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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 31 "Solutions and answers t
o Quiz 3" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "with(student):" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 
23 "Evaluate the integrals:" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "a)
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Int(ln(sqrt(x)), x=1..4)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%#lnG6#*$%\"xG#\"\"\"\"
\"#/F*;F,\"\"%" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Solution:" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "We first find the antiderivative:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "I =  Int(ln(sqrt(x)), x
); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/^#\"\"\"-%$IntG6$-%#lnG6#*$%\"
xG#F%\"\"#F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Use integration b
y parts:" }}{PARA 0 "" 0 "" {TEXT -1 3 "Let" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 21 "u=ln(sqrt(x)); dv=dx;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%\"uG-%#lnG6#*$%\"xG#\"\"\"\"\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%#dvG%#dxG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Henc
e" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "du=diff(ln(sqrt(x)), x
) *dx;v=x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#duG,$*(\"\"#!\"\"%\"x
GF(%#dxG\"\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"vG%\"xG" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 260 9 "Therefore" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 45 "I=intparts(Int(ln(sqrt(x)), x), ln(sqrt(x)));" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#/^#\"\"\",&*&-%#lnG6#*$%\"xG#F%\"\"#F
%F,F%F%-%$IntG6$F-F,!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
34 "I = x*ln(sqrt(x)) - int(1/2, x)+C;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/^#\"\"\",(*&-%#lnG6#*$%\"xG#F%\"\"#F%F,F%F%*&F.!\"\"F,F%F0%\"CG
F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Now   we can evaluate the d
efinite integral:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "Int(ln
(sqrt(x)), x=1..4) = 4*ln(sqrt(4)) - `4/2` - (1*'ln(1)'-`1/2`);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%#lnG6#*$%\"xG#\"\"\"\"\"#/
F+;F-\"\"%,**&F1F--F(6#F.F-F-%$4/2G!\"\"-F(6#F-F7%$1/2GF-" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 54 " Int(ln(sqrt(x)), x=1..4) =  int(ln(sqrt(x)), x=1..4)
;" }}{PARA 0 "" 0 "" {TEXT 257 7 "Answer:" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%$IntG6$-%#lnG6#*$%\"xG#\"\"\"\"\"#/F+;F-\"\"%,&*&F1F
--F(6#F.F-F-#\"\"$F.!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "____
_____________________________" }}}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 
2 "b)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Int((sin(x))^3*sqrt
(cos(x)), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&)-%$sinG6#
%\"xG\"\"$\"\"\"-%$cosGF*#F-\"\"#F+" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 
-1 9 "Solution:" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "This is a trigo
nometric integral with sin(x) in an odd power." }}{PARA 0 "" 0 "" 
{TEXT -1 21 "Factor out sin(x) dx:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 44 "I = Int(`(sin(x))`^2*sqrt(cos(x))*sin(x),x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/^#\"\"\"-%$IntG6$*()%)(sin(x))G\"\"#F
%-%$cosG6#%\"xG#F%F,-%$sinGF/F%F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
21 "Use the trig identity" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
27 "(sin(x))^2= 1 - (cos(x))^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$
)-%$sinG6#%\"xG\"\"#\"\"\",&F+F+*$)-%$cosGF(F*F+!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "I = Int((1-`cos(x)`^2)*sqrt(cos(x))
 *sin(x), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/^#\"\"\"-%$IntG6$*(,
&F%F%*$)%'cos(x)G\"\"#F%!\"\"F%-%$cosG6#%\"xG#F%F.-%$sinGF2F%F3" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Substitute" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 29 "u = cos(x); du = -sin(x) *dx;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/%\"uG-%$cosG6#%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/%#duG,$*&-%$sinG6#%\"xG\"\"\"%#dxGF+!\"\"" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 59 "I = changevar(cos(x) = u, Int((sin(x))^3*sqrt
(cos(x)), x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/^#\"\"\"-%$IntG6$,$
*&,&F%F%*$)%\"uG\"\"#F%!\"\"F%F.#F%F/F0F." }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 26 "Multiply the integrand out" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "I=Int(collect((u^2-1)*sqrt(u), u), u);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/^#\"\"\"-%$IntG6$,&*$)%\"uG#\"\"&\"\"#F%F%*$F,#
F%F/!\"\"F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Integrate" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "I=int(u^(5/2)-sqrt(u), u)+C;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/^#\"\"\",(*(\"\"#F%\"\"(!\"\"%\"u
G#F)F(F%*(F(F%\"\"$F*F+#F.F(F*%\"CGF%" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 28 "Re-substituting for u gives " }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 59 "Int((sin(x))^3*sqrt(cos(x)), x) = subs(u = cos(x), \+
rhs(%));" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 7 "Answer:" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&)-%$sinG6#%\"xG\"\"$\"\"\"
-%$cosGF+#F.\"\"#F,,(*&#F2\"\"(F.*$)F/#F6F2F.F.F.*&#F2F-F.*$)F/#F-F2F.
F.!\"\"%\"CGF." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "_______________
__________________" }}}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "c)" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Int(x^2*sin(4*x), x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&)%\"xG\"\"#\"\"\"-%$sinG6#,
$*&\"\"%F*F(F*F*F*F(" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Solution:
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "I=Int(x^2*sin(4*x), x);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/^#\"\"\"-%$IntG6$*&)%\"xG\"\"#F%-
%$sinG6#,$*&\"\"%F%F+F%F%F%F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "
Use integration by parts" }}{PARA 0 "" 0 "" {TEXT -1 3 "Let" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "u= x^2; dv= sin(4*x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"uG*$)%\"xG\"\"#\"\"\"" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#/%#dvG-%$sinG6#,$*&\"\"%\"\"\"%\"xGF+F+" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Hence" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "du=2*x*dx; v= -1/4*cos(4*x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%#duG,$*(\"\"#\"\"\"%\"xGF(%#dxGF(F(" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#/%\"vG,$*&#\"\"\"\"\"%F(-%$cosG6#,$*&F)F(%\"xGF(F(F(
!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Therefore" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "I=intparts(Int(x^2*sin(4*x), x), x^
2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/^#\"\"\",&*&#F%\"\"%F%*&)%\"xG
\"\"#F%-%$cosG6#,$*&F)F%F,F%F%F%F%!\"\"-%$IntG6$,$*&#F%F-F%*&F,F%F.F%F
%F3F,F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "The second integral " 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "I2=Int(1/2*x*cos(4*x), x)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#I2G-%$IntG6$,$*&#\"\"\"\"\"#F+
*&%\"xGF+-%$cosG6#,$*&\"\"%F+F.F+F+F+F+F+F." }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 30 "can be solved by parts as well" }}{PARA 0 "" 0 "" 
{TEXT -1 3 "let" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "unassign
(u, v);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "u=1/2*x; dv=cos(
4*x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"uG,$*&\"\"#!\"\"%\"xG\"\"
\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#dvG-%$cosG6#,$*&\"\"%\"\"\"
%\"xGF+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Hence" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "du= 1/2*dx; v= 1/4*sin(4*x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#duG,$*&\"\"#!\"\"%#dxG\"\"\"F*" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"vG,$*&#\"\"\"\"\"%F(-%$sinG6#,$*&F
)F(%\"xGF(F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Therefore" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "I2:= intparts(Int(1/2*x*cos(
4*x), x), x/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#I2G,&*&#\"\"\"\"
\")F(*&%\"xGF(-%$sinG6#,$*&\"\"%F(F+F(F(F(F(F(-%$IntG6$,$*&F'F(F,F(F(F
+!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "That gives" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "I = -1/4*x^2*cos(4*x) + I2;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/^#\"\"\",(*&#F%\"\"%F%*&)%\"xG\"\"#F%
-%$cosG6#,$*&F)F%F,F%F%F%F%!\"\"*&#F%\"\")F%*&F,F%-%$sinGF0F%F%F%-%$In
tG6$,$*&F5F%F8F%F%F,F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "And the
 last integral is easy to solve, giving" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 65 "I = -1/4*x^2*cos(4*x) + 1/8*x*sin(4*x) -1/8*int(sin(4
*x), x) + C;" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 6 "Answer" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/^#\"\"\",**&#F%\"\"%F%*&)%\"xG\"\"#F%
-%$cosG6#,$*&F)F%F,F%F%F%F%!\"\"*&#F%\"\")F%*&F,F%-%$sinGF0F%F%F%*&#F%
\"#KF%F.F%F%%\"CGF%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "__________
______________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}}}}{MARK "
2" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }