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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 11 "Calculus II" }}{PARA 297 
"" 0 "" {TEXT -1 12 "Lesson 13:  " }{TEXT 260 41 "Integration Techniqu
es Summary and Review" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 124 " You ca
n use the student package in Maple to practice your integration techni
ques.  First load the student package by typing" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student):" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 169 "Then rea
d over the help screens on  changevar,  intparts, and  value, paying p
articular attention to the examples at the bottom of the screens.  Her
e are some examples. " }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 262 24 " A Sub
stitution Problem:" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 
258 27 "Integration by substitution" }{TEXT -1 78 "    is based on the
 chain rule.  Thus if we have an integral which looks like " }
{XPPEDIT 18 0 "Int(f(g(x))*diff(g(x),x),x)" "6#-%$IntG6$*&-%\"fG6#-%\"
gG6#%\"xG\"\"\"-%%diffG6$-F+6#F-F-F.F-" }{TEXT -1 39 ",  then by make \+
the change of variable " }{XPPEDIT 18 0 "g(x)=u" "6#/-%\"gG6#%\"xG%\"u
G" }{TEXT -1 15 ",  and letting " }{XPPEDIT 18 0 "du=diff(g(x),x)*dx" 
"6#/%#duG*&-%%diffG6$-%\"gG6#%\"xGF,\"\"\"%#dxGF-" }{TEXT -1 41 " we h
ave a new, perhaps simpler integral " }{XPPEDIT 18 0 "Int(f(u),u)" "6#
-%$IntG6$-%\"fG6#%\"uGF)" }{TEXT -1 16 ", to work on.   " }}{PARA 0 "
" 0 "" {TEXT -1 45 "In Maple this is accomplished using the word " }
{TEXT 256 15 " changevar     " }{TEXT -1 25 "from the student package.
" }}{PARA 0 "" 0 "Integration by substitution" {TEXT -1 26 "Find an an
tiderivative of " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "F := In
t(1/sqrt(1+sqrt(x)),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG-%$In
tG6$*$,&\"\"\"F**$%\"xG#F*\"\"#F*#!\"\"F.F," }}}{EXCHG {PARA 0 "" 0 "c
hange of variable" {TEXT -1 14 "Let's try the " }{TEXT 258 18 "change \+
of variable" }{TEXT -1 17 "    sqrt(x) = u ." }}}{EXCHG {PARA 0 "> " 
0 "changevar" {MPLTEXT 1 0 28 "G := changevar(sqrt(x)=u,F);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"GG-%$IntG6$,$*&,&\"\"\"F+%\"uGF+#!\"\"\"
\"#F,F+F/F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "This does not seem
 to help.  Lets try 1 + sqrt(x)= u" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "G := changevar(1+sqrt(x)=u,F);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"GG-%$IntG6$,$*&%\"uG#!\"\"\"\"#,(\"\"\"F/F*!\"#*$F*
F-F/#F/F-F-F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "Now we can do it
 by inspection, so just finish it off." }}}{EXCHG {PARA 0 "> " 0 "valu
e" {MPLTEXT 1 0 14 "G := value(G);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%\"GG,$**%\"uG#\"\"\"\"\"#,&!\"$F)F'F)F)*$,&!\"\"F)F'F)F*F(F.F/#\"\"%
\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Now substitute back and \+
add in the constant." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "F :
= subs(u=sqrt(x),G) + C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG,&**
%\"xG#\"\"\"\"\"%,&!\"$F)*$F'#F)\"\"#F)F)*$,&!\"\"F)F-F)F/F.F1F2#F*\"
\"$%\"CGF)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "Integration by sub
stitution is the method use try after you decide you can't find the an
tiderivative by inspection." }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 263 34 
"An Integration by Parts Problem:  " }}{EXCHG {PARA 0 "" 0 "integratio
n by parts" {TEXT 258 20 "Integration by parts" }{TEXT -1 75 "    is b
ased on the product rule for derivatives.  It is usually written   " }
{XPPEDIT 18 0 "Int(u,v)= uv - Int(v,u)" "6#/-%$IntG6$%\"uG%\"vG,&%#uvG
\"\"\"-F%6$F(F'!\"\"" }{TEXT -1 160 ".   It turns one integration prob
lem into one which 'may' be more doable. Once you decide to use parts,
 the problem is what part of the integrand to let be u.  " }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 261 1 " " }{TEXT -1 10 "Integrate " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "F := Int(x^2*arctan(x),x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG-%$IntG6$*&%\"xG\"\"#-%'arctanG6
#F)\"\"\"F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "The word is " }
{TEXT 256 8 "intparts" }{TEXT -1 24 ".    Let's try letting  " }
{XPPEDIT 18 0 "u=x^2" "6#/%\"uG*$%\"xG\"\"#" }{TEXT -1 2 " ." }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 21 "G := intparts(F,x^2);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"GG,&*&%\"xG\"\"#,&*&F'\"\"\"-%'arctanG6#F'F+F+-%#
lnG6#,&F+F+*$F'F(F+#!\"\"F(F+F+-%$IntG6$,$*&F'F+F)F+F(F'F5" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 37 "That was a bad choice.  Try letting  " }
{XPPEDIT 18 0 "u = arctan(x)" "6#/%\"uG-%'arctanG6#%\"xG" }{TEXT -1 1 
" " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "G := intparts(F,arcta
n(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG,&*&-%'arctanG6#%\"xG
\"\"\"F*\"\"$#F+F,-%$IntG6$,$*&,&F+F+*$F*\"\"#F+!\"\"F*F,F-F*F6" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "This is much more promising.  Spli
t off the integral on the end." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "H := op(2,G);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"HG,$-%$IntG6$,$*&,&\"\"\"F,*$%\"xG\"\"#F,!\"\"F.\"
\"$#F,F1F.F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "Now do a partial \+
fractions decomposition of the integrand of H, using " }{TEXT 256 7 "p
arfrac" }{TEXT -1 6 ".     " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
46 "H:= Int(convert(integrand(H),parfrac,x),x);   " }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"HG-%$IntG6$,&%\"xG#\"\"\"\"\"$*&F)F+,&F,F+*$F)\"
\"#F,!\"\"F1F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Now we can do i
t by inspection. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "H1 := \+
1/6*x^2 - 1/3*1/2*ln(1+x^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#H1G
,&*$%\"xG\"\"##\"\"\"\"\"'-%#lnG6#,&F*F*F&F*#!\"\"F+" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 34 "Let's check this with the student " }{TEXT 256 
5 "value" }{TEXT -1 6 ".     " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 22 "simplify(value(H-H1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%#l
nG6#\"\"$#!\"\"\"\"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Note the \+
difference of a constant, which is fine for antiderivatives." }}{PARA 
0 "" 0 "ETAIL" {TEXT 258 5 "ETAIL" }{TEXT -1 433 ":    The problem of \+
choosing which part of the integrand to assign to u can often be solve
d quickly by following the etail convention.  If your integrand has an
 Exponential factor, choose that for u, otherwise if it has a Trigonom
etric factor, let that be u, otherwise choose an Algebraic factor for \+
u, otherwise chose an Inverse trig function, and as a last resort choo
se u to be a logarithmic factor.  Let dv be what's left over.  " }}}}
{SECT 0 {PARA 5 "" 0 "" {TEXT -1 20 "A Trig Substitution:" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 25 "Find an antiderivative of" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 28 "F := Int(x^3/sqrt(x^2+1),x);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%\"FG-%$IntG6$*&%\"xG\"\"$,&\"\"\"F,*$F)\"\"#F,#!\"
\"F.F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "The presence of  " }
{XPPEDIT 18 0 "x^2 + 1 " "6#,&*$%\"xG\"\"#\"\"\"F'F'" }{TEXT -1 18 " s
uggests letting " }{XPPEDIT 18 0 " x = tan(t)  " "6#/%\"xG-%$tanG6#%\"
tG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "G := \+
changevar(x=tan(t),F,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG-%$I
ntG6$*&-%$tanG6#%\"tG\"\"$,&\"\"\"F/*$F)\"\"#F/#F/F1F," }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 26 "Now use the trig identity " }{XPPEDIT 18 
0 "1 + tan(t)^2 = sec(t)^2" "6#/,&\"\"\"F%*$-%$tanG6#%\"tG\"\"#F%*$-%$
secG6#F*F+" }{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
37 "G := subs(sqrt(1+tan(t)^2)=sec(t),G);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"GG-%$IntG6$*&-%$tanG6#%\"tG\"\"$-%$secGF+\"\"\"F," 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Another substitution into the i
ntegrand." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "G := subs(tan(
t)^3 = (sec(t)^2-1)*tan(t),G);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"
GG-%$IntG6$*(,&*$-%$secG6#%\"tG\"\"#\"\"\"!\"\"F0F0-%$tanGF-F0F+F0F." 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Let's make a change of variable
," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "H := changevar(sec(t)=
u,G);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"HG-%$IntG6$,&*$%\"uG\"\"#
\"\"\"!\"\"F,F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "From here, we \+
can do it by inspection." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 
"H := value(H);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"HG,&*$%\"uG\"\"
$#\"\"\"F(F'!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Now unwind t
he substitutions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "G := s
ubs(u=sec(t),H);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG,&*$-%$secG6
#%\"tG\"\"$#\"\"\"F+F'!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
27 "F := subs(t = arctan(x),G);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"FG,&*$-%$secG6#-%'arctanG6#%\"xG\"\"$#\"\"\"F.F'!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "F := subs(sec(arctan(x))=sqrt(1+x^2
),F) + C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG,(*$,&\"\"\"F(*$%\"
xG\"\"#F(#\"\"$F+#F(F-*$F'#F(F+!\"\"%\"CGF(" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 26 "Checking this calculation:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 29 "F1 := int(x^3/sqrt(x^2+1),x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#F1G,&*&%\"xG\"\"#,&\"\"\"F**$F'F(F*#F*F(#F*\"\"$*$F)
F,#!\"#F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "It looks different, \+
but is it?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "simplify(F-F1
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"CG" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 28 "Yes, but only by a constant." }}}}{SECT 0 {PARA 5 "" 0 
"" {TEXT 265 27 "A Partial Fractions Problem" }}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 32 " Integrate the rational function" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "y :=(4*x^2+x -1 )/(x^2*(x-1)*(x^2+1));" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"yG**,(*$%\"xG\"\"#\"\"%F(\"\"\"!\"\"F+F+F(!
\"#,&F(F+F,F+F,,&F+F+F'F+F," }}}{EXCHG {PARA 0 "" 0 "partial fractions
 decomposition" {TEXT -1 14 "First get the " }{TEXT 258 31 "partial fr
actions decomposition" }{TEXT -1 9 "    of y." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 26 "y := convert(y,parfrac,x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"yG,(*$%\"xG!\"#\"\"\"*$,&F'F)!\"\"F)F,\"\"#*&,&\"\"
$F)F'F-F),&F)F)*$F'F-F)F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "We
 can almost do this by inspection, except for the last term." }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 15 "F := Int(y,x); " }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"FG-%$IntG6$,(*$%\"xG!\"#\"\"\"*$,&F*F,!\"\"F,F/\"\"
#*&,&\"\"$F,F*F0F,,&F,F,*$F*F0F,F/F/F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "F := expand(F);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"FG,*-%$IntG6$*$%\"xG!\"#F*\"\"\"-F'6$*$,&F*F,!\"\"F,F1F*\"\"#-F'6$*$
,&F,F,*$F*F2F,F1F*!\"$-F'6$*&F6F1F*F,F*F+" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 66 "Now we can do each one by inspection.  So we'll just use \+
  value ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "F := value(F) \+
+ C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG,,*$%\"xG!\"\"F(-%#lnG6#
,&F'\"\"\"F(F-\"\"#-%'arctanG6#F'!\"$-F*6#,&F-F-*$F'F.F-F(%\"CGF-" }}}
}{SECT 0 {PARA 5 "" 0 "" {TEXT 264 10 "Exercises:" }}{EXCHG {PARA 289 
"" 0 "" {TEXT -1 74 "Exercise: Use the student package to perform the \+
following integrations.  " }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#-
%$IntG6$*&-%$cosG6#%\"xG\"\"\",&F+F+-%$sinGF)F+#!\"\"\"\"#F*" }}}
{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*(,&%\"xG\"\"$!\"(\"\"
\"F+,&F(F+!\"\"F+!\"#,&F(F+F.F+!\"$F(" }}}{EXCHG {PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%$IntG6$*&%\"xG\"\"#-%$sinG6#*&%\"aG\"\"\"F'F.F.F'" }}
}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%#lnG6#,&%\"xG\"\"\"
*$F*#F+\"\"#F+F*" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*
&%\"zG\"\"&,&*$F'\"\"#\"\"\"F,F,#!\"\"F+F'" }}}{EXCHG {PARA 11 "" 1 "
" {XPPMATH 20 "6#-%$IntG6$*$,&-%$expG6#,$%\"xG\"\"$\"\"\"!\"\"F.F/F," 
}}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"xG\"\"\"-%'arc
sinG6#,$F'\"\"#F(F'" }}}{EXCHG {PARA 289 "" 0 "" {TEXT -1 77 "Exercise
:  Find the area of the region enclosed by the x-axis and the curve  \+
" }{XPPEDIT 18 0 "y = x sin(x) " "6#/%\"yG*&%\"xG\"\"\"-%$sinG6#F&F'" 
}{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[0 , Pi ]" "6#7$\"\"!
%#PiG" }{TEXT -1 53 ".  Sketch the region.  Then find the vertical lin
e   " }{XPPEDIT 18 0 "x = a" "6#/%\"xG%\"aG" }{TEXT -1 45 " that divid
es the region in half and plot it." }}}{EXCHG {PARA 289 "" 0 "" {TEXT 
-1 56 "Exercise:  Find the length of the graph of the parabola " }
{XPPEDIT 18 0 "y = x^2" "6#/%\"yG*$%\"xG\"\"#" }{TEXT -1 43 " from O(0
,0) to P(10,100).  Find the point " }{XPPEDIT 18 0 "Q(a,a^2)" "6#-%\"Q
G6$%\"aG*$F&\"\"#" }{TEXT -1 116 " on the graph which is 10 units from
 O along the graph.  Make a sketch, showing the points O, P, and Q on \+
the graph." }}}{EXCHG {PARA 289 "" 0 "" {TEXT -1 120 "Exercise:  Find \+
the volume of the solid of revolution obtained by revolving the region
 trapped between the the graph of " }{XPPEDIT 18 0 "y = exp(x)*sin(x) \+
" "6#/%\"yG*&-%$expG6#%\"xG\"\"\"-%$sinG6#F)F*" }{TEXT -1 3 "on " }
{XPPEDIT 18 0 "[0, n*Pi ] " "6#7$\"\"!*&%\"nG\"\"\"%#PiGF'" }{TEXT -1 
108 "and the x-axis about the x-axis.  Sketch a graph.  Does this volu
me approach a finite limit as n gets large?" }}}}}{MARK "4 20 0 0" 2 }
{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }