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ssignment 1 due date Feb 28, 2007" }}{PARA 0 "" 0 "" {TEXT 216 0 "" }}
{PARA 0 "" 0 "" {TEXT 216 75 "Student's name:                         \+
                                   " }}{PARA 0 "" 0 "" {TEXT 216 0 "" 
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restart; with(student);" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 217 9 "Probl
em 1" }}{PARA 0 "" 0 "" {TEXT 216 60 "Integrate the function f(x) = (1
+ln(x))*sqrt(1+(x*ln(x))^2)." }}{PARA 0 "" 0 "" {TEXT 216 0 "" }}
{PARA 0 "" 0 "" {TEXT 216 26 "a) First, try using using " }{TEXT 200 
3 "Int" }{TEXT 216 11 " and then  " }{TEXT 200 6 "value." }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "A:=Int((1+ln(x))*sqrt(1+(x*ln(x))^2
),x);value(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
216 0 "" }}{PARA 0 "" 0 "" {TEXT 216 45 "b) by using the substitution \+
u = x*ln(x) and " }{TEXT 200 87 "simplify.  You have to use the studen
t package to be able to use the command changevar." }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
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{TEXT 216 0 "" }}{PARA 0 "" 0 "" {TEXT 216 10 "c) Use the" }{TEXT 200 
6 " diff " }{TEXT 216 4 "and " }{TEXT 200 8 "simplify" }{TEXT 216 66 "
 commands to make sure that the answer obtained in (b) is correct." }}
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{PARA 0 "" 0 "" {TEXT 216 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 217 9 
"Problem 2" }}{PARA 0 "" 0 "" {TEXT 216 22 "Evaluate the integral " }
{XPPEDIT 18 0 "Int(x*ln(x+sqrt(1+x^2)), x);" "6#-%$IntG6$*&%\"xG\"\"\"
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}}{PARA 0 "" 0 "" {TEXT 216 12 "a) by using " }{TEXT 208 4 "Int " }
{TEXT 216 4 "and " }{TEXT 209 6 "value " }{TEXT 216 66 " - Again, you \+
will see that Maple doesn't know how to evaluate it." }}{EXCHG {PARA 0
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0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" 
{TEXT 216 0 "" }}{PARA 0 "" 0 "" {TEXT 216 38 "b) by using integration
 by parts with " }{XPPEDIT 18 0 "u = x;" "6#/%\"uG%\"xG" }{TEXT 216 
72 "  - you will see that, still, the integral doesn't evaluate comple
tely. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0
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0 "" }}}{PARA 0 "" 0 "" {TEXT 216 0 "" }}{PARA 0 "" 0 "" {TEXT 216 38 
"c) by using integration by parts with " }{XPPEDIT 18 0 "u = ln(x+sqrt
(1+x^2));" "6#/%\"uG-%#lnG6#,&%\"xG\"\"\"-%%sqrtG6#,&F*F**$)F)\"\"#F*F
*F*" }{TEXT 216 81 " - this should be successful. in order to eliminat
e \"arctanh\", use the command " }{TEXT 210 13 "convert(%,ln)" }{TEXT 
216 23 " like in the worksheet." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
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{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 217 9 "Problem 3" 
}}{PARA 0 "" 0 "" {TEXT 216 73 "Use Maple to find the integral of f(x)
 = (x^8+2*x^2+3)/((x+1)^2(x^2+4)^2 " }}{PARA 0 "" 0 "" {TEXT 216 84 "a
) First, find the partial fraction expansion of f(x) by using convert(
f,parfrac,x) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 216 0 "" }}{PARA 0 "" 0 "" 
{TEXT 216 47 "b) Now integrate the above and find its value. " }}
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{PARA 0 "" 0 "" {TEXT 216 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 217 9 
"Problem 4" }}{PARA 0 "" 0 "" {TEXT 216 22 "Consider the integral " }
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F'FjoF\\pF]pF^pF_pF`pFapFEFHFKFgpFhpF_q" }{TEXT 216 2 " :" }}{PARA 0 "
" 0 "" {TEXT 216 89 "a) Approximate the above by using LeftSum, RightS
um, MidSum, Trapezoidal, Simpson's rule " }}{PARA 0 "" 0 "" {TEXT 216 
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{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "%?;" "%#%?G" }}}{PARA 0 "" 0 "
" {TEXT 216 0 "" }}{PARA 0 "" 0 "" {TEXT 216 100 "b) How large should \+
'n' be to compute the integral to 4 digits of accuracy, using the Midp
oint Rule?" }}{PARA 0 "" 0 "" {TEXT 216 133 "Recall that the error bou
nd for Midpoint Rule is |E| < K(b-a)^3/(24n^2). (You can use the metho
d similar to what we used in lab1.mws)" }}{EXCHG {PARA 0 "> " 0 "" 
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216 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 217 0 "" }}{PARA 0 "" 0 "" 
{TEXT 200 0 "" }}{PARA 0 "" 0 "" {TEXT 216 86 " Integrate the function
 f(x) = x^2*arctan(x)/(1+x^2)^2.  The denominator is (1+x^2)^2." }}
{PARA 0 "" 0 "" {TEXT 216 0 "" }}{PARA 0 "" 0 "" {TEXT 216 13 "a)  by \+
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"b) by using substitution " }{XPPEDIT 18 0 "arctan(x) = theta;" "6#/-%
'arctanG6#%\"xG%&thetaG" }{TEXT 216 16 " and then using " }{TEXT 213 
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{TEXT 215 8 "simplify" }{TEXT 216 66 " commands to make sure that the \+
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