3_:<6_Ffn$\"3iWq&\\'f\"o8*F-7$$!3Jh\"z%*H8L%\\Ffn$\"3-
zyOx!pF,\"Ffn7$$!3-%*\\(o/ban%Ffn$\"3>yId\"yiI/\"Ffn7$$!3tE3F%z'f2WFfn
$\"3!o8d'oV'G+\"Ffn7$$!3WfmmT&Q(RTFfn$\"3'=C_8XBg&*)F-7$$!3!*f\"z%\\5w
$*QFfn$\"3G0h&Gq'*4X(F-7$$!3Qg;HdNyZOFfn$\"3#H[y4*\\[ZbF-7$$!3h5z>6[zC
NFfn$\"3Sj3Zuu%Q[%F-7$$!3%3;/^11=S$Ffn$\"3#>S)*zBo#oLF-7$$!336/,>t\")y
KFfn$\"3X#QF!f7'*=AF-7$$!3Ihm\"HdGe:$Ffn$\"3/u/[`_#[0\"F-7$$!3`6H#o#)R
G.$Ffn$!3yl7r\\m7[5F^r7$$!3xh\"H23^)4HFfn$!3PWMFwSXS7F-7$$!3+7ajMB'oy#
Ffn$!39J=g.?oKBF-7$$!3Ai;a)etQm#Ffn$!3))*>t@1/CO$F-7$$!3piTN'4'*yT#Ffn
$!3Q^he=G$4;&F-7$$!3;jm;/'=><#Ffn$!3%H%Gt]a\"z\\'F-7$$!3xz?IE'e#R?Ffn$
!3kFB&*4o6%)pF-7$$!3Q'\\P%[')f1>Ffn$!3]#Gb=3?SG(F-7$$!3o/_]f'o-%=Ffn$!
3EHZX?$R%ftF-7$$!3+8Hdq'QRx\"Ffn$!3ozIPq/\"HQ(F-7$$!3I@1k\"o3wq\"Ffn$!
3#>H$R*H%)GN(F-7$$!3gH$3Fpy7k\"Ffn$!3)*z>1(=HzE(F-7$$!3#G;zptefP\"Ffn$
!3]K=4%Q&[ajF-7$$!3/'**\\7yQ16\"Ffn$!3+@.<=Ze#[%F-7$$!3aET&Q.)yz(*F-$!
3Gbj^7-Q#=$F-7$$!3k#H3_D)=`%)F-$!3%*f]gg+W\\;F-7$$!3veCcw%)eErF-$\"3J`
g&)p_vL$*F[`l7$$!3'[i;zp))**z&F-$\"3P*>:1,nD+#F-7$$!3\"zq$f3))oO^F-$\"
3?Qncmg`)*HF-7$$!3'4zq#>*)QtWF-$\"3(Gz![Xp(o+%F-7$$!3-uy%*H!*35QF-$\"3
<W.\"yrsP,&F-7$$!31d\\iS\"*yYJF-$\"3<]qqo/)Q+'F-7$$!3=B\"z>O*=?=F-$\"3
y%p2)z.;pyF-7$$!3%G*GLLe*e$\\F^r$\"3)=0K)epI#[*F-7$$\"3o6@/EvuV))F^r$
\"3s&RZ&[<c!3\"Ffn7$$\"3i6<a)3RBE#F-$\"3?1#eeC>c<\"Ffn7$$\"327#zWU..k$
F-$\"34SOc\\9.R7Ffn7$$\"3_7nTgxE=]F-$\"3+!o([C$e)z7Ffn7$$\"3U8<HKk>uxF
-$\"3\\CO3(*etC8Ffn7$$\"3WrmT5D,`5Ffn$\"3_FEL:O6V8Ffn7$$\"3YYN@F2\">7
\"Ffn$\"3!4WSJ,nOM\"Ffn7$$\"3Y@/,W*33>\"Ffn$\"3`C4O3)HAM\"Ffn7$$\"3]'H
23;2(f7Ffn$\"3)*zPCs%p&Q8Ffn7$$\"3_rTgx`gG8Ffn$\"36;4xdLXK8Ffn7$$\"3e@
z>6=Sm9Ffn$\"3SXoJsC)>J\"Ffn7$$\"3hr;zW#)>/;Ffn$\"3G4O:gOGz7Ffn7$$\"3q
r\"z>6\"zz=Ffn$\"3m#Q*)oG9K<\"Ffn7$$\"3!=nm\"zRQb@Ffn$\"3U+Zy\"Hm5,\"F
fn7$$\"370vozhLvAFfn$\"3\"o'*pP#\\'zB*F-7$$\"3WQ$3-Q)G&R#Ffn$\"3]q!z6c
]NF)F-7$$\"3yr\"H2eS_^#Ffn$\"3MRYDwC&oA(F-7$$\"360+D\"y#>NEFfn$\"3geQ)
*yW44hF-7$$\"3WQ3x\")\\9bFFfn$\"3v>Y9Qr.L\\F-7$$\"3wr;H#=(4vGFfn$\"3B#
H/IN:Fr$F-7$$\"350D\"GQ\\]*HFfn$\"3qP*p+3qJY#F-7$$\"3VQLL$e,]6$Ffn$\"3
JyA%eMw,?\"F-7$$\"3wrT&Qy`\\B$Ffn$!3DDt)e/>I+'F[`l7$$\"330]P%)f!\\N$Ff
n$!3#=Dk$oc-,8F-7$$\"3UQe*[=e[Z$Ffn$!30i)f8\"oY1DF-7$$\"3urmT&Q5[f$Ffn
$!3=)eZBG`.m$F-7$$\"330v$feiZr$Ffn$!3UT0(3u!GZZF-7$$\"3TQ$eky9Z$QFfn$!
3oy]\"Hr\"o_dF-7$$\"3ur\"zp)pmaRFfn$!3s[dRG90jmF-7$$\"310+](=>Y2%Ffn$!
3g*=c#))e:muF-7$$\"35Q3-8DsLVFfn$!3S$o#)>o_gy)F-7$$\"39r;aQe#Gf%Ffn$!3
y8/]Ay*fZ*F-7$$\"3m(3-8]xBs%Ffn$!3q%Rv[CK.d*F-7$$\"3=/D1k\"H>&[Ffn$!3+
4i@'zA`\\*F-7$$\"3q?H#o#3[\")\\Ffn$!3ax*)=1zv_#*F-7$$\"3APLe*[K56&Ffn$
!3anb%QR)=Z))F-7$$\"3EqT5:e8q`Ffn$!3E8(R)3xdyvF-7$$\"3I.]iS\"R#HcFfn$!
3c>A`/@JxdF-7$$\"3#)>aQ.3zedFfn$!3qh#>\"Q5=9ZF-7$$\"3MOe9mCM))eFfn$!3)
>^\\+*QImNF-7$$\"3'GD1*GT*y,'Ffn$!3)oKU`@%>`BF-7$$\"3Qpmm\"zXu9'Ffn$!3
I.[9^%Ra4\"F-7$$\"3E-v$fL&G\"G'Ffn$\"3WM)G958>G#F^r7$$\"30O$3-)[7:kFfn
$\"35TyoEuz_:F-7$$\"3#)p\"zWUk*[lFfn$\"338(\\,&*eW&GF-7$$\"3q-+voR!Go'
Ffn$\"3eaC'Q[\"o4TF-7$$\"3gN3-8Nk;oFfn$\"3;x#>P4:eH&F-7$$\"3Qp;HdI[]pF
fn$\"3a2`,a#e9R'F-7$$\"3;.Dc,EK%3(Ffn$\"3w!z$*feRoP(F-7$$\"3/OL$e9i\"=
sFfn$\"3Fa\"QkaqTB)F-7$$\"3s-]PM7%e[(Ffn$\"3'o)[[0UI0&*F-7$$\"3Qpm\"HK
?Nv(Ffn$\"3J?O3W+K65Ffn7$$\"3q'3_]4S/#yFfn$\"3\"Rf,pBDb,\"Ffn7$$\"39.v
=n)ft)yFfn$\"3\"QrZ&oJE:5Ffn7$$\"3g>HKR'zU&zFfn$\"3*y$)yCfL0,\"Ffn7$$
\"3$pLe9T*>@!)Ffn$\"3CV'R(*Hc8+\"Ffn7$$\"3#)p\"Hd&*Q]:)Ffn$\"3)*3_usGP
)p*F-7$$\"3q-+++&y))G)Ffn$\"3wSah!o&f7#*F-7$$\"3%=+DcE]2b)Ffn$\"3/TYT/
Q7/yF-7$$\"3w-+DJ?i7))Ffn$\"3ykMc'*zXteF-7$$\"3L-D19zbV*)Ffn$\"3kpJuFj
3_ZF-7$$\"3o.](oz$\\u!*Ffn$\"3_[9)oUG;b$F-7$$\"3D.voz'Ha?*Ffn$\"3%3$>H
Cuc#H#F-7$$\"3#G++DcljL*Ffn$\"3;y*=q&z#R'**F^r7$$\"3S-DJX9In%*Ffn$!3#*
=f,YrnZJF^r7$$\"3'>+D\"GtB)f*Ffn$!3#fQtT3F&=;F-7$$\"3J.v$4@t\"H(*Ffn$!
35`^JHCi#*GF-7$$\"3)G+]P44,')*Ffn$!3W4?a&G'H:TF-7$$\"3Y-Dcw\\/\"***Ffn
$!3yL:9D#pcE&F-7$$\"3Q+v$f3)>75F1$!3#=#*pNV(4CjF-7$$\"3M](=Un\"HD5F1$!
3Y$=\\Q74DF(F-7$$\"3I++]i_QQ5F1$!3-xPMISr%4)F-7$$\"3Q]PM-Fsj5F1$!3g0zH
WS<x#*F-7$$\"3G+v=U,1*3\"F1$!3YcF2dcMh)*F-7$$\"3iP%[@]%R&4\"F1$!3BS`&*
pSo3**F-7$$\"38v$4@')G<5\"F1$!380j6m-$f\"**F-7$$\"3l7.2AK136F1$!3%\\lc
lJfI))*F-7$$\"3<]7.#e(R96F1$!3W5w)482-\")*F-7$$\"3@DJ&>Imq7\"F1$!3M%[Q
4)H!fa*F-7$$\"3D+](=-N(R6F1$!3SD'=\\=&GF\"*F-7$$\"3K](=<Ys];\"F1$!3[6%
*zC;VcyF-7$$\"3A+Dc,*4/>\"F1$!37o12tZ-zgF-7$$\"33vV[@'yI?\"F1$!3Gno^+:
#[.&F-7$$\"37]iSTtu:7F1$!3o'Q@*HWx3RF-7$$\"3;D\"G81;%G7F1$!3>PcJJl&*=F
F-7$$\"3>++D\"y%3T7F1$!3Eye\\B@Y%[\"F-7$$\"37DcwM7:b7F1$!3_jNOS>=]&)F[
`l7$$\"3/]7G)o<#p7F1$\"30!)es4.W;8F-7$$\"39vozTTG$G\"F1$\"3#H(e_66k$p#
F-7$$\"31+DJ&f]tH\"F1$\"3dGdkkr')=SF-7$$\"3)\\7G)[qT68F1$\"35[ni+F\"fE
&F-7$$\"33]PM-N[D8F1$\"38#>0\"Gs65kF-7$$\"3+v$fe&*\\&R8F1$\"3>#3xSj_)G
uF-7$$\"3#***\\P4kh`8F1$\"3O?\">*R,(>I)F-7$$\"3w\\iS;$\\<Q\"F1$\"3=+!R
81%[X&*F-7$$\"3y*\\PMA#))49F1$\"3T_yK=0F/5Ffn7$$\"3\")\\(o/8:!Q9F1$\"3
'pU>LRoVv*F-7$$\"3m****\\P![hY\"F1$\"3f+%*e>O)Hq)F-7$$\"3+$e9;P(yy9F1$
\"3e76F)>z?+)F-7$$\"3Om\"HdqE9\\\"F1$\"31Y9Y>aDurF-7$$\"3r\\P%)Rg1/:F1
$\"3g[QgUlqKiF-7$$\"32L$eRP0n^\"F1$\"3O]K>J:W#>&F-7$$\"3V;H23ZMH:F1$\"
3+j/6)eW+2%F-7$$\"3y*\\(=US)>a\"F1$\"3s3*Q+c5M)GF-7$$\"39$3-jPBYb\"F1$
\"3ECCIK%f9l\"F-7$$\"3]mmT5FEn:F1$\"3B;Bc!pW$QRF^r7$$\"3')\\7`W?!*z:F1
$!3u%fJJ&=6%p)F^r7$$\"3?Leky8a#f\"F1$!3:P>jVW8=@F-7$$\"3Q;/w72=0;F1$!3
+ew&zkBCL$F-7$$\"3#***\\(o/?yh\"F1$!3gy^,\\i\"H\\%F-7$$\"3X$e*)4Qf/j\"
F1$!3I%\\-[s26e&F-7$$\"3jmT5:()4V;F1$!3'*G%*))4okzlF-7$$\"3#)\\(=#\\!Q
dl\"F1$!3!f%*H`K7EZ(F-7$$\"3MLLL$Qx$o;F1$!3M#R_E>ndC)F-7$$\"3ym;HdxO&p
\"F1$!3&z0zM`u:W*F-7$$\"3@++DJ\"eBs\"F1$!3Wb`\")RV=^**F-7$$\"3mL$3_][$
\\<F1$!3ZYBz\\$yxt*F-7$$\"35nm;z)Qjx\"F1$!3d%>GB\"e)o\"))F-7$$\"3!Q$e9
mS$)*y\"F1$!3/7]3U$*45\")F-7$$\"3_+]7`#HL!=F1$!3'))pNuHf_D(F-7$$\"3CnT
5SW#o\"=F1$!3c'=#QS\">zE'F-7$$\"3'RL$3F'>.$=F1$!3K6iao:/m^F-7$$\"3o+D1
9[\"Q%=F1$!3?s_]!)=npRF-7$$\"3Sn;/,+Jd=F1$!3a**34qMd+FF-7$$\"37M3-)=03
(=F1$!3gn3(z2K=Q\"F-7$$\"3%3++]P+V)=F1$!3i(f)f'p*fVPF[`l7$$\"3U#H#=?!H
l*=F1$\"3_1_%>6'[#=\"F-7$$\"3+%ek`md(3>F1$\"3stHTM$o^Q#F-7$$\"3fvoa5j)
4#>F1$\"3+_U,UFk_NF-7$$\"3=n\"Hd&\\@L>F1$\"3avl'GNnum%F-7$$\"3we9\"4gV
a%>F1$\"3s()\\-Jm)Hr&F-7$$\"3O]P4YAnd>F1$\"3![L?,C!etmF-7$$\"3%>/w7*3!
*p>F1$\"3qSG,Oj*[`(F-7$$\"3_L$ek`H@)>F1$\"3K4-:9f1%G)F-7$$\"3p;H#o#oe1
?F1$\"30&o))Q)=..%*F-7$$\"3()*\\(=<T/J?F1$\"3QKC!f`TQ'**F-7$$\"3X\"zpB
wsK/#F1$\"3A;R2LHJ-5Ffn7$$\"3/$3_vS,b0#F1$\"3y6WYk,3L**F-7$$\"3kuVt_+t
n?F1$\"3/FhC4Y.&p*F-7$$\"3Amm\"zpe*z?F1$\"3j!4sJpVDJ*F-7$$\"3!H3-j(pp1
@F1$\"3]MvH\"*Qu.!)F-7$$\"3&**\\(oa_VL@F1$\"3g5Hyd$[x7'F-7$$\"3I3-)QR/
o9#F1$\"3d1R733P<]F-7$$\"3+<H2LN<g@F1$\"3^@C*>uSy\"QF-7$$\"3MDcEsEat@F
1$\"3-R(e9.h0b#F-7$$\"3qL$e9\"=\"p=#F1$\"3dKHw_P9Q7F-7$$\"3.U5l]4G+AF1
$!3iB/\"\\(oL*f*F[`l7$$\"3Q]P%)*3]O@#F1$!3KkY'=*[/G9F-7$$\"33fk.H#>qA#
F1$!3>?q:7DCMFF-7$$\"3Vn\"H#o$)QSAF1$!3#zL,=Dz7*RF-7$$\"3yv=U2vv`AF1$!
3uZJ8;asw^F-7$$\"3[%e9mkErE#F1$!3!emkM*)G%piF-7$$\"3#GH2ey&\\!G#F1$!3C
)o(4aE*)\\sF-7$$\"3;,++D\\'QH#F1$!3\"*)[#R6Fi+\")F-7$$\"3i<z>hN->BF1$!
3C[%*yt&=&)H*F-7$$\"32MeR(>#=WBF1$!3?l)*)pWu(4**F-7$$\"3I#z%\\::wcBF1$
!3Ov3H4w'=)**F-7$$\"3_]PfL3MpBF1$!3%H/UFcHf*)*F-7$$\"3u3Fp^,#>Q#F1$!3'
RJP)e9K`'*F-7$$\"3'pm\"zp%*\\%R#F1$!3/$))*yn-)yD*F-7$$\"3U$e*)f5e'>CF1
$!3/+!*=AO#e.)F-7$$\"3()*\\(=Un\"[W#F1$!3V#z4J_EnI'F-7$$\"35ekGggRdCF1
$!3O6^k0Y*\\G&F-7$$\"3K;aQy`(*pCF1$!33gBj/(z%zTF-7$$\"3cuV['paD[#F1$!3
%\\rf00bw+$F-7$$\"3yKLe9S8&\\#F1$!3U/e6I</)y\"F-7$$\"3`*\\il-$G3DF1$!3
p*>?'f:LI[F^r7$$\"3Hm;aQ?V@DF1$\"3S%p+``9fI)F^r7$$\"30L3_]5eMDF1$\"3KA
1%>z\\,8#F-7$$\"3!)****\\i+tZDF1$\"3Ur^.\\C?$R$F-7$$\"3cm\"zW2z3c#F1$
\"3q[ft?.%zf%F-7$$\"3KL$ek3GSd#F1$\"30)[Oq8hNs&F-7$$\"33+vV)4xre#F1$\"
3W]l>2'G1v'F-7$$\"3%om;/6E.g#F1$\"3GtGm\")zShwF-7$$\"3O+]PMTiEEF1$\"3!
)R>')*ftM2*F-7$$\"3)QLL$e@#Hl#F1$\"3C'4gOYUE')*F-7$$\"33<HKkm\\fEF1$\"
3?y+UB\"Q_&**F-7$$\"3k+DJq62mEF1$\"3pl'y],)[+5Ffn7$$\"3=%3-jnXEn#F1$\"
3_%HcWPN6+\"Ffn7$$\"3Sn;H#=?#zEF1$\"3].\\TI1ju**F-7$$\"39M3F%>pBp#F1$
\"3[4;sTPTs(*F-7$$\"3!4+]i?=bq#F1$\"3Abp\"RL=<S*F-7$$\"3V</^J$H7t#F1$
\"3IxWm`#*)p@)F-7$$\"3'R$3xc/%pv#F1$\"335\"Q6HRH\\'F-7$$\"3/U5S>gzpFF1
$\"3i(4z;\\))GY&F-7$$\"3[]7.#e^Ey#F1$\"3nLG#))y&*GM%F-7$$\"3#*e9mWr]&z
#F1$\"3;\")))e5LW^JF-7$$\"3*pm\"H2FO3GF1$\"3FE!*yqD>3>F-7$$\"33v=#*p#=
7#GF1$\"3)z65!)y*fOjF^r7$$\"3_$3_D$Q2MGF1$!3:@2s'f<7^'F^r7$$\"3'>H#=&R
Hp%GF1$!3$Gp#oi2&\\#>F-7$$\"3/+D\"y&\\yfGF1$!3c9)fT)e!o;$F-7$$\"373FW?
0ksGF1$!3!y***=VO>cVF-7$$\"3c;H2$3'\\&)GF1$!3)>+2[W'[taF-7$$\"3+DJqX;N
)*GF1$!3v0<O5lC+lF-7$$\"34LLL3s?6HF1$!3$QS!Rr,`>uF-7$$\"3`;zptV7QHF1$!
3)[%[h+GjJ*)F-7$$\"3)**\\i!R:/lHF1$!31pR=Yrm*z*F-7$$\"3mXOSI3xrHF1$!3Z
TkIuwh2**F-7$$\"3q\"zW<7+&yHF1$!3#=<j9co1(**F-7$$\"3sPf38%H_)HF1$!3o4j
qLY`))**F-7$$\"3S$3FWqe>*HF1$!3/F!G.DN6'**F-7$$\"37v$4rG<a+$F1$!3sIp%G
[T7x*F-7$$\"3&om\"zpe()=IF1$!3]#[ep]BWS*F-7$$\"3G]i:NIzXIF1$!3oDND`!R'
p\")F-7$$\"3sL3_+-rsIF1$!3K:\"H!*Q@dM'F-7$$\"3WDJ?$yoh3$F1$!3k.'G5#HZ_
_F-7$$\"3;<a)eOF'*4$F1$!3gH0POs.kSF-7$$\"3))3xc[f38JF1$!3e_5Rp+\">!GF-
7$$\"3h++DJXaEJF1$!3yG5S=*>*)[\"F-7$$\"3SUN'fu1&RJF1$!3/2y\\HLS%)>F^r7
$$\"3?%3x1'*oC:$F1$\"3))H9&)\\)Rb4\"F-7$$\"3kD1Rv6VlJF1$\"3EOFD;$38P#F
-7$$\"3WnT5!R$RyJF1$\"3Z%=wJLeug$F-7$$\"3C4x\"[gb8>$F1$\"3'y%>U=zC$y%F
-7$$\"3o]7`>yJ/KF1$\"39j>Vdy%*yeF-7$$\"3$GzWU.!G<KF1$\"3yT*\\ussh(oF-7
$$\"3GM$e*[ACIKF1$\"3K'3D]P*=exF-7$$\"3)yT&Qym;cKF1$\"3,j#)4L)y&>\"*F-
7$$\"3w+D\"y5\"4#G$F1$\"3k)4e/a@@()*F-7$$\"3%=Fp^@s&)G$F1$\"3g]7-_dFe*
*F-7$$\"3!H/EDK`]H$F1$\"3p6u$=Tl-+\"Ffn7$$\"3)R\"G))HW`,LF1$\"3=4R09q]
+5Ffn7$$\"3N%eRs`:!3LF1$\"3!o^[0_8b'**F-7$$\"3zDJ&>vx4K$F1$\"3S3Nj'o@8
w*F-7$$\"3&zmmm'*RRL$F1$\"3ST?A@M]$R*F-7$$\"3knT5S9XgLF1$\"3Kt9`UTWg\"
)F-7$$\"3Kn;a8H'pQ$F1$\"3)HSz8W%zdjF-7$$\"3_</E]'=-S$F1$\"3'GfE%\\@!4G
&F-7$$\"3sn\"zpQuMT$F1$\"3J3vuJ\\]6TF-7$$\"3#z\"zpB,tEMF1$\"3k.XaU&>,(
GF-7$$\"3TnmTge)*RMF1$\"37G9lNe_y:F-7$$\"3!pTNrfTKX$F1$\"3Y^=X1i%Qf#F^
r7$$\"35nT&QL(\\mMF1$!3)Q`f!Q*fT1\"F-7$$\"3I<HdqIvzMF1$!3MX-$HS'))oBF-
7$$\"3]n;H2)3I\\$F1$!3[*>Z3].>j$F-7$$\"3p</,WXE1NF1$!3/'\\]ww^5$[F-7$$
\"3=n\"H2G?&>NF1$!3;#)Hb_BHXfF-7$$\"3o;zW<gxKNF1$!3g@xbgh2bpF-7$$\"3)o
mmTvJga$F1$!3`0B+cooUyF-7$$\"3?](=#*pBBd$F1$!3IH;u2Y)G=*F-7$$\"3_L3FWc
h)f$F1$!3g*e0s,39*)*F-7$$\"3<aQ`I')=0OF1$!3Ye*)3`\">J'**F-7$$\"3`voz;;
w6OF1$!3sqAJ?ww\"***F-7$$\"3*o*)fIgM$=OF1$!3gVx#>\")Hs(**F-7$$\"3b<HK*
e2\\i$F1$!3V&*3yN'o&>**F-7$$\"3&)e*[=c`!QOF1$!3_u!z5&z0w'*F-7$$\"3)3+v
V`*>^OF1$!3aZ1[K(QaE*F-7$$\"3?%3F%z9\\xOF1$!3/rpg!z$)R(zF-7$$\"3Ao\"zW
U$y.PF1$!3%=)[Fk?(R8'F-7$$\"3_4_+(RHpr$F1$!3'yS#*e%*Gl/&F-7$$\"3a^7`p`
2IPF1$!3]+]'*)ft=(QF-7$$\"3c$Hd?M@Ku$F1$!3=]SJ%ox-j#F-7$$\"3'[L$e9tOcP
F1$!3I0.WK!oJM\"F-7$$\"3owo/)4Z%oPF1$!3?QXG*)om$R\"F^r7$$\"3]=/^\")o_!
y$F1$\"3!Gx<m#)ol1\"F-7$$\"3gfR(\\m1Ez$F1$\"3O0#e24iqD#F-7$$\"3T,vV[ko
/QF1$\"3@SFjZ<w9MF-7$$\"3BV5!>Bmn\"QF1$\"3o\\r:%R%zAXF-7$$\"3M%ek`,Y)G
QF1$\"3i1[N*\\5]c&F-7$$\"3:E\"G))zD4%QF1$\"3l#[,Px>i_'F-7$$\"3'zm\"H#e
0I&QF1$\"3UR,7eDT#R(F-7$$\"3g^(=#\\^;xQF1$\"37ika#3=3z)F-7$$\"3^Me9;ZK
,RF1$\"3K!>Umj&**y'*F-7$$\"3Lw$4'*\\/M\"RF1$\"3:a2)3FtV\"**F-7$$\"3W<H
2$G%[DRF1$\"332/Y_]`+5Ffn7$$\"3Dfk`mScPRF1$\"3Y8b-s3g]**F-7$$\"33,++]Q
k\\RF1$\"3I`?Hs;#4v*F-7$$\"3-oT5SMLxRF1$\"3O!f1+Olcw)F-7$$\"3BM$3-.B]+
%F1$\"3$zGX+(**48rF-7$$\"3q</EDy')=SF1$\"3+?=:X)pT2'F-7$$\"3Z+DJ?ErKSF
1$\"3+t=(Qu:\">\\F-7$$\"3%Rek`Tdl/%F1$\"3;([#e$RT+n$F-7$$\"3SnmT5ASgSF
1$\"3Q%e$*y\")\\3N#F-7$$\"3)3voa+ZU2%F1$\"3?=H&p)*ey')*F^r7$$\"3MM3_+=
4)3%F1$!3?f'HDTn/'RF^r7$$\"35<Hd&fO>5%F1$!351$f'*G&=r<F-7$$\"3d+]i!R\"
y:TF1$!3Ra+$*pTJ7JF-7$$\"3/%3xc=E'HTF1$!3#fN8(['oPR%F-7$$\"3!o;H2)4ZVT
F1$!3k%)**4))f-\"f&F-7$$\"3G]7yvdJdTF1$!3[TW=VZ<\"o'F-7$$\"3uLL$3dg6<%
F1$!3(H/FRA`Lk(F-7$$\"35+DJqk#f>%F1$!3=h!oUWX!*)*)F-7$$\"3=n;zpBp?UF1$
!3yk_>n%fey*F-7$$\"3[f9mWQ)oA%F1$!3SW(4qy:G*)*F-7$$\"32^7`>`2LUF1$!3I*
*H!QBR='**F-7$$\"3mU5S%zm#RUF1$!3#*fc;e`m#***F-7$$\"3EM3Fp#eaC%F1$!3M#
pN14w^)**F-7$$\"3W</,>7%yD%F1$!37!H1R:8b&)*F-7$$\"3i++voTAqUF1$!3/32*)
)fO[d*F-7$$\"3)p;H#o+*\\H%F1$!3=g***[r%))z&)F-7$$\"31M$3x'fv>VF1$!3y%H
&z!QQ51(F-7$$\"3&z\"zW<*Q@L%F1$!3,O*GkQ8I8'F-7$$\"38,v=n=_WVF1$!3e)QO>
i%)46&F-7$$\"3K%3Fp\"[!pN%F1$!3NKG6nSg5SF-7$$\"3]nmmmxGpVF1$!3Ge%z'*eB
([GF-7$$\"3'f7GQd!\\#Q%F1$!3%)G6(f/?Ac\"F-7$$\"3S%e*)4Q$p&R%F1$!3K)oJX
sMV[#F^r7$$\"3&G/^\")='*)3WF1$\"312p6xwwp5F-7$$\"3I,DJ&**)4AWF1$\"33kW
K8\"Q%pBF-7$$\"3wfRZ-=INWF1$\"3o^3;*4bzi$F-7$$\"3A=aj4Y][WF1$\"3ariqBA
TB[F-7$$\"3mwoz;uqhWF1$\"3V@iO22+NfF-7$$\"3SM$eRA5\\Z%F1$\"3j0Ci,;PVpF
-7$$\"3f]7GQeJ,XF1$\"3%)*)=t\\MN#e)F-7$$\"3]nTg_9sFXF1$\"3ag9k-rtE'*F-
7$$\"3&fil(fU#4a%F1$\"3]ZD*)G-c,**F-7$$\"3S%3Fp1FTb%F1$\"3ED'Hu.7/+\"F
fn7$$\"3'Ga)3u)Htc%F1$\"3#\\us](=jK**F-7$$\"3K,+D\"oK0e%F1$\"3kr7t!eQ$
)o*F-7$$\"3],vVBi!eg%F1$\"3+aXYjwgc()F-7$$\"3q,]il(z5j%F1$\"3sq))e)*y
\")osF-7$$\"3!=v=n`;Pk%F1$\"3;t8^a-rWjF-7$$\"3!>]7yI`jl%F1$\"3N[[+#4%
\\>`F-7$$\"3+_i!*y+**oYF1$\"3m;M2/(=&4UF-7$$\"3Q,++]oi\"o%F1$\"3hVVy*3
&[KIF-7$$\"3y]P4@OE%p%F1$\"3Jhr\\mO;2=F-7$$\"3(3](=#R+pq%F1$\"3?(*y\"[
F\\4`&F^r7$$\"3(4D\"Gjr`>ZF1$!33UA%H\"=A(4(F^r7$$\"32,]PMR<KZF1$!3)e,!
e&>\\6'>F-7$$\"3<^(oaq5[u%F1$!3i.6B#HH7=$F-7$$\"3G,DcwuWdZF1$!3Cej'=b0
0N%F-7$$\"3Q^ilZU3qZF1$!3EX&z#\\7L]aF-7$$\"3Y,+v=5s#y%F1$!3_U1BBs;jkF-
7$$\"3k^iS\"*3))4[F1$!3mBS\"3LWAF)F-7$$\"35,D1k2/P[F1$!3)>`X\"4k`u%*F-
7$$\"3[D1R+2i][F1$!3bD'3@%\\l=)*F-7$$\"3c](=nj+U'[F1$!3'z8FTu->)**F-7$
$\"358G)[g!*4([F1$!3)>2i0))GY***F-7$$\"3mvo/t0yx[F1$!3?Ar9[VFh**F-7$$
\"3?Q4@T0d%)[F1$!3?lXq=H*>))*F-7$$\"3u+]P40O\"*[F1$!3)[IKB-]rv*F-7$$\"
3\"4DJ?Q?&=\\F1$!39.%)[0\"fn\"))F-7$$\"3Q+voa-oX\\F1$!3&zl\"fhnnHsF-7$
$\"3uCc,\">g#f\\F1$!3k\"zkH4[$GiF-7$$\"3%)\\PMF,%G(\\F1$!3aFY6N4E7^F-7
$$\"3#\\(=nj+U')\\F1$!3`%y99Pm>!RF-7$$\"#]F*$!3,W&*RO,v>EF--%'COLOURG6
&%$RGBG$\"#5!\"\"$F*F*Fg_x-%*THICKNESSG6#\"\"#-%+AXESLABELSG6$Q\"x6\"Q
!F``x-%%VIEWG6$;F(F\\_x%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
151 "O gr\341fico indica que a integral sobre todo o eixo real pode n
\343o existir.  Deve-se ter um m\355nimo de cuidado, para que n\343o s
e calcule esta integral como " }{XPPEDIT 18 0 "Limit(Int((1+(1+x^2)*si
n(x))/(1+x^2),x = -a .. a),a = infinity);" "6#-%&LimitG6$-%$IntG6$*&,&
\"\"\"F+*&,&F+F+*$%\"xG\"\"#F+F+-%$sinG6#F/F+F+F+,&F+F+*$F/F0F+!\"\"/F
/;,$%\"aGF6F:/F:%)infinityG" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 59 "finiteint := int((1 + (1+x^2)*sin(x))/(1+x^2),x = \+
-a .. a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*finiteintG,$*&\"\"#\"
\"\"-%'arctanG6#%\"aGF(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
66 "Int(f(x) ,x = -infinity..infinity) = limit(finiteint, a=infinity);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&,&\"\"\"F)*&,&*$)%\"xG
\"\"#F)F)F)F)F)-%$sinG6#F.F)F)F)F+!\"\"/F.;,$%)infinityGF3F7%#PiG" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 287 26 "ESTA RESPOSTA EST
\301 ERRADA!" }{TEXT -1 96 "  Para calcular corretamente a integral, d
eve-se fazer com que as duas extremidades tendam a +/-" }{XPPEDIT 18 
0 "infinity;" "6#%)infinityG" }{TEXT -1 32 " separadamente.  Ao escolh
ermos " }{XPPEDIT 18 0 "c = 0;" "6#/%\"cG\"\"!" }{TEXT -1 27 ", a inte
gral impr\363pria at\351 " }{XPPEDIT 18 0 "infinity;" "6#%)infinityG" 
}{TEXT -1 17 " \351 calculada por " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "inttob := int(f(x), x=0..b);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'inttobG,(-%'arctanG6#%\"bG\"\"\"-%$cosGF(!\"\"F*F*" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "Int(f(x), x=0..infinity) \+
= limit(inttob, b=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$In
tG6$*&,&\"\"\"F)*&,&*$)%\"xG\"\"#F)F)F)F)F)-%$sinG6#F.F)F)F)F+!\"\"/F.
;\"\"!%)infinityG;,$*&F/F3%#PiGF)F),&F/F)*&F/F3F;F)F)" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 111 "A apar\352ncia um tanto estranha da resp
osta mostra que algo de errado aconteceu. De fato, tem-se que a expres
s\343o " }{TEXT 288 6 "inttob" }{TEXT -1 26 " n\343o possui limite qua
ndo " }{XPPEDIT 18 0 "proc (b) options operator, arrow; infinity end;
" "6#f*6#%\"bG7\"6$%)operatorG%&arrowG6\"%)infinityGF*F*F*" }{TEXT -1 
105 " .  N\343o h\341 necessidade de continuar: quando uma das integra
is impr\363pria semi-infinitas n\343o existe, ent\343o " }{XPPEDIT 18 
0 "Int((1+(1+x^2)*sin(x))/(1+x^2),x = -infinity .. infinity);" "6#-%$I
ntG6$*&,&\"\"\"F(*&,&F(F(*$%\"xG\"\"#F(F(-%$sinG6#F,F(F(F(,&F(F(*$F,F-
F(!\"\"/F,;,$%)infinityGF3F7" }{TEXT -1 19 " tamb\351m n\343o existe.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 
9 "Quest\343o 1" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 144 "Encontrar as i
ntegrais impr\363pria a seguir.  Caso n\343o seja poss\355vel, mostre \+
que n\343o existem.  (N\343o \351 aconselh\341vel calcular com um \372
nico passo do " }{TEXT 289 5 "Maple" }{TEXT -1 49 ": calcule-as como l
imites de integrais pr\363prias.)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "Int((2*x + 1)/(x^4 + 1), x=0..infinity);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#-%$IntG6$*&,&*&\"\"#\"\"\"%\"xGF*F*F*F*F*,&*$)F
+\"\"%F*F*F*F*!\"\"/F+;\"\"!%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 43 "inttob := int((2*x + 1)/(x^4 + 1), x=0..b);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%'inttobG,**&#\"\"\"\"\")F(*&\"\"##F(F+-%#l
nG6#*&,(*$)%\"bGF+F(F(*&F4F(F+F,F(F(F(F(,(F2F(F5!\"\"F(F(F7F(F(F(*&#F(
\"\"%F(*&F+F,-%'arctanG6#,&F5F(F(F(F(F(F(*&F9F(*&F+F,-F=6#,&F5F(F(F7F(
F(F(-F=6#F2F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "limit(intt
ob, b=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&#\"\"\"\"\"%F&
*&\"\"##F&F)%#PiGF&F&F&*&F*F&F+F&F&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "Int(1/(x*ln(x)), x=2..infinity);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'*&%\"xGF'-%#lnG6#F)F'!\"\"/F);\"\"#%
)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "inttob := int
(1/(x*ln(x)), x=2..b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'inttobG,&
-%#lnG6#-F'6#%\"bG\"\"\"-F'6#-F'6#\"\"#!\"\"" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 26 "limit(inttob, b=infinity);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "(Ve
rifica-se que esta integral impr\363pria n\343o existe.)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "Int((2*x + 1)/(x^4 + 1), x=-infinit
y..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&,&*&\"\"#
\"\"\"%\"xGF*F*F*F*F*,&*$)F+\"\"%F*F*F*F*!\"\"/F+;,$%)infinityGF0F4" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "part1 := int((2*x + 1)/(x^
4 + 1), x=-a..0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&part1G,**&#\"
\"\"\"\")F(*&\"\"##F(F+-%#lnG6#*&,(*$)%\"aGF+F(F(*&F4F(F+F,!\"\"F(F(F(
,(F2F(F5F(F(F(F6F(F(F6*&#F(\"\"%F(*&F+F,-%'arctanG6#,&F5F(F(F6F(F(F(*&
F9F(*&F+F,-F=6#,&F5F(F(F(F(F(F(-F=6#F2F6" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "limit1 := limit(part1, a=infinity);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%'limit1G,&*&#\"\"\"\"\"%F(*&\"\"##F(F+%#PiGF(F(F(*
&#F(F+F(F-F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "part2 \+
:= int((2*x + 1)/(x^4 + 1), x=0..b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%&part2G,**&#\"\"\"\"\")F(*&\"\"##F(F+-%#lnG6#*&,(*$)%\"bGF+F(F(*&F
4F(F+F,F(F(F(F(,(F2F(F5!\"\"F(F(F7F(F(F(*&#F(\"\"%F(*&F+F,-%'arctanG6#
,&F5F(F(F(F(F(F(*&F9F(*&F+F,-F=6#,&F5F(F(F7F(F(F(-F=6#F2F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "limit2 := limit(part2, b=infinity);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'limit2G,&*&#\"\"\"\"\"%F(*&\"\"
##F(F+%#PiGF(F(F(*&F,F(F-F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 134 
"Como ambas as partes das integral impr\363pria possuem limites, separ
adamente, deve-se adicion\341-las para que se obtenha a resposta final
." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "Int((2*x + 1)/(x^4 + 1
), x=-infinity..infinity) = limit1 +    limit2;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%$IntG6$*&,&*&\"\"#\"\"\"%\"xGF+F+F+F+F+,&*$)F,\"\"%F
+F+F+F+!\"\"/F,;,$%)infinityGF1F5,$*(F*F1F*#F+F*%#PiGF+F+" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 45 "Int(sin(x)/(x^2 + 1), x=-infinity..infinity);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%$sinG6#%\"xG\"\"\",&*$)F*
\"\"#F+F+F+F+!\"\"/F*;,$%)infinityGF0F4" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "part1 := int(sin(x)/(x^2 + 1), x=-a..0);" }}{PARA 12 
"" 1 "" {XPPMATH 20 "6#>%&part1G,0*(-%#SiG6#^#\"\"\"F+-%%coshG6#F+F+F*
F+F+*&-%#CiGF)F+-%%sinhGF.F+F+*(^##!\"\"\"\"#F+F2F+%#PiGF+F+*(F5F+-F(6
#,&%\"aGF+F*F+F+F,F+F+*&#F+F8F+*&-F16#,&F>F7^#F7F+F+F2F+F+F7*(^##F+F8F
+-F(6#,&F>F+FEF+F+F,F+F+*&#F+F8F+*&-F16#,&F>F7F*F+F+F2F+F+F7" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "limit1 := limit(part1, a=inf
inity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'limit1G,(*(-%#SiG6#^#\"
\"\"F+-%%coshG6#F+F+F*F+F+*&-%#CiGF)F+-%%sinhGF.F+F+*(^##!\"\"\"\"#F+F
2F+%#PiGF+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "part2 := in
t(sin(x)/(x^2 + 1), x=0..b);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&par
t2G,0*(^##!\"\"\"\"#\"\"\"-%#SiG6#,&%\"bGF+^#F)F+F+-%%coshG6#F+F+F+*&#
F+F*F+*&-%#CiGF.F+-%%sinhGF4F+F+F+*(^#F6F+-F-6#,&F0F+^#F+F+F+F2F+F+*&F
6F+*&-F9F?F+F:F+F+F+*(F1F+-F-6#FAF+F2F+F+*&-F9FGF+F:F+F)*(F=F+F:F+%#Pi
GF+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "limit2 := limit(pa
rt2, b=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'limit2G,(*(^#!
\"\"\"\"\"-%#SiG6#^#F)F)-%%coshG6#F)F)F)*&-%#CiGF,F)-%%sinhGF0F)F(*(^#
#F)\"\"#F)F4F)%#PiGF)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 132 "Estes
 limites envolvem algumas fun\347\365es n\343o familiares. Por\351m, (
ambas) existem, e podem ser adicionadas a fim de obter-se a resposta.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "Int(sin(x)/(x^2 + 1), x
=-infinity..infinity) = limit1 + limit2;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%$IntG6$*&-%$sinG6#%\"xG\"\"\",&*$)F+\"\"#F,F,F,F,!\"\"/F+;,$%
)infinityGF1F5\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 146 "A respost
a final \351 menos surpreendente quando compreendemos que a fun\347
\343o integrada \351 \355mpar, e os limites da integral s\343o sim\351
tricos em torno de 0." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "In
t(x*sin(x)/(x^2 + 1), x=-infinity..infinity);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%$IntG6$*(%\"xG\"\"\"-%$sinG6#F'F(,&*$)F'\"\"#F(F(F(F(
!\"\"/F';,$%)infinityGF0F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
42 "part1 := int(x*sin(x)/(x^2 + 1), x=-a..0);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%&part1G,,*&#\"\"\"\"\"#F(*&-%%sinhG6#F(F(%#PiGF(F(F(*
&F'F(*&-%#SiG6#,&%\"aGF(^#F(F(F(-%%coshGF-F(F(F(*(^##!\"\"F)F(-%#CiG6#
,&F5F<^#F<F(F(F+F(F(*&F'F(*&-F26#,&F5F(FAF(F(F7F(F(F(*(^#F'F(-F>6#,&F5
F<F6F(F(F+F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "limit1 :=
 limit(part1, a=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'limit
1G,&*&#\"\"\"\"\"#F(*&-%%sinhG6#F(F(%#PiGF(F(!\"\"*&#F(F)F(*&F.F(-%%co
shGF-F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "part2 := int
(x*sin(x)/(x^2 + 1), x=0..b);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&pa
rt2G,,*&#\"\"\"\"\"#F(*&-%#SiG6#,&%\"bGF(^#!\"\"F(F(-%%coshG6#F(F(F(F(
*(^#F'F(-%#CiGF-F(-%%sinhGF4F(F(*&F'F(*&-F,6#,&F/F(^#F(F(F(F2F(F(F(*(^
##F1F)F(-F8F>F(F9F(F(*&#F(F)F(*&F9F(%#PiGF(F(F1" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 35 "limit2 := limit(part2, b=infinity);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%'limit2G,&*&#\"\"\"\"\"#F(*&-%%sinhG6#F(F(
%#PiGF(F(!\"\"*&#F(F)F(*&F.F(-%%coshGF-F(F(F(" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 68 "Int(x*sin(x)/(x^2 + 1), x=-infinity..infinity) =
 limit1 +    limit2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*(%
\"xG\"\"\"-%$sinG6#F(F),&*$)F(\"\"#F)F)F)F)!\"\"/F(;,$%)infinityGF1F5,
&*&-%%sinhG6#F)F)%#PiGF)F1*&F;F)-%%coshGF:F)F)" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 42 "Int(2*x/(x^2 + 5), x=-infinity..infinity);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*(\"\"#\"\"\"%\"xGF),&*$)F*
F(F)F)\"\"&F)!\"\"F)/F*;,$%)infinityGF/F3" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 37 "part1 := int(2*x/(x^2 + 5), x=-a..0);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%&part1G,&-%#lnG6#\"\"&\"\"\"-F'6#,&*$)%\"aG\"
\"#F*F*F)F*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "limit1 \+
:= limit(part1, a=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'lim
it1G,$%)infinityG!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "Como um
a das duas partes da integral impr\363pria n\343o existe, deve-se ence
rrar os c\341lculos:  " }{XPPEDIT 18 0 "Int(2*x/(x^2+5),x = -infinity \+
.. infinity);" "6#-%$IntG6$*(\"\"#\"\"\"%\"xGF(,&*$F)F'F(\"\"&F(!\"\"/
F);,$%)infinityGF-F1" }{TEXT -1 13 "  n\343o existe." }}}}}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 26 "Tipo 2: Fun\347\365es Ilimitadas" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "O segundo tipo de integral impr
\363pria \351 aquele em que a fun\347\343o \351 ilimitada no intervalo
  " }{XPPEDIT 18 0 "[a, b];" "6#7$%\"aG%\"bG" }{TEXT -1 237 " . Um exe
mplo comum ocorre quando a fun\347\343o torna-se ilimitada em uma das \+
extremidades do intervalo. Examinaremos este caso. (Como explicado ant
eriormente, outros casos podem ser reduzidos a este.)  Suponhamos inic
ialmente que, para todo " }{XPPEDIT 18 0 "0 < epsilon;" "6#2\"\"!%(eps
ilonG" }{TEXT -1 10 " pequeno, " }{TEXT 290 1 "f" }{TEXT -1 35 " \351 \+
integr\341vel em todo o intervalo (" }{XPPEDIT 18 0 "a+epsilon,b;" "6$
,&%\"aG\"\"\"%(epsilonGF%%\"bG" }{TEXT -1 50 "] . Portanto, a integral
 impr\363pria \351 definida como" }}}{EXCHG {PARA 256 "" 0 "" 
{XPPEDIT 18 0 "Int(f(x),x = a .. b) := limit(Int(f(x),x = a+epsilon ..
 b),epsilon = 0,right);" "6#>-%$IntG6$-%\"fG6#%\"xG/F*;%\"aG%\"bG-%&li
mitG6%-F%6$-F(6#F*/F*;,&F-\"\"\"%(epsilonGF9F./F:\"\"!%&rightG" }
{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Apresentamos um
 exemplo a seguir.  Verifica-se que o " }{TEXT 291 5 "Maple" }{TEXT 
-1 54 " tamb\351m pode resolver este tipo de integral impr\363pria." }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "Int(1/sqrt(x), x=0..2) = i
nt(1/sqrt(x), x=0..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&
\"\"\"F(*$%\"xG#F(\"\"#!\"\"/F*;\"\"!F,,$*&F,F(F,#F(F,F(" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 52 "Para verificar a resposta, resolvemos a i
ntegral de " }{XPPEDIT 18 0 "epsilon;" "6#%(epsilonG" }{TEXT -1 17 " a
t\351 2 e fazemos " }{XPPEDIT 18 0 "epsilon;" "6#%(epsilonG" }{TEXT 
-1 59 " aproximar-se de 0 pela direita.  (Note o uso do argumento " }
{TEXT 292 5 "right" }{TEXT -1 19 " no comando limit.)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "ad := int(1/sqrt(x), x);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#adG,$*&\"\"#\"\"\"%\"xG#F(F'F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "properint := subs(x=2, ad) -
 subs(x=epsilon, ad);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*properintG
,&*&\"\"#\"\"\"F'#F(F'F(*&F'F(%(epsilonGF)!\"\"" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 59 "Int(1/sqrt(x), x=0..2) = limit(properint, epsi
lon=0,right);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(*
$%\"xG#F(\"\"#!\"\"/F*;\"\"!F,,$*&F,F(F,#F(F,F(" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 46 "A prop\363sito, note que o limite de dois lados, " }
{XPPEDIT 18 0 "limit(sqrt(epsilon),epsilon = 0);" "6#-%&limitG6$-%%sqr
tG6#%(epsilonG/F)\"\"!" }{TEXT -1 45 ", n\343o existe.  Por\351m, ao m
enos neste caso, o " }{TEXT 293 6 "Maple " }{TEXT -1 16 "n\343o nos in
forma:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "limit(properint, \+
epsilon=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"#\"\"\"F%#F&F%F
&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "\n\nNo caso em que " }{TEXT 
294 1 "f" }{TEXT -1 53 " torna-se ilimitado na extremidade direita, de
fine-se" }}}{EXCHG {PARA 256 "" 0 "" {XPPEDIT 18 0 "Int(f(x),x = a .. \+
b) := limit(Int(f(x),x = a .. b-epsilon),epsilon = 0,right);" "6#>-%$I
ntG6$-%\"fG6#%\"xG/F*;%\"aG%\"bG-%&limitG6%-F%6$-F(6#F*/F*;F-,&F.\"\"
\"%(epsilonG!\"\"/F:\"\"!%&rightG" }{TEXT -1 2 " ." }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 48 "Int(1/(x-2)^2, x=0..2) = int(1/(x-2)^2, x=0
..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(*$),&%\"x
GF(\"\"#!\"\"F-F(F./F,;\"\"!F-%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 14 "A resposta do " }{TEXT 295 5 "Maple" }{TEXT -1 58 " suger
e que a integral impr\363pria n\343o exite.  Verifiquemos:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "ad := int(1/(x-2)^2, x);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#adG,$*&\"\"\"F',&%\"xGF'\"\"#!\"\"F+F+" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "properint := subs(x=2-epsi
lon, ad) - subs(x=0, ad);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*proper
intG,&*&\"\"\"F'%(epsilonG!\"\"F'#F'\"\"#F)" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 34 "Pode-se notar que o limite quando " }{XPPEDIT 18 0 "pro
c (epsilon) options operator, arrow; 0 end;" "6#f*6#%(epsilonG7\"6$%)o
peratorG%&arrowG6\"\"\"!F*F*F*" }{TEXT -1 19 " n\343o existe.  Pelo " 
}{TEXT 296 5 "Maple" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "limit(properint, epsilon=0, right);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#%)infinityG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "
\n\nFinalmente, se " }{TEXT 297 1 "f" }{TEXT -1 95 " \351 ilimitada em
 ambas as extremidades, deve-se dividir o intervalo em algum ponto con
veniente, " }{TEXT 298 1 "c" }{TEXT -1 13 ", e define-se" }}}{EXCHG 
{PARA 256 "" 0 "" {XPPEDIT 18 0 "Int(f(x),x = a .. b) := Int(f(x),x = \+
a .. c)+Int(f(x),x = c .. b);" "6#>-%$IntG6$-%\"fG6#%\"xG/F*;%\"aG%\"b
G,&-F%6$-F(6#F*/F*;F-%\"cG\"\"\"-F%6$-F(6#F*/F*;F6F.F7" }{TEXT -1 2 " \+
," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "contanto que ambas as integr
ais impr\363prias do lado direito existam." }}}{SECT 0 {PARA 4 "" 0 "
" {TEXT -1 9 "Quest\343o 2" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "Reso
lver as integrais impr\363prias a seguir.  Caso n\343o seja poss\355ve
l, mostrar que n\343o existem." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 25 "Int(1/sqrt(1-x), x=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-
%$IntG6$*&\"\"\"F'*$,&F'F'%\"xG!\"\"#F'\"\"#F+/F*;\"\"!F'" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "inttoe := int(1/sqrt(1-x), x=0..1-e
psilon);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'inttoeG,&*&\"\"#\"\"\"%
(epsilonG#F(F'!\"\"F'F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "
Int(1/sqrt(1-x), x=0..1) = limit(inttoe, epsilon=0, right);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(*$,&F(F(%\"xG!\"\"#F(\"
\"#F,/F+;\"\"!F(F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Int(1
/sqrt(4-x^2), x=0..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&
\"\"\"F'*$,&\"\"%F'*$)%\"xG\"\"#F'!\"\"#F'F.F//F-;\"\"!F." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "inttoe := int(1/sqrt(4-x^2), x=0..2
-epsilon);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'inttoeG,$-%'arcsinG6#
,&\"\"\"!\"\"*&\"\"#F+%(epsilonGF*F*F+" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 61 "Int(1/sqrt(4-x^2), x=0..2) = limit(inttoe, epsilon=0,
 right);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&\"\"\"F(*$,&\"
\"%F(*$)%\"xG\"\"#F(!\"\"#F(F/F0/F.;\"\"!F/,$*&F/F0%#PiGF(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 25 "Int(1/(x^2 - 1), x=1..2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%$IntG6$*&\"\"\"F',&*$)%\"xG\"\"#F'F'F'!\"\"F-/F+;F'F,
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "inttoe := int(1/(x^2 - \+
1), x=1+epsilon..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'inttoeG,(*&
^##\"\"\"\"\"#F)%#PiGF)F)*&#F)F*F)-%#lnG6#\"\"$F)!\"\"-%(arctanhG6#,&F
)F)%(epsilonGF)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "limit(
inttoe, epsilon=0, right);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%)infini
tyG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Esta integral impr\363pria
 n\343o existe." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Int(exp(
sqrt(x))/sqrt(x), x=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6
$*&-%$expG6#*$%\"xG#\"\"\"\"\"#F-F+#!\"\"F./F+;\"\"!F-" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "inttoe := int(exp(sqrt(x))/sqrt(x),
 x=epsilon..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'inttoeG,&*&\"\"#
\"\"\"-%$expG6#F(F(F(*&F'F(-F*6#*$%(epsilonG#F(F'F(!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "Int(exp(sqrt(x))/sqrt(x), x=0..1) =
 limit(inttoe, epsilon=0,   right);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#/-%$IntG6$*&-%$expG6#*$%\"xG#\"\"\"\"\"#F.F,#!\"\"F//F,;\"\"!F.,&*&F/
F.-F)6#F.F.F.F/F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Int(1/
(x^2 - 5*x + 6), x=1..4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$
*&\"\"\"F',(*$)%\"xG\"\"#F'F'*&\"\"&F'F+F'!\"\"\"\"'F'F//F+;F'\"\"%" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "Note que o denominador desta fun
\347\343o \351 igual a 0 em dois lugares: " }{TEXT 299 1 "x" }{TEXT 
-1 7 " = 2 e " }{TEXT 300 1 "x" }{TEXT -1 196 " = 3.  Portanto, deve-s
e dividir a integral em (pelo menos) 3 partes, resolv\352-las separada
mente e ent\343o adicion\341-las.  Primeiramente, deve-se tratar a int
egral de 1 a 2, sendo esta impr\363pria em 2." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 49 "inttoe := int(1/(x^2 - 5*x + 6), x=1..2-epsilon)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'inttoeG,(-%#lnG6#,&\"\"\"!\"\"
%(epsilonGF+F*-F'6#,$F,F+F+-F'6#\"\"#F+" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "part1 := limit(inttoe, epsilon=0, right);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%&part1G%)infinityG" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 120 "Deve-se parar neste momento: a primeira parte da in
tegral n\343o existe e, portanto, a integral completa tamb\351m n\343o
 existe." }}}}}}{MARK "5 22 20 0 0" 120 }{VIEWOPTS 1 1 0 1 1 1803 1 1 
1 1 }{PAGENUMBERS 0 1 2 33 1 1 }