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{SECT 0 {PARA 3 "" 0 "" {TEXT -1 49 "Integrals involving inverse hyper
bolic functions " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanai
mo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version:  26.3.2007" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 "An integral involving arcsinh " }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 
-1 34 "The formula for the derivative of " }{XPPEDIT 18 0 "f(x)=arcsin
h*x" "6#/-%\"fG6#%\"xG*&%(arcsinhG\"\"\"F'F*" }{TEXT -1 32 " leads to \+
the standard integral:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "Int(1/sqrt(x^2+1),x) = arcsinh*x+c;" "6#/-%$IntG6$*&\"\"\"F(-%%s
qrtG6#,&*$%\"xG\"\"#F(F(F(!\"\"F.,&*&%(arcsinhGF(F.F(F(%\"cGF(" }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 40 "More generally, for a positive constant " }{TEXT 262 1 "a
" }{TEXT -1 3 ",  " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 
"Int(1/sqrt(x^2+a^2),x) = arcsinh(x/a)+c[1];" "6#/-%$IntG6$*&\"\"\"F(-
%%sqrtG6#,&*$%\"xG\"\"#F(*$%\"aGF/F(!\"\"F.,&-%(arcsinhG6#*&F.F(F1F2F(
&%\"cG6#F(F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = ln((x+sqrt(x^2+a^2))/a)+c[1];" "6#/%!G,&-%#lnG6#*&
,&%\"xG\"\"\"-%%sqrtG6#,&*$F+\"\"#F,*$%\"aGF2F,F,F,F4!\"\"F,&%\"cG6#F,
F," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 
"``=ln((x+sqrt(x^2+a^2)))+c[2]" "6#/%!G,&-%#lnG6#,&%\"xG\"\"\"-%%sqrtG
6#,&*$F*\"\"#F+*$%\"aGF1F+F+F+&%\"cG6#F1F+" }{TEXT -1 2 ", " }}{PARA 
0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "c[2]=c[1]-ln(a)" "6#/&%
\"cG6#\"\"#,&&F%6#\"\"\"F+-%#lnG6#%\"aG!\"\"" }{TEXT -1 1 "." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 "An int
egral involving arccosh " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";
" }}}{PARA 0 "" 0 "" {TEXT -1 35 "The formula for the derivative of  \+
" }{XPPEDIT 18 0 "f(x)=arccosh(x)" "6#/-%\"fG6#%\"xG-%(arccoshG6#F'" }
{TEXT -1 32 " leads to the standard integral:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1
/sqrt(x^2-1),x) = arccosh*x+c;" "6#/-%$IntG6$*&\"\"\"F(-%%sqrtG6#,&*$%
\"xG\"\"#F(F(!\"\"F0F.,&*&%(arccoshGF(F.F(F(%\"cGF(" }{TEXT -1 2 ", " 
}}{PARA 257 "" 0 "" {TEXT -1 7 "where  " }{XPPEDIT 18 0 "x>= 1" "6#1\"
\"\"%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 40 "More genera
lly, for a positive constant " }{TEXT 265 1 "a" }{TEXT -1 2 ", " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/sqrt(x^2-a^2),x
) = arccosh(x/a)+c[1];" "6#/-%$IntG6$*&\"\"\"F(-%%sqrtG6#,&*$%\"xG\"\"
#F(*$%\"aGF/!\"\"F2F.,&-%(arccoshG6#*&F.F(F1F2F(&%\"cG6#F(F(" }{TEXT 
-1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ln((x
+sqrt(x^2-a^2))/a)+c[1];" "6#/%!G,&-%#lnG6#*&,&%\"xG\"\"\"-%%sqrtG6#,&
*$F+\"\"#F,*$%\"aGF2!\"\"F,F,F4F5F,&%\"cG6#F,F," }{TEXT -1 1 " " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ln(x+sqrt(x^2-a^
2))+c[2];" "6#/%!G,&-%#lnG6#,&%\"xG\"\"\"-%%sqrtG6#,&*$F*\"\"#F+*$%\"a
GF1!\"\"F+F+&%\"cG6#F1F+" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 
6 "where " }{XPPEDIT 18 0 "x>=a" "6#1%\"aG%\"xG" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "c[2]=c[1]-ln(a)" "6#/&%\"cG6#\"\"#,&&F%6#\"\"\"F+-%#lnG
6#%\"aG!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 "An integral involving arcsech " }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 
-1 34 "The formula for the derivative of " }{XPPEDIT 18 0 "f(x)=arcsec
h*x" "6#/-%\"fG6#%\"xG*&%(arcsechG\"\"\"F'F*" }{TEXT -1 32 " leads to \+
the standard integral:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "Int(1/(x*sqrt(1-x^2)),x) = -arcsech*x+c;" "6#/-%$IntG6$*&\"\"\"F
(*&%\"xGF(-%%sqrtG6#,&F(F(*$F*\"\"#!\"\"F(F1F*,&*&%(arcsechGF(F*F(F1%
\"cGF(" }{TEXT -1 1 "," }}{PARA 257 "" 0 "" {TEXT -1 6 "where " }
{XPPEDIT 18 0 "0<x" "6#2\"\"!%\"xG" }{XPPEDIT 18 0 "``<1" "6#2%!G\"\"
\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 40 "More generally, fo
r a positive constant " }{TEXT 263 1 "a" }{TEXT -1 3 ",  " }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/(x*sqrt(a^2-x^2)),x) \+
= -1/a;" "6#/-%$IntG6$*&\"\"\"F(*&%\"xGF(-%%sqrtG6#,&*$%\"aG\"\"#F(*$F
*F1!\"\"F(F3F*,$*&F(F(F0F3F3" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arcsech(
x/a)+c" "6#,&-%(arcsechG6#*&%\"xG\"\"\"%\"aG!\"\"F)%\"cGF)" }{TEXT -1 
1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= -1/a" "6#
/%!G,$*&\"\"\"F'%\"aG!\"\"F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln((a+sq
rt(a^2-x^2))/x)+c;" "6#,&-%#lnG6#*&,&%\"aG\"\"\"-%%sqrtG6#,&*$F)\"\"#F
**$%\"xGF0!\"\"F*F*F2F3F*%\"cGF*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" 
{TEXT -1 6 "where " }{XPPEDIT 18 0 "0<x" "6#2\"\"!%\"xG" }{XPPEDIT 18 
0 "``<a" "6#2%!G%\"aG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 "An integral involving arccsc
h " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" 
{TEXT -1 34 "The formula for the derivative of " }{XPPEDIT 18 0 "f(x)=
arccsch*x" "6#/-%\"fG6#%\"xG*&%(arccschG\"\"\"F'F*" }{TEXT -1 32 " lea
ds to the standard integral:" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }
{XPPEDIT 18 0 "Int(1/(x*sqrt(1+x^2)),x) = -arccsch(abs(x))+c;" "6#/-%$
IntG6$*&\"\"\"F(*&%\"xGF(-%%sqrtG6#,&F(F(*$F*\"\"#F(F(!\"\"F*,&-%(arcc
schG6#-%$absG6#F*F1%\"cGF(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "More generally, for a positive \+
constant " }{TEXT 264 1 "a" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/(x*sqrt(a^2+x^2)),x) = -1/a" "6#/
-%$IntG6$*&\"\"\"F(*&%\"xGF(-%%sqrtG6#,&*$%\"aG\"\"#F(*$F*F1F(F(!\"\"F
*,$*&F(F(F0F3F3" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arccsch(abs(x)/a)+c" 
"6#,&-%(arccschG6#*&-%$absG6#%\"xG\"\"\"%\"aG!\"\"F,%\"cGF," }{TEXT 
-1 2 "  " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= -1/a
" "6#/%!G,$*&\"\"\"F'%\"aG!\"\"F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(
(a+sqrt(a^2+x^2))/abs(x))+c" "6#,&-%#lnG6#*&,&%\"aG\"\"\"-%%sqrtG6#,&*
$F)\"\"#F**$%\"xGF0F*F*F*-%$absG6#F2!\"\"F*%\"cGF*" }{TEXT -1 2 ". " }
}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Summary " }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 
18 0 "Int(1/sqrt(x^2+a^2),x)=arcsinh(x/a)+c[1]" "6#/-%$IntG6$*&\"\"\"F
(-%%sqrtG6#,&*$%\"xG\"\"#F(*$%\"aGF/F(!\"\"F.,&-%(arcsinhG6#*&F.F(F1F2
F(&%\"cG6#F(F(" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "ln(x+sqrt(x^2+a^2))+
c[2]" "6#,&-%#lnG6#,&%\"xG\"\"\"-%%sqrtG6#,&*$F(\"\"#F)*$%\"aGF/F)F)F)
&%\"cG6#F/F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/sqrt(x^2-a^2),x) = ar
ccosh(x/a)+c[1]" "6#/-%$IntG6$*&\"\"\"F(-%%sqrtG6#,&*$%\"xG\"\"#F(*$%
\"aGF/!\"\"F2F.,&-%(arccoshG6#*&F.F(F1F2F(&%\"cG6#F(F(" }{TEXT -1 3 " \+
= " }{XPPEDIT 18 0 "ln(x+sqrt(x^2-a^2))+c[2]" "6#,&-%#lnG6#,&%\"xG\"\"
\"-%%sqrtG6#,&*$F(\"\"#F)*$%\"aGF/!\"\"F)F)&%\"cG6#F/F)" }{TEXT -1 1 "
 " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+
" }{XPPEDIT 18 0 "Int(1/(x*sqrt(a^2-x^2)),x) = -1/a;" "6#/-%$IntG6$*&
\"\"\"F(*&%\"xGF(-%%sqrtG6#,&*$%\"aG\"\"#F(*$F*F1!\"\"F(F3F*,$*&F(F(F0
F3F3" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arcsech(x/a)+c = -1/a;" "6#/,&-%
(arcsechG6#*&%\"xG\"\"\"%\"aG!\"\"F*%\"cGF*,$*&F*F*F+F,F," }{TEXT -1 
1 " " }{XPPEDIT 18 0 "ln((a+sqrt(a^2-x^2))/x)+c" "6#,&-%#lnG6#*&,&%\"a
G\"\"\"-%%sqrtG6#,&*$F)\"\"#F**$%\"xGF0!\"\"F*F*F2F3F*%\"cGF*" }{TEXT 
-1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 
1 " " }{XPPEDIT 18 0 "Int(1/(x*sqrt(a^2+x^2)),x) = -1/a;" "6#/-%$IntG6
$*&\"\"\"F(*&%\"xGF(-%%sqrtG6#,&*$%\"aG\"\"#F(*$F*F1F(F(!\"\"F*,$*&F(F
(F0F3F3" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arccsch(abs(x)/a)+c = -1/a;" 
"6#/,&-%(arccschG6#*&-%$absG6#%\"xG\"\"\"%\"aG!\"\"F-%\"cGF-,$*&F-F-F.
F/F/" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln((a+sqrt(a^2+x^2))/abs(x))+c;
" "6#,&-%#lnG6#*&,&%\"aG\"\"\"-%%sqrtG6#,&*$F)\"\"#F**$%\"xGF0F*F*F*-%
$absG6#F2!\"\"F*%\"cGF*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "
" 0 "" {TEXT -1 9 "Examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 31 "C
onsider the definite integral " }{XPPEDIT 18 0 "Int(1/sqrt(x^2+4),x = \+
0 .. 1);" "6#-%$IntG6$*&\"\"\"F'-%%sqrtG6#,&*$%\"xG\"\"#F'\"\"%F'!\"\"
/F-;\"\"!F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 53 "Using the \+
formula above, this integral is equal to:  " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/sqrt(x^2+4),x = 0 .. 1) =arcsinh(
x/2)" "6#/-%$IntG6$*&\"\"\"F(-%%sqrtG6#,&*$%\"xG\"\"#F(\"\"%F(!\"\"/F.
;\"\"!F(-%(arcsinhG6#*&F.F(F/F1" }{TEXT -1 2 "  " }{XPPEDIT 18 0 "PIEC
EWISE([1, ``],[``, ``],[0, ``]) = arcsinh(1/2);" "6#/-%*PIECEWISEG6%7$
\"\"\"%!G7$F)F)7$\"\"!F)-%(arcsinhG6#*&F(F(\"\"#!\"\"" }{TEXT -1 1 " \+
" }{TEXT 267 1 "~" }{TEXT -1 17 " 0.4812118251,   " }}{PARA 257 "" 0 "
" {TEXT -1 4 "or  " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 
"Int(1/sqrt(x^2+4),x = 0 .. 1)=ln(x+sqrt(x^2+4))" "6#/-%$IntG6$*&\"\"
\"F(-%%sqrtG6#,&*$%\"xG\"\"#F(\"\"%F(!\"\"/F.;\"\"!F(-%#lnG6#,&F.F(-F*
6#,&*$F.F/F(F0F(F(" }{TEXT -1 2 "  " }{XPPEDIT 18 0 "PIECEWISE([1, ``]
,[``, ``],[0, ``]);" "6#-%*PIECEWISEG6%7$\"\"\"%!G7$F(F(7$\"\"!F(" }
{TEXT -1 3 "   " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``
 = ln(1+sqrt(5))-ln*2;" "6#/%!G,&-%#lnG6#,&\"\"\"F*-%%sqrtG6#\"\"&F*F*
*&F'F*\"\"#F*!\"\"" }{XPPEDIT 18 0 "``= ln((1+sqrt(5))/2)" "6#/%!G-%#l
nG6#*&,&\"\"\"F*-%%sqrtG6#\"\"&F*F*\"\"#!\"\"" }{TEXT -1 1 " " }{TEXT 
268 1 "~" }{TEXT -1 15 " 0.4812118251. " }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "ln((1+sqrt(5))/2);\neva
lf(evalf(%,13));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#,&#\"\"\"
\"\"#F(*&F'F(-%%sqrtG6#\"\"&F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$
\"+^#=@\"[!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 132 "Maple's symbolic integration procedure gives the result \+
using the arcsinh function, as in the first part of the formula given \+
above." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 56 "Int(1/sqrt(x^2+4),x=0..1);\nvalue(%);\nevalf(evalf(%,
13));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'*$-%%sqrtG
6#,&*$)%\"xG\"\"#F'F'\"\"%F'F'!\"\"/F/;\"\"!F'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%(arcsinhG6##\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#$\"+^#=@\"[!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 
10 "Example 2 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
{PARA 0 "" 0 "" {TEXT -1 31 "Consider the definite integral " }
{XPPEDIT 18 0 "Int(1/sqrt(x^2+4*x+5),x = -2 .. 2);" "6#-%$IntG6$*&\"\"
\"F'-%%sqrtG6#,(*$%\"xG\"\"#F'*&\"\"%F'F-F'F'\"\"&F'!\"\"/F-;,$F.F2F.
" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 65 "By completing the squ
are the integral can be written in the form:" }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/sqrt((x+2)^2+1),x = -2 .. 2)" "6#
-%$IntG6$*&\"\"\"F'-%%sqrtG6#,&*$,&%\"xGF'\"\"#F'F/F'F'F'!\"\"/F.;,$F/
F0F/" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "Substituting " }
{XPPEDIT 18 0 "u=x+2" "6#/%\"uG,&%\"xG\"\"\"\"\"#F'" }{TEXT -1 22 " gi
ves the integral:  " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "Int(1/sqrt(u^2+1),u=0..4)" "6#-%$IntG6$*&\"\"\"F'-%%sqrtG6#,&*$%\"u
G\"\"#F'F'F'!\"\"/F-;\"\"!\"\"%" }{TEXT -1 3 ",  " }}{PARA 0 "" 0 "" 
{TEXT -1 14 "which is then " }}{PARA 256 "" 0 "" {TEXT -1 3 "   " }
{XPPEDIT 18 0 "arcsinh*u;" "6#*&%(arcsinhG\"\"\"%\"uGF%" }{TEXT -1 1 "
 " }{XPPEDIT 18 0 "PIECEWISE([4 ,`` ],[0 , ``])" "6#-%*PIECEWISEG6$7$
\"\"%%!G7$\"\"!F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = arcsinh*4;" "6#/
%!G*&%(arcsinhG\"\"\"\"\"%F'" }{XPPEDIT 18 0 "``=ln(4+sqrt(17))" "6#/%
!G-%#lnG6#,&\"\"%\"\"\"-%%sqrtG6#\"#<F*" }{TEXT -1 1 " " }{TEXT 266 1 
"~" }{TEXT -1 14 " 2.094712547. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "ln(4+sqrt(17));\nevalf(evalf
[13](%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#,&\"\"%\"\"\"*$\"
#<#F(\"\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ZDr%4#!\"*" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "Maple's s
ymbolic integration procedure gives essentially the same result." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
61 "Int(1/sqrt(x^2+4*x+5),x=-2..2);\nvalue(%);\nevalf(evalf(%,13));" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'*$-%%sqrtG6#,(*$)%
\"xG\"\"#F'F'*&\"\"%F'F/F'F'\"\"&F'F'!\"\"/F/;!\"#F0" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#,$-%#lnG6#,&!\"%\"\"\"*$-%%sqrtG6#\"#<F)F)!\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ZDr%4#!\"*" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 
1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 3 " }}{PARA 0 "" 0 "" {TEXT -1 
31 "Consider the definite integral " }{XPPEDIT 18 0 "Int(1/sqrt(4*x^2-
9),x = 2 .. 5);" "6#-%$IntG6$*&\"\"\"F'-%%sqrtG6#,&*&\"\"%F'*$%\"xG\"
\"#F'F'\"\"*!\"\"F2/F/;F0\"\"&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 66 "We can use the formula above by writing the integral in t
he form  " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 1 "
 " }{XPPEDIT 18 0 "Int(1/sqrt(x^2-9/4),x = 2 .. 5)" "6#-%$IntG6$*&\"\"
\"F'-%%sqrtG6#,&*$%\"xG\"\"#F'*&\"\"*F'\"\"%!\"\"F2F2/F-;F.\"\"&" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 28 "This integral is equal \+
to:  " }}{PARA 256 "" 0 "" {TEXT -1 4 "    " }{XPPEDIT 18 0 "1/2" "6#*
&\"\"\"F$\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arccosh(2*x/3);" 
"6#-%(arccoshG6#*(\"\"#\"\"\"%\"xGF(\"\"$!\"\"" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "PIECEWISE([5, ``],[``, ``],[2, ``]);" "6#-%*PIECEWISEG6
%7$\"\"&%!G7$F(F(7$\"\"#F(" }{TEXT -1 2 "  " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/2;" "6#/%!G*&\"\"\"F&\"\"#!\"\"
" }{TEXT -1 2 "  " }{XPPEDIT 18 0 "[arccosh(10/3) - arccosh(4/3)]" "6#
7#,&-%(arccoshG6#*&\"#5\"\"\"\"\"$!\"\"F*-F&6#*&\"\"%F*F+F,F," }{TEXT 
-1 1 " " }{TEXT 269 1 "~" }{TEXT -1 15 " 0.5392273907, " }}{PARA 257 "
" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "1/2" "6#*&
\"\"\"F$\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(x+sqrt(x^2-9/4)
);" "6#-%#lnG6#,&%\"xG\"\"\"-%%sqrtG6#,&*$F'\"\"#F(*&\"\"*F(\"\"%!\"\"
F2F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([5, ``],[``, ``],[2, `
`]);" "6#-%*PIECEWISEG6%7$\"\"&%!G7$F(F(7$\"\"#F(" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`` = 1/2;" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 1 " \+
" }{XPPEDIT 18 0 "[ln(5+sqrt(91/4))-ln(2+sqrt(7/4))]=1/2" "6#/7#,&-%#l
nG6#,&\"\"&\"\"\"-%%sqrtG6#*&\"#\"*F+\"\"%!\"\"F+F+-F'6#,&\"\"#F+-F-6#
*&\"\"(F+F1F2F+F2*&F+F+F6F2" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[ln((10+s
qrt(91))/(4+sqrt(7)))]" "6#7#-%#lnG6#*&,&\"#5\"\"\"-%%sqrtG6#\"#\"*F*F
*,&\"\"%F*-F,6#\"\"(F*!\"\"" }{TEXT -1 1 " " }{TEXT 270 1 "~" }{TEXT 
-1 15 " 0.5392273907. " }}{PARA 0 "" 0 "" {TEXT -1 62 "We can obtain a
 numerical value for this integral thus . . . ." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "(arccosh(10/
3)-arccosh(4/3))/2;\nevalf(evalf[13](%));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,&*&#\"\"\"\"\"#F&-%(arccoshG6##\"#5\"\"$F&F&*&#F&F'F&-
F)6##\"\"%F-F&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+2RF#R&!#5" 
}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 " . . .
 or thus . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 53 "ln((10+sqrt(91))/(4+sqrt(7)))/2;\nevalf(evalf[
13](%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"#F&-%#lnG6#
*&,&\"#5F&*$\"#\"*F%F&F&,&\"\"%F&*$\"\"(F%F&!\"\"F&F&" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#$\"+2RF#R&!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 61 "Maple's symbolic integration procedure gi
ves the same result." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 73 "Int(1/sqrt(4*x^2-9),x=2..5);\nvalue(%);\n
combine(%,ln);\nevalf(evalf(%,13));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#-%$IntG6$*&\"\"\"F'*$-%%sqrtG6#,&*$)%\"xG\"\"#F'\"\"%\"\"*!\"\"F'F3/F
/;F0\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%#lnG6#,&\"#5\"\"\"*
$-%%sqrtG6#\"#\"*F*F*F*-F-6#\"\"%F*#F*F2*&#F*F2F**&-F&6#,&F2F**$-F-6#
\"\"(F*F*F*F0F*F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%%sqrtG
6#\"\"%\"\"\"-%#lnG6#*&,&\"#5F)*&-F&6#\"#8F)-F&6#\"\"(F)F)F),&F(F)*$F4
F)F)!\"\"F)#F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+2RF#R&!#5" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 4 " }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 33 "C
onsider the indefinite integral " }{XPPEDIT 18 0 "Int(1/sqrt(x^2-2*x),
x);" "6#-%$IntG6$*&\"\"\"F'-%%sqrtG6#,&*$%\"xG\"\"#F'*&F.F'F-F'!\"\"F0
F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 66 "By completing this \+
square the integral can be written in the form:" }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/sqrt((x-1)^2-1),x);" "6#-%$IntG6$
*&\"\"\"F'-%%sqrtG6#,&*$,&%\"xGF'F'!\"\"\"\"#F'F'F/F/F." }{TEXT -1 1 "
." }}{PARA 0 "" 0 "" {TEXT -1 13 "Substituting " }{XPPEDIT 18 0 "u = x
-1;" "6#/%\"uG,&%\"xG\"\"\"F'!\"\"" }{TEXT -1 22 " gives the integral:
  " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/sqrt(u^2-
1),u);" "6#-%$IntG6$*&\"\"\"F'-%%sqrtG6#,&*$%\"uG\"\"#F'F'!\"\"F/F-" }
{TEXT -1 3 ",  " }}{PARA 0 "" 0 "" {TEXT -1 14 "which is then " }}
{PARA 256 "" 0 "" {TEXT -1 3 "   " }{XPPEDIT 18 0 "arccosh*u+c;" "6#,&
*&%(arccoshG\"\"\"%\"uGF&F&%\"cGF&" }{TEXT -1 3 "   " }}{PARA 256 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = arccosh(x-1)+c;" "6#/%!G,&-%(
arccoshG6#,&%\"xG\"\"\"F+!\"\"F+%\"cGF+" }{XPPEDIT 18 0 "`` = ln(x-1+s
qrt(x^2-2*x))+c;" "6#/%!G,&-%#lnG6#,(%\"xG\"\"\"F+!\"\"-%%sqrtG6#,&*$F
*\"\"#F+*&F2F+F*F+F,F+F+%\"cGF+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Maple's symbolic integrat
ion procedure gives the same result." }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Int(1/sqrt(x^2-2*x),x);\nv
alue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'*$-%%sq
rtG6#,&*$)%\"xG\"\"#F'F'*&F0F'F/F'!\"\"F'F2F/" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%#lnG6#,(%\"xG\"\"\"F(!\"\"*$-%%sqrtG6#,&*$)F'\"\"#F(F
(*&F1F(F'F(F)F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "E
xample 5 " }}{PARA 0 "" 0 "" {TEXT -1 31 "Consider the definite integr
al " }{XPPEDIT 18 0 "Int(1/(x*sqrt(9-x^2)),x=1..2)" "6#-%$IntG6$*&\"\"
\"F'*&%\"xGF'-%%sqrtG6#,&\"\"*F'*$F)\"\"#!\"\"F'F1/F);F'F0" }{TEXT -1 
1 "." }}{PARA 0 "" 0 "" {TEXT -1 53 "Using the formula above, this int
egral is equal to:  " }}{PARA 256 "" 0 "" {TEXT -1 4 "    " }{XPPEDIT 
18 0 "-1/3" "6#,$*&\"\"\"F%\"\"$!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 18 
0 "arcsech(x/3);" "6#-%(arcsechG6#*&%\"xG\"\"\"\"\"$!\"\"" }{TEXT -1 
1 " " }{XPPEDIT 18 0 "PIECEWISE([2, ``],[``, ``],[1, ``]);" "6#-%*PIEC
EWISEG6%7$\"\"#%!G7$F(F(7$\"\"\"F(" }{TEXT -1 4 "    " }}{PARA 256 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -1/3;" "6#/%!G,$*&\"\"\"F'\"
\"$!\"\"F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[ arcsech(2/3)-arcsech(1/3
)]" "6#7#,&-%(arcsechG6#*&\"\"#\"\"\"\"\"$!\"\"F*-F&6#*&F*F*F+F,F," }
{TEXT -1 1 " " }{TEXT 271 1 "~" }{TEXT -1 15 " 0.2667745080, " }}
{PARA 257 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 3 "   \+
" }{XPPEDIT 18 0 "-1/3" "6#,$*&\"\"\"F%\"\"$!\"\"F'" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "ln((3+sqrt(9-x^2))/x);" "6#-%#lnG6#*&,&\"\"$\"\"\"-%%sq
rtG6#,&\"\"*F)*$%\"xG\"\"#!\"\"F)F)F0F2" }{TEXT -1 1 " " }{XPPEDIT 18 
0 "PIECEWISE([2, ``],[``, ``],[1, ``]);" "6#-%*PIECEWISEG6%7$\"\"#%!G7
$F(F(7$\"\"\"F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "``= -1/3" "6#/%!G,$*&\"\"\"F'\"\"$!\"\"F)" }{TEXT -1 2 
"  " }{XPPEDIT 18 0 "[ln((3+sqrt(5))/2)-ln(3+sqrt(8))];" "6#7#,&-%#lnG
6#*&,&\"\"$\"\"\"-%%sqrtG6#\"\"&F+F+\"\"#!\"\"F+-F&6#,&F*F+-F-6#\"\")F
+F1" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "``= -1/3" "6#/%!G,$*&\"\"\"F'\"\"$!\"\"F)" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "[ ln((3+sqrt(5))/(2*(3+sqrt(8))) ]" "6#7#-%#lnG6#*&,&\"
\"$\"\"\"-%%sqrtG6#\"\"&F*F**&\"\"#F*,&F)F*-F,6#\"\")F*F*!\"\"" }
{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "``=
 1/3" "6#/%!G*&\"\"\"F&\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[ l
n((2*(3+sqrt(8))/(3+sqrt(5))) ]" "6#7#-%#lnG6#*(\"\"#\"\"\",&\"\"$F)-%
%sqrtG6#\"\")F)F),&F+F)-F-6#\"\"&F)!\"\"" }{TEXT -1 1 " " }{TEXT 272 
1 "~" }{TEXT -1 15 " 0.2667745080. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 60 "We can obtain a decimal value for this i
ntegral thus . . . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 51 "-(arcsech(2/3)-arcsech(1/3))/3;\nevalf(ev
alf(%,13));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%(arcsechG6##\"\"#\"
\"$#!\"\"F)*&#\"\"\"F)F.-F%6#F-F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#$\"+!3Xxm#!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 21 " . . . or thus . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "ln(2*(3+sqrt(8))/(3+sqrt(5))
)/3;\nevalf(evalf(%,13));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%#lnG6
#,$*&,&\"\"$\"\"\"*&\"\"#F+-%%sqrtG6#F-F+F+F+,&F*F+*$-F/6#\"\"&F+F+!\"
\"F-#F+F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+!3Xxm#!#5" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "Maple's symboli
c integration procedure gives a result involving arctanh. The function
  " }{XPPEDIT 18 0 "Re" "6#%#ReG" }{TEXT -1 24 " (with Maple input for
m " }{TEXT 0 2 "Re" }{TEXT -1 71 ") which takes the real part of a com
plex number is clearly superfluous." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "Int(1/(x*sqrt(9-x^2)),x=1..
2);\nvalue(%);\nconvert(%,ln);\nevalf(evalf(%,13));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'*&%\"xGF',&\"\"*F'*$)F)\"\"#F'!\"
\"#F'F.F//F);F'F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&#\"\"\"\"\"$F
&-%#ReG6#-%(arctanhG6#,$*(F'F&\"\"&!\"\"F0#F&\"\"#F&F&F1*&#F&F'F&-F)6#
-F,6#,$*(F'F&\"\"%F1F3F2F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&
#\"\"\"\"\"'F&-%#lnG6#,&*(\"\"$F&\"\"&!\"\"F.#F&\"\"#F&F&F&F&F/*&#F&F'
F&-F)6#,&F&F/*(F-F&F.F/F.F0F&F&F&*&F3F&-F)6#,&*(F-F&\"\"%F/F1F0F&F&F&F
&F&*&#F&F'F&-F)6#,&F&F/*(F-F&F=F/F1F0F&F&F/" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+!3Xxm#!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 10 "Example 6 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "
;" }}}{PARA 0 "" 0 "" {TEXT -1 31 "Consider the definite integral " }
{XPPEDIT 18 0 "Int(1/((2*x-3)*sqrt(7+12*x-4*x^2)),x = 2 .. 3);" "6#-%$
IntG6$*&\"\"\"F'*&,&*&\"\"#F'%\"xGF'F'\"\"$!\"\"F'-%%sqrtG6#,(\"\"(F'*
&\"#7F'F,F'F'*&\"\"%F'*$F,F+F'F.F'F./F,;F+F-" }{TEXT -1 2 ". " }}
{PARA 0 "" 0 "" {TEXT -1 65 "By completing the square the integral can
 be written in the form:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "Int(1/((2*x-3)*sqrt(16-(2*x-3)^2)),x = 2 .. 3);" "6#-%$
IntG6$*&\"\"\"F'*&,&*&\"\"#F'%\"xGF'F'\"\"$!\"\"F'-%%sqrtG6#,&\"#;F'*$
,&*&F+F'F,F'F'F-F.F+F.F'F./F,;F+F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 13 "Substituting " }{XPPEDIT 18 0 "u = 2*x-3;" "6#/%\"uG,&*&
\"\"#\"\"\"%\"xGF(F(\"\"$!\"\"" }{TEXT -1 22 " gives the integral:  " 
}}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/2;" "6#*&\"\"\"F$
\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/(u*sqrt(16-u^2)),u =
 1 .. 3);" "6#-%$IntG6$*&\"\"\"F'*&%\"uGF'-%%sqrtG6#,&\"#;F'*$F)\"\"#!
\"\"F'F1/F);F'\"\"$" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 14 "w
hich is then " }}{PARA 256 "" 0 "" {TEXT -1 3 "   " }{XPPEDIT 18 0 "-1
/8;" "6#,$*&\"\"\"F%\"\")!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln((
4+sqrt(16-u^2))/u);" "6#-%#lnG6#*&,&\"\"%\"\"\"-%%sqrtG6#,&\"#;F)*$%\"
uG\"\"#!\"\"F)F)F0F2" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([3, ``
],[``, ``],[1, ``]);" "6#-%*PIECEWISEG6%7$\"\"$%!G7$F(F(7$\"\"\"F(" }
{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "``=
 -1/8" "6#/%!G,$*&\"\"\"F'\"\")!\"\"F)" }{TEXT -1 1 " " }{XPPEDIT 18 
0 "[ ln((4+sqrt(7))/3) - ln((4+sqrt(15)))]" "6#7#,&-%#lnG6#*&,&\"\"%\"
\"\"-%%sqrtG6#\"\"(F+F+\"\"$!\"\"F+-F&6#,&F*F+-F-6#\"#:F+F1" }{TEXT 
-1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 
1 " " }{XPPEDIT 18 0 "`` = -1/8" "6#/%!G,$*&\"\"\"F'\"\")!\"\"F)" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "[ln((4+sqrt(7))/(3*(4+sqrt(15)))]" "6#7
#-%#lnG6#*&,&\"\"%\"\"\"-%%sqrtG6#\"\"(F*F**&\"\"$F*,&F)F*-F,6#\"#:F*F
*!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "
" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/8;" "6#/%!G*&\"\"\"F&\"\"
)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[ln(3*(4+sqrt(15))/(4+sqrt(7))
)];" "6#7#-%#lnG6#*(\"\"$\"\"\",&\"\"%F)-%%sqrtG6#\"#:F)F),&F+F)-F-6#
\"\"(F)!\"\"" }{TEXT -1 1 " " }{TEXT 273 1 "~" }{TEXT -1 15 " 0.158508
9510. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "ln(3*(4+sqrt(15))/(4+sqr
t(7)))/8;\nevalf(evalf(%,13));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%
#lnG6#,$*&,&\"\"%\"\"\"*$-%%sqrtG6#\"#:F+F+F+,&F*F+*$-F.6#\"\"(F+F+!\"
\"\"\"$#F+\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+5&*3&e\"!#5" }}
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "Maple's \+
symbolic integration procedure gives a result involving arctanh." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
73 "Int(1/((2*x-3)*sqrt(7-4*x^2+12*x)),x=2..3);\nvalue(%);\nevalf(eval
f(%,13));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'*&,&%
\"xG\"\"#\"\"$!\"\"F'-%%sqrtG6#,(\"\"(F'*&\"\"%F')F*F+F'F-*&\"#7F'F*F'
F'F'F-/F*;F+F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%#ReG6#-%(arctanh
G6#,$*$-%%sqrtG6#\"\"(\"\"\"#\"\"%F/#!\"\"\"\")*&#F0F5F0-F%6#-F(6#,$*$
-F-6#\"#:F0#F2F@F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+5&*3&e\"!#
5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 7 \+
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" 
{TEXT -1 31 "Consider the definite integral " }{XPPEDIT 18 0 "Int(1/(x
*sqrt(4+x^2)),x = 1 .. 3);" "6#-%$IntG6$*&\"\"\"F'*&%\"xGF'-%%sqrtG6#,
&\"\"%F'*$F)\"\"#F'F'!\"\"/F);F'\"\"$" }{TEXT -1 1 "." }}{PARA 0 "" 0 
"" {TEXT -1 53 "Using the formula above, this integral is equal to:  \+
" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(1/(x*sqrt(4+x
^2)),x = 1 .. 3)=-1/2 " "6#/-%$IntG6$*&\"\"\"F(*&%\"xGF(-%%sqrtG6#,&\"
\"%F(*$F*\"\"#F(F(!\"\"/F*;F(\"\"$,$*&F(F(F1F2F2" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "arccsch(x/2);" "6#-%(arccschG6#*&%\"xG\"\"\"\"\"#!\"\"
" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([3, ``],[``, ``],[1, ``]);
" "6#-%*PIECEWISEG6%7$\"\"$%!G7$F(F(7$\"\"\"F(" }{TEXT -1 3 "   " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -1/2;" "6#/%!G,$
*&\"\"\"F'\"\"#!\"\"F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[arccsch(3/2)-
arccsch(1/2)];" "6#7#,&-%(arccschG6#*&\"\"$\"\"\"\"\"#!\"\"F*-F&6#*&F*
F*F+F,F," }{TEXT -1 1 " " }{TEXT 276 1 "~" }{TEXT -1 15 " 0.4092451790
, " }}{PARA 257 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 
1 " " }{XPPEDIT 18 0 "Int(1/(x*sqrt(4+x^2)),x = 1 .. 3) = -1/2 " "6#/-
%$IntG6$*&\"\"\"F(*&%\"xGF(-%%sqrtG6#,&\"\"%F(*$F*\"\"#F(F(!\"\"/F*;F(
\"\"$,$*&F(F(F1F2F2" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln((2+sqrt(4+x^2)
)/abs(x));" "6#-%#lnG6#*&,&\"\"#\"\"\"-%%sqrtG6#,&\"\"%F)*$%\"xGF(F)F)
F)-%$absG6#F0!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([3, ``],
[``, ``],[1, ``]);" "6#-%*PIECEWISEG6%7$\"\"$%!G7$F(F(7$\"\"\"F(" }
{TEXT -1 2 "  " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` \+
= -1/2;" "6#/%!G,$*&\"\"\"F'\"\"#!\"\"F)" }{TEXT -1 2 "  " }{XPPEDIT 
18 0 "[ln((2+sqrt(13))/3)-ln(2+sqrt(5))];" "6#7#,&-%#lnG6#*&,&\"\"#\"
\"\"-%%sqrtG6#\"#8F+F+\"\"$!\"\"F+-F&6#,&F*F+-F-6#\"\"&F+F1" }{TEXT 
-1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 
-1 1 " " }{XPPEDIT 18 0 "``=-1/2" "6#/%!G,$*&\"\"\"F'\"\"#!\"\"F)" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "ln((2+sqrt(13))/(3*(2+sqrt(5)))" "6#-%#
lnG6#*&,&\"\"#\"\"\"-%%sqrtG6#\"#8F)F)*&\"\"$F),&F(F)-F+6#\"\"&F)F)!\"
\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "
" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= 1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"
" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln((3*(2+sqrt(5)))/(2+sqrt(13)))" "6
#-%#lnG6#*(\"\"$\"\"\",&\"\"#F(-%%sqrtG6#\"\"&F(F(,&F*F(-F,6#\"#8F(!\"
\"" }{TEXT -1 1 " " }{TEXT 275 1 "~" }{TEXT -1 15 " 0.4092451790. " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "We can ob
tain a numerical value for this integral thus . . . ." }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "-(arccsch
(3/2)-arccsch(1/2))/2;\nevalf(evalf[13](%));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,&*&#\"\"\"\"\"#F&-%(arccschG6##\"\"$F'F&!\"\"*&#F&F'F&
-F)6#F/F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+!z^C4%!#5" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 " . . . or
 thus . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 56 "ln((3*(2+sqrt(5)))/(2+sqrt(13)))/2;\nevalf(evalf[1
3](%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"#F&-%#lnG6#,
$*(\"\"$F&,&F'F&*$\"\"&F%F&F&,&F'F&*$\"#8F%F&!\"\"F&F&F&" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#$\"+!z^C4%!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 73 "Maple's symbolic integration procedure \+
gives a result involving arctanh. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "Int(1/(x*sqrt(4+x^2)),x=1..3
);\nvalue(%);\nevalf(evalf(%,13));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
-%$IntG6$*&\"\"\"F'*&%\"xGF'-%%sqrtG6#,&\"\"%F'*$)F)\"\"#F'F'F'!\"\"/F
);F'\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&-%(arctanhG6#,$*$-%%sqr
tG6#\"#8\"\"\"#\"\"#F,#!\"\"F/*&#F-F/F--F%6#,$*$-F*6#\"\"&F-#F/F:F-F-
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+!z^C4%!#5" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 
1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 8 " }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 33 "Consider the indefi
nite integral " }{XPPEDIT 18 0 "Int(tan*x/sqrt(1+cos^2*x),x);" "6#-%$I
ntG6$*(%$tanG\"\"\"%\"xGF(-%%sqrtG6#,&F(F(*&%$cosG\"\"#F)F(F(!\"\"F)" 
}{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "In
t(tan*x/sqrt(1+cos^2*x),x)=Int(sin*x/(cos*x*sqrt(1+cos^2*x)),x)" "6#/-
%$IntG6$*(%$tanG\"\"\"%\"xGF)-%%sqrtG6#,&F)F)*&%$cosG\"\"#F*F)F)!\"\"F
*-F%6$*(%$sinGF)F*F)*(F0F)F*F)-F,6#,&F)F)*&F0F1F*F)F)F)F2F*" }{TEXT 
-1 10 "    ...   " }{XPPEDIT 18 0 "PIECEWISE([u=cos*x,``],[du=-sin*x*d
x,``])" "6#-%*PIECEWISEG6$7$/%\"uG*&%$cosG\"\"\"%\"xGF+%!G7$/%#duG,$*(
%$sinGF+F,F+%#dxGF+!\"\"F-" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT 
-1 2 "  " }{XPPEDIT 18 0 "`` = -Int(1/(u*sqrt(1+u^2)),u);" "6#/%!G,$-%
$IntG6$*&\"\"\"F**&%\"uGF*-%%sqrtG6#,&F*F**$F,\"\"#F*F*!\"\"F,F3" }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = arccsch(abs(u))+c;" "6#/%!G,&-%(ar
ccschG6#-%$absG6#%\"uG\"\"\"%\"cGF-" }{XPPEDIT 18 0 " ``= arccsch(abs(
cos*x))+c" "6#/%!G,&-%(arccschG6#-%$absG6#*&%$cosG\"\"\"%\"xGF.F.%\"cG
F." }{TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT -1 16 "Alternatively,  \+
" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-Int(1/(u*sqrt(1+
u^2)),u)=ln((1+sqrt(1+u^2))/abs(u))+c" "6#/,$-%$IntG6$*&\"\"\"F)*&%\"u
GF)-%%sqrtG6#,&F)F)*$F+\"\"#F)F)!\"\"F+F2,&-%#lnG6#*&,&F)F)-F-6#,&F)F)
*$F+F1F)F)F)-%$absG6#F+F2F)%\"cGF)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = \+
ln((1+sqrt(1+cos^2*x))/abs(cos*x))+c;" "6#/%!G,&-%#lnG6#*&,&\"\"\"F+-%
%sqrtG6#,&F+F+*&%$cosG\"\"#%\"xGF+F+F+F+-%$absG6#*&F1F+F3F+!\"\"F+%\"c
GF+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 44 "We can check this
 result by differentiation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "Diff(ln((1+sqrt(1+cos(x)^2))/cos(x)
),x);\nsimplify(value(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG
6$-%#lnG6#*&,&\"\"\"F+*$,&F+F+*$)-%$cosG6#%\"xG\"\"#F+F+#F+F4F+F+F0!\"
\"F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%$sinG6#%\"xG\"\"\"-%$cosGF
&!\"\",&F(F(*$)F)\"\"#F(F(#F+F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "Diff(ln((1+sqrt(1+cos(x)^2))
/(-cos(x))),x);\nsimplify(value(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#-%%DiffG6$-%#lnG6#,$*&,&\"\"\"F,*$-%%sqrtG6#,&F,F,*$)-%$cosG6#%\"xG
\"\"#F,F,F,F,F,F4!\"\"F9F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%$sin
G6#%\"xG\"\"\",&F(F(*$)-%$cosGF&\"\"#F(F(#!\"\"F.F,F0" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "Letting the arbitrar
y constant " }{TEXT 274 1 "c" }{TEXT -1 67 " be zero we obtain the fol
lowing graph of the indefinite integral. " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 234 "g := x -> ln((1+sqrt
(1+cos(x)^2))/abs(cos(x)));\np1 := plot(g(x),x=-6..6,y=0..4):\np2 := p
lots[implicitplot](\{x=-3*Pi/2,x=-Pi/2,x=Pi/2,x=3*Pi/2\},\n           \+
x=-6..6,y=0..4,linestyle=3,color=COLOR(RGB,.2,.2,.2)):\nplots[display]
([p1,p2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)op
eratorG%&arrowGF(-%#lnG6#*&,&\"\"\"F1-%%sqrtG6#,&F1F1*$)-%$cosG6#9$\"
\"#F1F1F1F1-%$absG6#F8!\"\"F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 400 
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yF^iq-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F(Ffgn;F_hnFhhq" 1 2 0 1 10 0 2 9 
1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "
Curve 4" "Curve 5" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 72 "Maple's symbolic integration procedure gives a result i
nvolving arctanh." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 41 "Int(tan(x)/sqrt(1+cos(x)^2),x);\nvalue(%);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%$tanG6#%\"xG\"\"\",&F+F+
*$)-%$cosGF)\"\"#F+F+#!\"\"F1F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%(
arctanhG6#*&\"\"\"F'*$-%%sqrtG6#,&F'F'*$)-%$cosG6#%\"xG\"\"#F'F'F'!\"
\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "We
 can also check this result by differentiation." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "Diff(arctanh
(1/(sqrt(1+cos(x)^2))),x);\nsimplify(value(%));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%%DiffG6$-%(arctanhG6#*&\"\"\"F**$-%%sqrtG6#,&F*F**$)-
%$cosG6#%\"xG\"\"#F*F*F*!\"\"F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-
%$sinG6#%\"xG\"\"\",&F(F(*$)-%$cosGF&\"\"#F(F(#!\"\"F.F,F0" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "The graph of th
e second form for the indefinite integral appears to be the same as be
fore. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 227 "h := x -> arctanh(1/(sqrt(1+cos(x)^2)));\np1 := plot
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x=Pi/2,x=3*Pi/2\},\n           x=-6..6,y=0..4,linestyle=3,color=COLOR(
RGB,.2,.2,.2)):\nplots[display]([p1,p2]);" }}{PARA 11 "" 1 "" 
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+KF1F\\fq7$7$F[]qF^do7$F[]q$\"3;(*[zEjz]KF17$7$F[]qFh`pFefq7$7$F[]qFfd
o7$F[]q$\"3I(*[zEjz5MF17$7$F[]qFaeoF\\gq7$7$F[]qFeeo7$F[]q$\"3W(*[zEjz
qNF17$7$F[]q$\"3g++++++!o$F1Fcgq7$7$F[]qF\\fo7$F[]q$\"3f(*[zEjzIPF17$7
$F[]qF_\\qF\\hq7$7$F[]qFdfo7$F[]q$\"3s(*[zEjz!*QF17$7$F[]q$\"\"%F*Fchq
FagoFfgo-%+AXESLABELSG6%Q\"x6\"Q\"yF^iq-%%FONTG6#%(DEFAULTG-%%VIEWG6$;
F(Ffgn;F_hnFhhq" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 
0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" }}}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 9 " }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 279 8 "Question" }
{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 10 " Find (i) " }{XPPEDIT 
18 0 "Int(sqrt(x^2+a^2),x);" "6#-%$IntG6$-%%sqrtG6#,&*$%\"xG\"\"#\"\"
\"*$%\"aGF,F-F+" }{TEXT -1 13 "   and  (ii) " }{XPPEDIT 18 0 "Int(x^2/
sqrt(x^2+a^2),x)" "6#-%$IntG6$*&%\"xG\"\"#-%%sqrtG6#,&*$F'F(\"\"\"*$%
\"aGF(F.!\"\"F'" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "a>0" "6#2\"\"!
%\"aG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT 280 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }{TEXT 282 0 "" }{TEXT -1 8 "(i) Let " }{XPPEDIT 18 0 "x = a*s
inh*t;" "6#/%\"xG*(%\"aG\"\"\"%%sinhGF'%\"tGF'" }{TEXT -1 9 " so that \+
" }{XPPEDIT 18 0 "t=arcsinh(x/a)" "6#/%\"tG-%(arcsinhG6#*&%\"xG\"\"\"%
\"aG!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "dx = a*cosh*t*dt;" "6#/
%#dxG**%\"aG\"\"\"%%coshGF'%\"tGF'%#dtGF'" }{TEXT -1 2 ". " }}{PARA 0 
"" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "Int(sqrt(x^2+a^2),x) = Int(sqrt(a^2*sinh^2*t+a^2)*cosh*
t,t);" "6#/-%$IntG6$-%%sqrtG6#,&*$%\"xG\"\"#\"\"\"*$%\"aGF-F.F,-F%6$*(
-F(6#,&*(F0F-%%sinhGF-%\"tGF.F.*$F0F-F.F.%%coshGF.F9F.F9" }{TEXT -1 1 
" " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = a^2*Int(co
sh^2*t,t);" "6#/%!G*&%\"aG\"\"#-%$IntG6$*&%%coshGF'%\"tG\"\"\"F-F." }
{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` =
 a^2*Int((cosh*2*t+1)/2,t);" "6#/%!G*&%\"aG\"\"#-%$IntG6$*&,&*(%%coshG
\"\"\"F'F/%\"tGF/F/F/F/F/F'!\"\"F0F/" }{TEXT -1 1 " " }}{PARA 256 "" 
0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = a^2*(sinh*2*t/4+t/2)+c;" "6#/
%!G,&*&%\"aG\"\"#,&**%%sinhG\"\"\"F(F,%\"tGF,\"\"%!\"\"F,*&F-F,F(F/F,F
,F,%\"cGF," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "``=a^2/2" "6#/%!G*&%\"aG\"\"#F'!\"\"" }{XPPEDIT 18 0 "`
`(sinh*t*cosh*t+t)+c;" "6#,&-%!G6#,&**%%sinhG\"\"\"%\"tGF*%%coshGF*F+F
*F*F+F*F*%\"cGF*" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "``=a^2/2" "6#/%!G*&%\"aG\"\"#F'!\"\"" }{XPPEDIT 18 0 "`
`(sinh*t*sqrt(sinh^2*t+1)+t)+c;" "6#,&-%!G6#,&*(%%sinhG\"\"\"%\"tGF*-%
%sqrtG6#,&*&F)\"\"#F+F*F*F*F*F*F*F+F*F*%\"cGF*" }{TEXT -1 1 " " }}
{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = a*sinh*t/2;" "6#
/%!G**%\"aG\"\"\"%%sinhGF'%\"tGF'\"\"#!\"\"" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "sqrt(a^2*sinh^2*t+a^2)+a^2*t/2+c;" "6#,(-%%sqrtG6#,&*(%
\"aG\"\"#%%sinhGF*%\"tG\"\"\"F-*$F)F*F-F-*(F)F*F,F-F*!\"\"F-%\"cGF-" }
{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=x
*sqrt(x^2+a^2)/2+a^2/2" "6#/%!G,&*(%\"xG\"\"\"-%%sqrtG6#,&*$F'\"\"#F(*
$%\"aGF.F(F(F.!\"\"F(*&F0F.F.F1F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arc
sinh(x/a) + c" "6#,&-%(arcsinhG6#*&%\"xG\"\"\"%\"aG!\"\"F)%\"cGF)" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 212 "Int(sqrt(x^2+a^2),x);\nstudent[changevar](x=a*s
inh(t),%,t);\nassume(a_>=0); assume(t_,real):\nsubs(\{t_=t,a_=a\},simp
lify(subs(\{t=t_,a=a_\},%)));\ncombine(%);\nvalue(%);\nsimplify(expand
(subs(t=arcsinh(x/a),%)),symbolic);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#-%$IntG6$*$,&*$)%\"xG\"\"#\"\"\"F,*$)%\"aGF+F,F,#F,F+F*" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%$IntG6$*(,&*&)%\"aG\"\"#\"\"\")-%%sinhG6#%\"tG
F+F,F,*$F)F,F,#F,F+F*F,-%%coshGF0F,F1" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#*&)%\"aG\"\"#\"\"\"-%$IntG6$*$)-%%coshG6#%\"tGF&F'F0F'" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#-%$IntG6$,&*&#\"\"\"\"\"#F)*&)%\"aGF*F)-%%coshG
6#,$*&F*F)%\"tGF)F)F)F)F)*&F*!\"\"F-F*F)F3" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,&*&#\"\"\"\"\"%F&*&)%\"aG\"\"#F&-%%sinhG6#,$*&F+F&%\"t
GF&F&F&F&F&*(F+!\"\"F*F+F1F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(
\"\"#!\"\"%\"xG\"\"\",&*$)F'F%F(F(*$)%\"aGF%F(F(#F(F%F(*&F/F(*&F-F(-%(
arcsinhG6#*&F'F(F.F&F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 281 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 74 "(ii) The integral can be found by using the integratio
n by parts formula: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "Int(u*``(dv/dx),x) = u*v-Int(v*``(du/dx),x);" "6#/-%$IntG6$*&%\"uG
\"\"\"-%!G6#*&%#dvGF)%#dxG!\"\"F)%\"xG,&*&F(F)%\"vGF)F)-F%6$*&F4F)-F+6
#*&%#duGF)F/F0F)F1F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x^2/sqrt(x^2+
a^2),x);" "6#-%$IntG6$*&%\"xG\"\"#-%%sqrtG6#,&*$F'F(\"\"\"*$%\"aGF(F.!
\"\"F'" }{TEXT -1 9 "  ...    " }{XPPEDIT 18 0 "PIECEWISE([u = x, v = \+
sqrt(x^2+a^2)],[du/dx = 1, dv/dx = x/sqrt(a^2+x^2)]);" "6#-%*PIECEWISE
G6$7$/%\"uG%\"xG/%\"vG-%%sqrtG6#,&*$F)\"\"#\"\"\"*$%\"aGF1F27$/*&%#duG
F2%#dxG!\"\"F2/*&%#dvGF2F9F:*&F)F2-F-6#,&*$F4F1F2*$F)F1F2F:" }{TEXT 
-1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 
1 " " }{XPPEDIT 18 0 "`` = x*sqrt(x^2+a^2)-Int(sqrt(x^2+a^2),x);" "6#/
%!G,&*&%\"xG\"\"\"-%%sqrtG6#,&*$F'\"\"#F(*$%\"aGF.F(F(F(-%$IntG6$-F*6#
,&*$F'F.F(*$F0F.F(F'!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = x*sqrt(x^
2+a^2)-x*sqrt(x^2+a^2)/2-a^2/2;" "6#/%!G,(*&%\"xG\"\"\"-%%sqrtG6#,&*$F
'\"\"#F(*$%\"aGF.F(F(F(*(F'F(-F*6#,&*$F'F.F(*$F0F.F(F(F.!\"\"F7*&F0F.F
.F7F7" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arcsinh(x/a) + c" "6#,&-%(arcsi
nhG6#*&%\"xG\"\"\"%\"aG!\"\"F)%\"cGF)" }{TEXT -1 2 ", " }}{PARA 0 "" 
0 "" {TEXT -1 38 "incorporating the result of part (i), " }}{PARA 256 
"" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = x*sqrt(x^2+a^2)/2-a^2/2;" 
"6#/%!G,&*(%\"xG\"\"\"-%%sqrtG6#,&*$F'\"\"#F(*$%\"aGF.F(F(F.!\"\"F(*&F
0F.F.F1F1" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arcsinh(x/a) + c" "6#,&-%(a
rcsinhG6#*&%\"xG\"\"\"%\"aG!\"\"F)%\"cGF)" }{TEXT -1 2 ". " }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "stud
ent[intparts](Int(x^2/sqrt(x^2+a^2),x),x);\nvalue(%);\n" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,&*&%\"xG\"\"\",&*$)F%\"\"#F&F&*$)%\"aGF*F&F&#F&
F*F&-%$IntG6$*$F'F.F%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(\"\"
#!\"\"%\"xG\"\"\",&*$)F'F%F(F(*$)%\"aGF%F(F(#F(F%F(*&#F(F%F(*&F-F(-%#l
nG6#,&F'F(*$F)F/F(F(F(F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 11 "Example 10 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";
" }}}{PARA 0 "" 0 "" {TEXT 277 8 "Question" }{TEXT -1 2 ": " }}{PARA 
0 "" 0 "" {TEXT -1 6 " Find " }{XPPEDIT 18 0 "Int(arcsinh*x,x);" "6#-%
$IntG6$*&%(arcsinhG\"\"\"%\"xGF(F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 278 8 "Solution" }{TEXT -1 2 "
: " }}{PARA 0 "" 0 "" {TEXT -1 66 "The integral can be found using the
 integration by parts formula: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "Int(u*``(dv/dx),x) = u*v-Int(v*``(du/dx),x);" "6#/-%$In
tG6$*&%\"uG\"\"\"-%!G6#*&%#dvGF)%#dxG!\"\"F)%\"xG,&*&F(F)%\"vGF)F)-F%6
$*&F4F)-F+6#*&%#duGF)F/F0F)F1F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(a
rcsinh*x,x)" "6#-%$IntG6$*&%(arcsinhG\"\"\"%\"xGF(F)" }{TEXT -1 11 "  \+
...      " }{XPPEDIT 18 0 "PIECEWISE([u=arcsinh*x,v=x],[du/dx = 1/sqrt
(x^2+1),dv/dx=1])" "6#-%*PIECEWISEG6$7$/%\"uG*&%(arcsinhG\"\"\"%\"xGF+
/%\"vGF,7$/*&%#duGF+%#dxG!\"\"*&F+F+-%%sqrtG6#,&*$F,\"\"#F+F+F+F4/*&%#
dvGF+F3F4F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = x*arcsinh*x-Int(x/sqrt
(x^2+1),x);" "6#/%!G,&*(%\"xG\"\"\"%(arcsinhGF(F'F(F(-%$IntG6$*&F'F(-%
%sqrtG6#,&*$F'\"\"#F(F(F(!\"\"F'F4" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = \+
x*arcsinh*x-sqrt(x^2+1)+c;" "6#/%!G,(*(%\"xG\"\"\"%(arcsinhGF(F'F(F(-%
%sqrtG6#,&*$F'\"\"#F(F(F(!\"\"%\"cGF(" }{TEXT -1 1 " " }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "Int(arcsi
nh(x),x);\nstudent[intparts](%,arcsinh(x));\nvalue(%);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#-%$IntG6$-%(arcsinhG6#%\"xGF)" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,&*&-%(arcsinhG6#%\"xG\"\"\"F(F)F)-%$IntG6$*&,&F)F)*$)F
(\"\"#F)F)#!\"\"F1F(F)F(F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%(a
rcsinhG6#%\"xG\"\"\"F(F)F)*$,&F)F)*$)F(\"\"#F)F)#F)F.!\"\"" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" 
}}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "Tasks 1" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 
-1 3 "Q1 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "
" 0 "" {TEXT -1 6 "Find  " }{XPPEDIT 18 0 "Int(1/sqrt(9+x^2),x = 0 .. \+
2);" "6#-%$IntG6$*&\"\"\"F'-%%sqrtG6#,&\"\"*F'*$%\"xG\"\"#F'!\"\"/F.;
\"\"!F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 
{PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 2 "  " }
{XPPEDIT 18 0 "arcsinh(2/3)=ln((2+sqrt(13))/3)" "6#/-%(arcsinhG6#*&\"
\"#\"\"\"\"\"$!\"\"-%#lnG6#*&,&F(F)-%%sqrtG6#\"#8F)F)F*F+" }{TEXT -1 
1 " " }{TEXT 284 1 "~" }{TEXT -1 15 " 0.6251451173. " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "Int(1/sqrt
(9+x^2),x = 0 .. 2);\nvalue(%);\nconvert(%,ln);\nevalf(evalf[13](%));
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'*$,&\"\"*F'*$)%
\"xG\"\"#F'F'#F'F.!\"\"/F-;\"\"!F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
-%(arcsinhG6##\"\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#,&
#\"\"#\"\"$\"\"\"*&F)!\"\"\"#8#F*F(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#$\"+t6X^i!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 34 "___________
_______________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "_______________________________
___" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 
4 "" 0 "" {TEXT -1 3 "Q2 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "
;" }}}{PARA 0 "" 0 "" {TEXT -1 6 "Find  " }{XPPEDIT 18 0 "Int(1/sqrt(x
^2-4),x = 2 .. 4);" "6#-%$IntG6$*&\"\"\"F'-%%sqrtG6#,&*$%\"xG\"\"#F'\"
\"%!\"\"F0/F-;F.F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 
2 "  " }}{PARA 0 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "arccosh*2 = l
n(2+sqrt(3)" "6#/*&%(arccoshG\"\"\"\"\"#F&-%#lnG6#,&F'F&-%%sqrtG6#\"\"
$F&" }{TEXT -1 1 " " }{TEXT 283 1 "~" }{TEXT -1 14 " 1.316957897. " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
61 "Int(1/sqrt(x^2-4),x=2..4);\nvalue(%);\nconvert(%,ln);\nevalf(%);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'*$,&*$)%\"xG\"\"#
F'F'\"\"%!\"\"#F'F-F//F,;F-F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&
\"\"#\"\"\"-%(arcsinhG6#,$*&F%!\"\"F%#F&F%F&F&F&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$*&\"\"#\"\"\"-%#lnG6#,&*&F%!\"\"F%#F&F%F&*&F%F,\"\"'F
-F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+(*y&pJ\"!\"*" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "ar
ccosh(2)-arccosh(1);\nevalf(evalf[13](%));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%(arccoshG6#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$
\"+(*y&pJ\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 35 "ln(2+sqrt(3));\nevalf(evalf[13](%));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%#lnG6#,&\"\"#\"\"\"*$\"\"$#F(F'F(" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+(*y&pJ\"!\"*" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 
0 "" 0 "" {TEXT -1 34 "__________________________________" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 34 "__________________________________" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 
-1 5 "Find " }{XPPEDIT 18 0 "Int(cos*x/sqrt(1+sin^2*x),x)" "6#-%$IntG6
$*(%$cosG\"\"\"%\"xGF(-%%sqrtG6#,&F(F(*&%$sinG\"\"#F)F(F(!\"\"F)" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 
0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 3 "   " }{XPPEDIT 
18 0 "arcsinh(sin*x)+c;" "6#,&-%(arcsinhG6#*&%$sinG\"\"\"%\"xGF)F)%\"c
GF)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 56 "Int(cos(x)/sqrt(1+sin(x)^2),x);\nvalue(%);
\nconvert(%,ln);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&-%$cosG
6#%\"xG\"\"\",&F+F+*$)-%$sinGF)\"\"#F+F+#!\"\"F1F*" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#-%(arcsinhG6#-%$sinG6#%\"xG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%#lnG6#,&-%$sinG6#%\"xG\"\"\"*$,&F+F+*$)F'\"\"#F+F+#F+
F0F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 34 "__________________
________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 3 "Q4 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
{PARA 0 "" 0 "" {TEXT -1 5 "Find " }{XPPEDIT 18 0 "Int(1/((x-2)*sqrt(4
*x-x^2)),x);" "6#-%$IntG6$*&\"\"\"F'*&,&%\"xGF'\"\"#!\"\"F'-%%sqrtG6#,
&*&\"\"%F'F*F'F'*$F*F+F,F'F,F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "
" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "-1/2" "6#,$*&\"\"\"F%\"\"#!\"\"
F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "arcsech((x-2)/2)+c =-1/2" "6#/,&-%
(arcsechG6#*&,&%\"xG\"\"\"\"\"#!\"\"F+F,F-F+%\"cGF+,$*&F+F+F,F-F-" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "ln((2+sqrt(4*x-x^2))/(x-2))+c" "6#,&-%#
lnG6#*&,&\"\"#\"\"\"-%%sqrtG6#,&*&\"\"%F*%\"xGF*F**$F1F)!\"\"F*F*,&F1F
*F)F3F3F*%\"cGF*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "Int(1/((x-2)*sqrt(4*x-x^2))
,x);\nmap(student[completesquare],%,x);\nstudent[changevar](x-2=u,%,u)
;\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'*&,
&%\"xGF'\"\"#!\"\"F',&*&\"\"%F'F*F'F'*$)F*F+F'F,#F'F+F,F*" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'*&,&%\"xGF'\"\"#!\"\"F',&*$)
F)F+F'F,\"\"%F'#F'F+F,F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*
&\"\"\"F'*&%\"uGF',&*$)F)\"\"#F'!\"\"\"\"%F'#F'F-F.F)" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"\"#F&-%(arctanhG6#,$*&F'F&,&*$)%\"uGF
'F&!\"\"\"\"%F&#F1F'F&F&F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "Diff( -ln((2+sqrt(4*x-x^2))/(x-2))/
2,x);\n``=simplify(value(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%Di
ffG6$,$*&#\"\"\"\"\"#F)-%#lnG6#*&,&F*F)*$,&*&\"\"%F)%\"xGF)F)*$)F4F*F)
!\"\"#F)F*F)F),&F4F)F*F7F7F)F7F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%
!G*&\"\"\"F&*&,$*&%\"xGF&,&\"\"%!\"\"F*F&F&F-#F&\"\"#,&F*F&F/F-F&F-" }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 34 "_____________________________
_____" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 34 "__________________________________" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 3 "Q5 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
{PARA 0 "" 0 "" {TEXT -1 6 "Find  " }{XPPEDIT 18 0 "Int(1/(x*sqrt(1+x^
4)),x)" "6#-%$IntG6$*&\"\"\"F'*&%\"xGF'-%%sqrtG6#,&F'F'*$F)\"\"%F'F'!
\"\"F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 
{PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 3 "   " }
{XPPEDIT 18 0 "-1/2" "6#,$*&\"\"\"F%\"\"#!\"\"F'" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "arccsch(x^2)+c= -1/2" "6#/,&-%(arccschG6#*$%\"xG\"\"#\"
\"\"%\"cGF+,$*&F+F+F*!\"\"F/" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln((1+sq
rt(1+x^4))/x^2)+c" "6#,&-%#lnG6#*&,&\"\"\"F)-%%sqrtG6#,&F)F)*$%\"xG\"
\"%F)F)F)*$F/\"\"#!\"\"F)%\"cGF)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "Int(1/(x*sqr
t(1+x^4)),x);\nstudent[changevar](x^2=u,%,u);\nvalue(%);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#-%$IntG6$*&\"\"\"F'*&%\"xGF',&F'F'*$)F)\"\"%F'F'
#F'\"\"#!\"\"F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*&\"\"\"
F(*(\"\"#F(%\"uGF(,&F(F(*$)F+F*F(F(#F(F*!\"\"F(F+" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$*&#\"\"\"\"\"#F&-%(arctanhG6#*&F&F&*$,&F&F&*$)%\"uGF'
F&F&#F&F'!\"\"F&F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 69 "Diff(-ln((1+sqrt(1+x^4))/x^2)/2,x);\n``=val
ue(%);\n``=simplify(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%Dif
fG6$,$*&#\"\"\"\"\"#F)-%#lnG6#*&,&F)F)*$,&F)F)*$)%\"xG\"\"%F)F)#F)F*F)
F)F4!\"#F)!\"\"F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$**\"\"#!\"
\",&*(F'\"\"\",&F+F+*$)%\"xG\"\"%F+F+#F(F'F/F+F+*(F'F+,&F+F+*$F,#F+F'F
+F+F/!\"$F(F+F3F(F/F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*&\"\"
\"F&*&%\"xGF&,&F&F&*$)F(\"\"%F&F&#F&\"\"#!\"\"" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 
0 "" 0 "" {TEXT -1 34 "__________________________________" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 34 "__________________________________" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 "Tasks 2" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 
-1 26 "Show how the substitution " }{XPPEDIT 18 0 "t = a*sinh*t;" "6#/
%\"tG*(%\"aG\"\"\"%%sinhGF'F$F'" }{TEXT -1 21 " can be used to find " 
}{XPPEDIT 18 0 "Int(x^2/sqrt(x^2+a^2),x);" "6#-%$IntG6$*&%\"xG\"\"#-%%
sqrtG6#,&*$F'F(\"\"\"*$%\"aGF(F.!\"\"F'" }{TEXT -1 8 ", where " }
{XPPEDIT 18 0 "a>0" "6#2\"\"!%\"aG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "
" {TEXT -1 34 "__________________________________" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "__
________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 26 "Show ho
w the substitution " }{XPPEDIT 18 0 "t = a*sinh*t;" "6#/%\"tG*(%\"aG\"
\"\"%%sinhGF'F$F'" }{TEXT -1 21 " can be used to find " }{XPPEDIT 18 
0 "Int(x^3/((x^2+a^2)^(3/2)),x);" "6#-%$IntG6$*&%\"xG\"\"$),&*$F'\"\"#
\"\"\"*$%\"aGF,F-*&F(F-F,!\"\"F1F'" }{TEXT -1 8 ", where " }{XPPEDIT 
18 0 "a>0" "6#2\"\"!%\"aG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 
-1 34 "__________________________________" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "____________
______________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" 
}}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 17 "Show how to find \+
" }{XPPEDIT 18 0 "Int(arccosh*x,x);" "6#-%$IntG6$*&%(arccoshG\"\"\"%\"
xGF(F)" }{TEXT -1 44 ", by using the integration by parts formula." }}
{PARA 0 "" 0 "" {TEXT -1 34 "__________________________________" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 34 "__________________________________" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q4 \+
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" 
{TEXT -1 17 "Show how to find " }{XPPEDIT 18 0 "Int(arctanh*x,x);" "6#
-%$IntG6$*&%(arctanhG\"\"\"%\"xGF(F)" }{TEXT -1 44 ", by using the int
egration by parts formula." }}{PARA 0 "" 0 "" {TEXT -1 34 "___________
_______________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 34 "_______________________________
___" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 
1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }