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{SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT -1 7 "MAT 156" }}{PARA 259 "" 
0 "" {TEXT -1 6 "Lab 10" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 30 "Topic
 1 : Integration by Parts" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
14 "with(student):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "The steps t
o perform integration by parts on  int(f(x),x) are :" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "1)  choose a function u
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "2)  f
actor f as   f = u dv" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 63 "3)  Integrate  dv   to find v    and differentiate u t
o find du" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
17 "4)   Replace     " }{XPPEDIT 18 0 "int(udv,x) " "6#-%$intG6$%$udvG
%\"xG" }{TEXT -1 15 "     with      " }{XPPEDIT 18 0 "uv -int(v, u)" "
6#,&%#uvG\"\"\"-%$intG6$%\"vG%\"uG!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 104 "Maple has a command that do
es steps 2, 3 and 4.  You must still choose the function u.  The comma
nd is  " }}{PARA 257 "" 0 "" {TEXT -1 9 "intparts(" }{TEXT 35 1 "f" }
{TEXT -1 2 ", " }{TEXT 35 1 "u" }{TEXT -1 1 ")" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 11 "Example    " }{XPPEDIT 18 0 "Int(exp(2*x)*(2*x+3), x
);" "6#-%$IntG6$*&-%$expG6#*&\"\"#\"\"\"%\"xGF,F,,&*&F+F,F-F,F,\"\"$F,
F,F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "We can try   u = 2x+3" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 31 "Fint:=Int(exp(2*x)*(2*x+3), x);" }}{PARA 11 "" 1 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Gint:=intparts(Fint, 2*
x+3);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 74 "This is now an easier integral so the integration by part
s was successful." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "value(
Gint);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 265 17 "Assignment Part 1" }{TEXT -1 53 " :  Perform the integra
tion by parts in the integral " }{XPPEDIT -1 0 "int(x^10*ln(x),x);" "6
#-%$intG6$*&%\"xG\"#5-%#lnG6#F'\"\"\"F'" }{TEXT -1 59 ". Verify your a
nswer by asking Maple to integrate directly." }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "Many times we a
re forced to integrate by parts repeatedly. We do this one step at a t
ime. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 7 "
Example" }{TEXT -1 24 ": Calculate the integral" }{XPPEDIT 19 1 "int(e
xp(2*x)*cos(3*x),x);" "6#-%$intG6$*&-%$expG6#*&\"\"#\"\"\"%\"xGF,F,-%$
cosG6#*&\"\"$F,F-F,F,F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 32 "Lint:=Int
(exp(2*x)*cos(3*x), x);" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 44 "Now do integration by parts with u = exp(
2x)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "Mint:=intparts(Lint, exp(2*x));" }}{PARA 11 "" 1 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "We define the new
 integral to be Nint:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Nint:=Int(2/3*exp(2*x)*sin(3*x), x)
;" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 46 "Do integration by parts again with u = exp(2x)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Oint:=intparts(Nint, exp(2*x));" }}
{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 
"Here we see that we got back the integral we started with with coeffi
cient 4/9 in front. The result is " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "Lint:=1/3*exp(2*x)*sin(3*x)-Oint;" }}{PARA 11 "" 1 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "Lint:=(1+
4/9)^\{-1\}*(1/3*exp(2*x)*sin(3*x)+2/9*exp(2*x)*cos(3*x));" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 76 "We can check the answer by redefining the
 integral and integrating directly:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "lint:=Int(exp(2*x)*cos(3*x), x);" }}{PARA 11 "" 1 "" 
{TEXT -1 0 "" }}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "value(lint);" }}
{EXCHG {PARA 0 "" 0 "" {TEXT 258 24 "Assignment Part 2  :    " }{TEXT 
-1 56 "Perform 2 integrations by parts to compute the integral " }
{XPPEDIT -1 0 "int((x^2+5*x+9)*e^(2*x),x);" "6#-%$intG6$*&,(*$%\"xG\"
\"#\"\"\"*&\"\"&F+F)F+F+\"\"*F+F+)%\"eG*&F*F+F)F+F+F)" }{TEXT -1 44 ".
 Check your answer with direct integration." }}}{EXCHG {PARA 0 "" 0 "
" {TEXT 259 22 "Assignment Part 3 :   " }{TEXT -1 22 "Evaluate the int
egral " }{XPPEDIT -1 1 "int(x*ln(x+sqrt(1+x^2)),x);" "6#-%$intG6$*&%\"
xG\"\"\"-%#lnG6#,&F'F(-%%sqrtG6#,&F(F(*$F'\"\"#F(F(F(F'" }{TEXT -1 1 "
." }{TEXT 257 1 " " }{TEXT -1 107 "If you try to compute the integral \+
directly you will see that Maple gets stuck and does not know what to \+
do" }{TEXT 260 2 ". " }{TEXT -1 64 "You will need two integrations by \+
parts and the second is tricky" }{TEXT 261 2 ". " }{TEXT -1 54 "Ask Ma
ple to simplify the second integrand and to use " }{XPPEDIT 262 0 "u =
 x;" "6#/%\"uG%\"xG" }{TEXT 264 2 ". " }{TEXT -1 59 "This shows that s
ometimes we need to help Maple in its work" }{TEXT 263 1 "." }}}}
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