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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 31 "Solutions and answers to
 Quiz 4" }{TEXT 256 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 22 "Evalu
ate the integrals" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 2 "a)" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "I= Int((5*t^2 + 3*t -2)/(t^3+2*t^2)
, t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/^#\"\"\"-%$IntG6$*&,(*&\"\"&
F%)%\"tG\"\"#F%F%*&\"\"$F%F.F%F%F/!\"\"F%,&*$)F.F1F%F%*&F/F%F-F%F%F2F.
" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Solution:" }}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 111 "Since the integrand is a rational function we use
 Partial Fractions to express it as a sum of simple fractions:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "g(t):= (5*t^2+3*t-2)/(t^3+2*
t^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"gG6#%\"tG*&,(*&\"\"&\"\"
\")F'\"\"#F,F,*&\"\"$F,F'F,F,F.!\"\"F,,&*$)F'F0F,F,*&F.F,F-F,F,F1" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Factor the denominator" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "g(t) = numer(g(t))/factor(denom(g(t
)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,(*&\"\"&\"\"\")%\"tG\"\"#F
(F(*&\"\"$F(F*F(F(F+!\"\"F(,&*$)F*F-F(F(*&F+F(F)F(F(F.*(F%F(F*!\"#,&F*
F(F+F(F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Partial Fractions dec
omposition:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "g(t) = A/t +
 B/t^2 + C/(t+2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,(*&\"\"&\"\"
\")%\"tG\"\"#F(F(*&\"\"$F(F*F(F(F+!\"\"F(,&*$)F*F-F(F(*&F+F(F)F(F(F.,(
*&%\"AGF(F*F.F(*&%\"BGF(F*!\"#F(*&%\"CGF(,&F*F(F+F(F.F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "(5*t^2+3*t-2)/(t^3+2*t^2) = A/t+B/t
^2+C/(t+2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,(*&\"\"&\"\"\")%\"t
G\"\"#F(F(*&\"\"$F(F*F(F(F+!\"\"F(,&*$)F*F-F(F(*&F+F(F)F(F(F.,(*&%\"AG
F(F*F.F(*&%\"BGF(F*!\"#F(*&%\"CGF(,&F*F(F+F(F.F(" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 24 "Find common denominator:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 58 "g(t) = (A*t*(t+2) + B*(t+2) + C*t^2 )/factor(den
om(g(t)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,(*&\"\"&\"\"\")%\"tG
\"\"#F(F(*&\"\"$F(F*F(F(F+!\"\"F(,&*$)F*F-F(F(*&F+F(F)F(F(F.*(,(*(%\"A
GF(F*F(,&F*F(F+F(F(F(*&%\"BGF(F7F(F(*&%\"CGF(F)F(F(F(F*!\"#F7F." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Compare the numerators of both sid
es:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "numer(g(t)) = A*t*(t
+2) + B*(t+2) + C*t^2 ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&\"\"&
\"\"\")%\"tG\"\"#F'F'*&\"\"$F'F)F'F'F*!\"\",(*(%\"AGF'F)F',&F)F'F*F'F'
F'*&%\"BGF'F1F'F'*&%\"CGF'F(F'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
35 "Solve by \"choice of convenient t\" :" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 " t=0        ->     -2 = 2B        \+
                   ->         B = -1" }}{PARA 0 "" 0 "" {TEXT -1 60 " \+
t= -2     ->     5*4 -3*2 - 2 = 12 = 4C     ->        C = 3" }}{PARA 
0 "" 0 "" {TEXT -1 44 " t = -1    ->     0 = -A + B + C            " }
}{PARA 0 "" 0 "" {TEXT -1 68 "                         A = B + C = 3-1
 = 2       ->         A = 2 " }}{PARA 0 "" 0 "" {TEXT -1 9 "therefore
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "g(t) = convert(g(t), pa
rfrac, t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&,(*&\"\"&\"\"\")%\"tG
\"\"#F(F(*&\"\"$F(F*F(F(F+!\"\"F(,&*$)F*F-F(F(*&F+F(F)F(F(F.,(*&F-F(,&
F*F(F+F(F.F(*&F(F(*$F)F(F.F.*&F+F(F*F.F(" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 9 "Therefore" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "I \+
= Int(rhs(%), t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/^#\"\"\"-%$IntG6
$,(*&\"\"$F%,&%\"tGF%\"\"#F%!\"\"F%*&F%F%*$)F-F.F%F/F/*&F.F%F-F/F%F-" 
}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 
7 "Answer:" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "____________________
______________________________________________________________________
_" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Int( g(t), t) = int(rh
s(%%), t)+ C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&,(*&\"\"&
\"\"\")%\"tG\"\"#F+F+*&\"\"$F+F-F+F+F.!\"\"F+,&*$)F-F0F+F+*&F.F+F,F+F+
F1F-,**&F0F+-%#lnG6#,&F-F+F.F+F+F+*&F+F+F-F1F+*&F.F+-F96#F-F+F+%\"CGF+
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "_____________________________
_______________________________________________________________" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 3 "
b) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "I = Int(1/(x^3*sqrt(x
^2-1)), x= sqrt(2)..2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/^#\"\"\"-%
$IntG6$*&F%F%*&)%\"xG\"\"$F%,&*$)F,\"\"#F%F%F%!\"\"#F%F1F2/F,;*$F1#F%F
1F1" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Solution:" }}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 31 "Use trigonometric substitution:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "x = sec(theta); dx = diff(sec(theta
), theta)*d*theta;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"xG-%$secG6#%
&thetaG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#dxG**-%$secG6#%&thetaG\"
\"\"-%$tanGF(F*%\"dGF*F)F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "New
 bounds:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "x=sqrt(2), sec(
theta)=sqrt(2),cos(theta)='1/sqrt(2)', theta=arccos(1/sqrt(2)); " }}
{PARA 11 "" 1 "" {XPPMATH 20 "6&/%\"xG*$\"\"##\"\"\"F&/-%$secG6#%&thet
aGF%/-%$cosGF,*&F(F(-%%sqrtG6#F&!\"\"/F-,$*&\"\"%F5%#PiGF(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "x=2, sec(theta) = 2, cos(the
ta) = 1/2, theta=arccos(1/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&/%\"x
G\"\"#/-%$secG6#%&thetaGF%/-%$cosGF)#\"\"\"F%/F*,$*&\"\"$!\"\"%#PiGF/F
/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Therefore:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 94 "I = Int(sec(theta)*tan(theta)/((sec(theta
))^`3`* sqrt((sec(theta))^2-1)), theta = Pi/4..Pi/3);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/^#\"\"\"-%$IntG6$**-%$secG6#%&thetaGF%-%$tanGF,F%
)F*%\"3G!\"\",&*$)F*\"\"#F%F%F%F2#F2F6/F-;,$*&\"\"%F2%#PiGF%F%,$*&\"\"
$F2F=F%F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Use the identity:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "(tan(x))^2=(sec(x))^2-1 ;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)-%$tanG6#%\"xG\"\"#\"\"\",&*$)
-%$secGF(F*F+F+F+!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "to obta
in:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "I = Int(sec(theta)*t
an(theta)/((sec(theta))^3* sqrt((tan(theta))^2)), theta = Pi/4..Pi/3);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/^#\"\"\"-%$IntG6$*(-%$secG6#%&the
taG!\"#-%$tanGF,F%*$)F/\"\"#F%#!\"\"F3/F-;,$*&\"\"%F5%#PiGF%F%,$*&\"\"
$F5F;F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "I = Int(1/(sec
(theta))^2, theta= Pi/4.. Pi/3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/^
#\"\"\"-%$IntG6$*&F%F%*$)-%$secG6#%&thetaG\"\"#F%!\"\"/F/;,$*&\"\"%F1%
#PiGF%F%,$*&\"\"$F1F7F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
44 "I = Int((cos(theta))^2, theta = Pi/4..Pi/3);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/^#\"\"\"-%$IntG6$*$)-%$cosG6#%&thetaG\"\"#F%/F.;,$*&\"
\"%!\"\"%#PiGF%F%,$*&\"\"$F5F6F%F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 6 "Since " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "(cos(x))^2 \+
=1/2*((cos(2*x)+1)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)-%$cosG6#
%\"xG\"\"#\"\"\",&*&#F+F*F+-F'6#,$*&F*F+F)F+F+F+F+F.F+" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 9 "We obtain" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 70 "1/2*Int((cos(2*theta))+1, theta) = 1/2*int((cos(2*the
ta))+1, theta)+C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&#\"\"\"\"\"#
F'-%$IntG6$,&-%$cosG6#,$*&F(F'%&thetaGF'F'F'F'F'F2F'F',(*&#F'\"\"%F'-%
$sinGF/F'F'*&F(!\"\"F2F'F'%\"CGF'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
4 "Thus" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "1/2*Int((cos(2*
theta))+1, theta=Pi/4.. Pi/3) = 1/4*'sin(2*Pi/3)' + Pi/6 - 1/4* 'sin(`
2`*Pi/4)' - Pi/`8`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&#\"\"\"\"
\"#F'-%$IntG6$,&-%$cosG6#,$*&F(F'%&thetaGF'F'F'F'F'/F2;,$*&\"\"%!\"\"%
#PiGF'F',$*&\"\"$F8F9F'F'F'F',**&#F'F7F'-%$sinG6#,$*(F(F'F<F8F9F'F'F'F
'*&\"\"'F8F9F'F'*&#F'F7F'-FA6#,$*(F7F8%\"2GF'F9F'F'F'F8*&F9F'%\"8GF8F8
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "I=1/4*sin(2*Pi/3) + Pi/
6 - 1/4* sin(2*Pi/4) - Pi/`8`;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/^#
\"\"\",**&\"\")!\"\"\"\"$#F%\"\"#F%*&\"\"'F)%#PiGF%F%#F%\"\"%F)*&F/F%%
\"8GF)F)" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 7 "Answer:" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 91 "_________________________________________
__________________________________________________" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 81 "Int(1/(x^3*sqrt(x^2-1)), x= sqrt(2)..2)= in
t(1/(x^3*sqrt(x^2-1)), x= sqrt(2)..2);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%$IntG6$*&\"\"\"F(*&)%\"xG\"\"$F(,&*$)F+\"\"#F(F(F(!\"\"#F(F0F
1/F+;*$F0#F(F0F0,(*&\"\")F1F,F6F(*&\"#CF1%#PiGF(F(#F(\"\"%F1" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "__________________________________
_________________________________________________________" }}}}}}}
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