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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 256 25 "Calculus Single Variable
 " }{TEXT 259 2 " \n" }{TEXT 257 35 "Brian E. Blank and Steven G. Kran
tz" }{TEXT 258 35 "\n\nSection 7.1\nIntegration by Parts\n" }}}{SECT 
0 {PARA 3 "" 0 "" {TEXT -1 19 "1. The Formula for " }{TEXT 260 20 "Int
egration by Parts" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 5 " If  " }{XPPEDIT 18 0 "u;" "6#%\"uG" }{TEXT -1 7 "  and  \+
" }{XPPEDIT 18 0 "v;" "6#%\"vG" }{TEXT -1 22 "  are functions,  then" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "       \+
            " }{XPPEDIT 18 0 "diff(u(x)*v(x),x) = diff(u(x),x)*v(x)+u(
x)*diff(v(x),x);" "6#/-%%diffG6$*&-%\"uG6#%\"xG\"\"\"-%\"vG6#F+F,F+,&*
&-F%6$-F)6#F+F+F,-F.6#F+F,F,*&-F)6#F+F,-F%6$-F.6#F+F+F,F," }{TEXT -1 
1 " " }{TEXT 261 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 11 "As a result" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 26 "                          " }{XPPEDIT 18 0 "int
(diff(u(x)*v(x),x),x) = int(diff(u(x),x)*v(x),x)+int(u(x)*diff(v(x),x)
,x);" "6#/-%$intG6$-%%diffG6$*&-%\"uG6#%\"xG\"\"\"-%\"vG6#F.F/F.F.,&-F
%6$*&-F(6$-F,6#F.F.F/-F16#F.F/F.F/-F%6$*&-F,6#F.F/-F(6$-F16#F.F.F/F.F/
" }{TEXT -1 1 " " }{TEXT 262 1 "," }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 " or" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "                         \+
  " }{XPPEDIT 18 0 "u(x)*v(x) = int(diff(u(x),x)*v(x),x)+int(u(x)*diff
(v(x),x),x);" "6#/*&-%\"uG6#%\"xG\"\"\"-%\"vG6#F(F),&-%$intG6$*&-%%dif
fG6$-F&6#F(F(F)-F+6#F(F)F(F)-F/6$*&-F&6#F(F)-F36$-F+6#F(F(F)F(F)" }
{TEXT -1 3 "  ," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 2 "or" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 22 "                      " }{XPPEDIT 18 0 "int(u(x)*diff(v(x
),x),x) = u(x)*v(x)-int(diff(u(x),x)*v(x),x);" "6#/-%$intG6$*&-%\"uG6#
%\"xG\"\"\"-%%diffG6$-%\"vG6#F+F+F,F+,&*&-F)6#F+F,-F16#F+F,F,-F%6$*&-F
.6$-F)6#F+F+F,-F16#F+F,F+!\"\"" }{TEXT -1 2 "  " }{TEXT 263 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 68 "It is convenient to remember the shorthand vers
ion of this equation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "                      \+
                    " }{TEXT 264 1 " " }{TEXT 266 1 " " }{XPPEDIT 265 
0 "int(u,v) = u*v-int(v,u);" "6#/-%$intG6$%\"uG%\"vG,&*&F'\"\"\"F(F+F+
-F%6$F(F'!\"\"" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 156 "Notice t
hat integration by parts does not by itself result in the evaluation o
f an integral. Instead, integration by parts transforms one integral, \+
namely  " }{XPPEDIT 18 0 "int(u,v);" "6#-%$intG6$%\"uG%\"vG" }{TEXT 
-1 32 ", into  a new integral, namely  " }{XPPEDIT 18 0 "int(v,u);" "6
#-%$intG6$%\"vG%\"uG" }{TEXT -1 90 ". In general, it is desirable that
 the new integral be easier than the original integral. " }}}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 42 "2. Strategy for Using Integration by Part
s" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "In g
eneral, we think of using integration by parts when we see a product i
n the integrand." }}{PARA 0 "" 0 "" {TEXT -1 67 "However, when we deci
de to try integration by parts on the integral" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "                         \+
        " }{XPPEDIT 18 0 "int(f(x)*g(x),x);" "6#-%$intG6$*&-%\"fG6#%\"
xG\"\"\"-%\"gG6#F*F+F*" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 
0 "" {TEXT -1 26 "we must decide which of   " }{XPPEDIT 18 0 "f(x);" "
6#-%\"fG6#%\"xG" }{TEXT -1 8 "  and   " }{XPPEDIT 18 0 "g(x);" "6#-%\"
gG6#%\"xG" }{TEXT -1 20 "   to set equal to  " }{XPPEDIT 18 0 "u;" "6#
%\"uG" }{TEXT -1 36 ".  (The other factor together with  " }{XPPEDIT 
18 0 "dx;" "6#%#dxG" }{TEXT -1 16 "  then  becomes " }{XPPEDIT 18 0 "d
v;" "6#%#dvG" }{TEXT -1 2 ".)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 47 "The guiding principle is that in transfor
ming  " }{XPPEDIT 18 0 "int(u,v);" "6#-%$intG6$%\"uG%\"vG" }{TEXT -1 
26 "  into the new integral   " }{XPPEDIT 18 0 "int(v,u);" "6#-%$intG6
$%\"vG%\"uG" }{TEXT -1 21 ",  we differentiate  " }{XPPEDIT 18 0 "u;" 
"6#%\"uG" }{TEXT -1 13 "  to obtain  " }{XPPEDIT 18 0 "du;" "6#%#duG" 
}{TEXT -1 15 " and integrate " }{XPPEDIT 18 0 "dv;" "6#%#dvG" }{TEXT 
-1 13 "  to obtain  " }{XPPEDIT 18 0 "v;" "6#%\"vG" }{TEXT -1 23 ".  W
e generally choose " }{XPPEDIT 18 0 "u;" "6#%\"uG" }{TEXT -1 5 " and \+
" }{XPPEDIT 18 0 "dv;" "6#%#dvG" }{TEXT -1 96 "  so that at least one \+
of these operations results in a simpler expression.  For example, whe
n  " }{XPPEDIT 18 0 "ln(x);" "6#-%#lnG6#%\"xG" }{TEXT -1 43 "  appears
 in an integrand, we usually set  " }{XPPEDIT 18 0 "u = ln(x);" "6#/%
\"uG-%#lnG6#%\"xG" }{TEXT -1 17 "  because then   " }{XPPEDIT 18 0 "du
 = 1/x;" "6#/%#duG*&\"\"\"F&%\"xG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 
0 "dx;" "6#%#dxG" }{TEXT -1 26 "  and the algebraic term  " }{XPPEDIT 
18 0 "1/x;" "6#*&\"\"\"F$%\"xG!\"\"" }{TEXT -1 37 "  is much simpler t
han the original  " }{XPPEDIT 18 0 "ln(x);" "6#-%#lnG6#%\"xG" }{TEXT 
-1 45 ".  The same is true for expressions such as  " }{XPPEDIT 18 0 "
arcsin(x);" "6#-%'arcsinG6#%\"xG" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" 
{TEXT -1 5 "and  " }{XPPEDIT 18 0 "arctan(x);" "6#-%'arctanG6#%\"xG" }
{TEXT -1 8 " :  if  " }{XPPEDIT 18 0 "u = arcsin(x);" "6#/%\"uG-%'arcs
inG6#%\"xG" }{TEXT -1 8 "  then  " }{XPPEDIT 18 0 "du = 1/sqrt(1-x^2);
" "6#/%#duG*&\"\"\"F&-%%sqrtG6#,&F&F&*$%\"xG\"\"#!\"\"F." }{XPPEDIT 
18 0 "dx;" "6#%#dxG" }{TEXT -1 11 "  and if   " }{XPPEDIT 18 0 "u = ar
ctan(x);" "6#/%\"uG-%'arctanG6#%\"xG" }{TEXT -1 8 "  then  " }
{XPPEDIT 18 0 "du = 1/(1+x^2);" "6#/%#duG*&\"\"\"F&,&F&F&*$%\"xG\"\"#F
&!\"\"" }{XPPEDIT 18 0 "dx;" "6#%#dxG" }{TEXT -1 3 "  ." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "If  the integrand co
ntains  " }{XPPEDIT 18 0 "x^n;" "6#)%\"xG%\"nG" }{TEXT -1 11 "  but no
t  " }{XPPEDIT 18 0 "ln(x);" "6#-%#lnG6#%\"xG" }{TEXT -1 3 ",  " }
{XPPEDIT 18 0 "arcsin(x);" "6#-%'arcsinG6#%\"xG" }{TEXT -1 7 ", or   \+
" }{XPPEDIT 18 0 "arctan(x);" "6#-%'arctanG6#%\"xG" }{TEXT -1 27 ",  t
hen we generally set   " }{XPPEDIT 18 0 "u = x^n;" "6#/%\"uG)%\"xG%\"n
G" }{TEXT -1 21 " . The reason is that" }}{PARA 0 "" 0 "" {XPPEDIT 18 
0 "du = n*x^(n-1);" "6#/%#duG*&%\"nG\"\"\")%\"xG,&F&F'F'!\"\"F'" }
{XPPEDIT 18 0 "dx;" "6#%#dxG" }{TEXT -1 38 "    is simpler than the or
iginal term " }{XPPEDIT 18 0 "x^n;" "6#)%\"xG%\"nG" }{TEXT -1 64 ", th
e power having been decreased by 1.\n\nSince the expressions  " }
{XPPEDIT 18 0 "cos(a*x);" "6#-%$cosG6#*&%\"aG\"\"\"%\"xGF(" }{TEXT -1 
3 ",  " }{XPPEDIT 18 0 "sin(a*x);" "6#-%$sinG6#*&%\"aG\"\"\"%\"xGF(" }
{TEXT -1 7 ", and  " }{XPPEDIT 18 0 "exp(a*x);" "6#-%$expG6#*&%\"aG\"
\"\"%\"xGF(" }{TEXT -1 161 "   neither become neither simpler nor more
 complicated after differentiation or integration, they are usually no
t factors in our consideration of how to divide  " }{XPPEDIT 18 0 "f(x
)*g(x)*dx;" "6#*(-%\"fG6#%\"xG\"\"\"-%\"gG6#F'F(%#dxGF(" }{TEXT -1 9 "
   into  " }{XPPEDIT 18 0 "u;" "6#%\"uG" }{TEXT -1 7 "  and  " }
{XPPEDIT 18 0 "dv;" "6#%#dvG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 17 "2. Basic Examples" }}
{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 267 7 "Example
" }}{PARA 0 "" 0 "" {TEXT -1 12 "Calculate   " }{XPPEDIT 18 0 "int(x*c
os(a*x),x);" "6#-%$intG6$*&%\"xG\"\"\"-%$cosG6#*&%\"aGF(F'F(F(F'" }
{TEXT -1 7 "  for  " }{XPPEDIT 18 0 "a <> 0;" "6#0%\"aG\"\"!" }{TEXT 
-1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 268 
8 "Solution" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Set  " }
{XPPEDIT 18 0 "u = x;" "6#/%\"uG%\"xG" }{TEXT -1 3 ",  " }{XPPEDIT 18 
0 "du = dx;" "6#/%#duG%#dxG" }{TEXT -1 7 "  and  " }{XPPEDIT 18 0 "dv \+
= cos(a*x)*dx;" "6#/%#dvG*&-%$cosG6#*&%\"aG\"\"\"%\"xGF+F+%#dxGF+" }
{TEXT -1 3 ",  " }{XPPEDIT 18 0 "v = 1/a;" "6#/%\"vG*&\"\"\"F&%\"aG!\"
\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin(a*x);" "6#-%$sinG6#*&%\"aG\"\"
\"%\"xGF(" }{TEXT -1 6 ". Then" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 8 "        " }{XPPEDIT 18 0 "int(x*cos(a*x),x
) = 1/a;" "6#/-%$intG6$*&%\"xG\"\"\"-%$cosG6#*&%\"aGF)F(F)F)F(*&F)F)F.
!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x*sin(a*x)-1/a;" "6#,&*&%\"xG\"
\"\"-%$sinG6#*&%\"aGF&F%F&F&F&*&F&F&F+!\"\"F-" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "int(sin(a*x),x);" "6#-%$intG6$-%$sinG6#*&%\"aG\"\"\"%\"
xGF+F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/a;" "6#/%!G*&\"\"\"F&%
\"aG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x*sin(a*x)+1/(a^2);" "6#,&*
&%\"xG\"\"\"-%$sinG6#*&%\"aGF&F%F&F&F&*&F&F&*$F+\"\"#!\"\"F&" }{TEXT 
-1 1 " " }{XPPEDIT 18 0 "cos(a*x);" "6#-%$cosG6#*&%\"aG\"\"\"%\"xGF(" 
}{TEXT -1 7 "  +  C." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 3 "   " }}{PARA 0 "" 0 "" {TEXT -1 13 "Verification:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
56 "testeq(diff(x*sin(a*x)/a+cos(a*x)/a^2, x) = x*cos(a*x));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 3 "   \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 275 7 "Example" }}{PARA 0 "" 0 "" {TEXT -1 12 "C
alculate   " }{XPPEDIT 18 0 "int(x*exp(a*x),x);" "6#-%$intG6$*&%\"xG\"
\"\"-%$expG6#*&%\"aGF(F'F(F(F'" }{TEXT -1 7 "  for  " }{XPPEDIT 18 0 "
a <> 0;" "6#0%\"aG\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT 276 8 "Solution" }{TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 5 "Set  " }{XPPEDIT 18 0 "u = x;" "6#/%\"uG%\"xG" }
{TEXT -1 3 ",  " }{XPPEDIT 18 0 "du = dx;" "6#/%#duG%#dxG" }{TEXT -1 
7 "  and  " }{XPPEDIT 18 0 "dv = exp(a*x)*dx;" "6#/%#dvG*&-%$expG6#*&%
\"aG\"\"\"%\"xGF+F+%#dxGF+" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "v = 1/a;
" "6#/%\"vG*&\"\"\"F&%\"aG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(a
*x);" "6#-%$expG6#*&%\"aG\"\"\"%\"xGF(" }{TEXT -1 6 ". Then" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "        " }
{XPPEDIT 18 0 "int(x*exp(a*x),x) = 1/a;" "6#/-%$intG6$*&%\"xG\"\"\"-%$
expG6#*&%\"aGF)F(F)F)F(*&F)F)F.!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "
x*exp(a*x)-1/a;" "6#,&*&%\"xG\"\"\"-%$expG6#*&%\"aGF&F%F&F&F&*&F&F&F+!
\"\"F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "int(exp(a*x),x);" "6#-%$intG6$
-%$expG6#*&%\"aG\"\"\"%\"xGF+F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = \+
1/a;" "6#/%!G*&\"\"\"F&%\"aG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x*e
xp(a*x)-1/(a^2);" "6#,&*&%\"xG\"\"\"-%$expG6#*&%\"aGF&F%F&F&F&*&F&F&*$
F+\"\"#!\"\"F/" }{TEXT -1 1 " " }{XPPEDIT 18 0 "exp(a*x);" "6#-%$expG6
#*&%\"aG\"\"\"%\"xGF(" }{TEXT -1 7 "  +  C." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "   " }}{PARA 0 "" 0 "" {TEXT -1 
13 "Verification:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 56 "testeq(diff(x*exp(a*x)/a-exp(a*x)/a^2, x) = x
*exp(a*x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 1 " " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 55 "4. Antiderivati
ves of  ln(x), arcsin(x), and  arctan(x)" }}{PARA 0 "" 0 "" {TEXT 269 
7 "Example" }}{PARA 0 "" 0 "" {TEXT -1 12 "Calculate   " }{XPPEDIT 18 
0 "int(ln(x),x);" "6#-%$intG6$-%#lnG6#%\"xGF)" }{TEXT -1 2 " ." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 270 8 "Solution
" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Set  " }{XPPEDIT 18 0 "
u = ln(x);" "6#/%\"uG-%#lnG6#%\"xG" }{TEXT -1 5 ",    " }{XPPEDIT 18 
0 "du = ``;" "6#/%#duG%!G" }{TEXT -1 1 " " }{XPPEDIT 18 0 "1/x;" "6#*&
\"\"\"F$%\"xG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dx;" "6#%#dxG" }
{TEXT -1 8 "   and  " }{XPPEDIT 18 0 "dv = 1*dx;" "6#/%#dvG*&\"\"\"F&%
#dxGF&" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "v = x;" "6#/%\"vG%\"xG" }
{TEXT -1 7 " . Then" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 8 "        " }{XPPEDIT 18 0 "int(ln(x),x) = x*ln(x)-int(x*`
(1/x)`,x);" "6#/-%$intG6$-%#lnG6#%\"xGF*,&*&F*\"\"\"-F(6#F*F-F--F%6$*&
F*F-%&(1/x)GF-F*!\"\"" }{XPPEDIT 18 0 "`` = x*ln(x)-x+C;" "6#/%!G,(*&%
\"xG\"\"\"-%#lnG6#F'F(F(F'!\"\"%\"CGF(" }{TEXT -1 2 " ." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "   " }}{PARA 0 "" 0 "
" {TEXT -1 13 "Verification:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "testeq(diff(x*ln(x) - x + C,
 x) = ln(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 271 7 "Example" }}{PARA 0 "" 
0 "" {TEXT -1 12 "Calculate   " }{XPPEDIT 18 0 "int(arcsin(x),x);" "6#
-%$intG6$-%'arcsinG6#%\"xGF)" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 272 8 "Solution" }{TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 5 "Set  " }{XPPEDIT 18 0 "u = arcsin(x);" "
6#/%\"uG-%'arcsinG6#%\"xG" }{TEXT -1 5 ",    " }{XPPEDIT 18 0 "du = ``
;" "6#/%#duG%!G" }{TEXT -1 1 " " }{XPPEDIT 18 0 "1/sqrt(1-x^2);" "6#*&
\"\"\"F$-%%sqrtG6#,&F$F$*$%\"xG\"\"#!\"\"F," }{TEXT -1 1 " " }
{XPPEDIT 18 0 "dx;" "6#%#dxG" }{TEXT -1 8 "   and  " }{XPPEDIT 18 0 "d
v = 1*dx;" "6#/%#dvG*&\"\"\"F&%#dxGF&" }{TEXT -1 3 ",  " }{XPPEDIT 18 
0 "v = x;" "6#/%\"vG%\"xG" }{TEXT -1 7 " . Then" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "        " }{XPPEDIT 18 0 "
int(arcsin(x),x) = x*arcsin(x)-int(x*1/sqrt(1-x^2),x);" "6#/-%$intG6$-
%'arcsinG6#%\"xGF*,&*&F*\"\"\"-F(6#F*F-F--F%6$*(F*F-F-F--%%sqrtG6#,&F-
F-*$F*\"\"#!\"\"F9F*F9" }{XPPEDIT 18 0 "`` = x*arcsin(x)+(1-x^2)^(1/2)
+C;" "6#/%!G,(*&%\"xG\"\"\"-%'arcsinG6#F'F(F(),&F(F(*$F'\"\"#!\"\"*&F(
F(F/F0F(%\"CGF(" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 49 " (The final integration is done by substi
tuting  " }{XPPEDIT 18 0 "u = 1-x^2;" "6#/%\"uG,&\"\"\"F&*$%\"xG\"\"#!
\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "du = -2*x*dx;" "6#/%#duG,$*(\"
\"#\"\"\"%\"xGF(%#dxGF(!\"\"" }{TEXT -1 2 ".)" }}{PARA 0 "" 0 "" 
{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 13 "Verification:" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "te
steq(diff(x*arcsin(x)+(1-x^2)^(1/2) + C, x) = arcsin(x));" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT 273 7 "Example" }}{PARA 0 "" 0 "" {TEXT -1 12 "Calcula
te   " }{XPPEDIT 18 0 "int(arctan(x),x);" "6#-%$intG6$-%'arctanG6#%\"x
GF)" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT 274 8 "Solution" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "
Set  " }{XPPEDIT 18 0 "u = arctan(x);" "6#/%\"uG-%'arctanG6#%\"xG" }
{TEXT -1 5 ",    " }{XPPEDIT 18 0 "du = ``;" "6#/%#duG%!G" }{TEXT -1 
1 " " }{XPPEDIT 18 0 "1/(1+x^2);" "6#*&\"\"\"F$,&F$F$*$%\"xG\"\"#F$!\"
\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dx;" "6#%#dxG" }{TEXT -1 8 "   and
  " }{XPPEDIT 18 0 "dv = 1*dx;" "6#/%#dvG*&\"\"\"F&%#dxGF&" }{TEXT -1 
3 ",  " }{XPPEDIT 18 0 "v = x;" "6#/%\"vG%\"xG" }{TEXT -1 7 " . Then" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "        \+
" }{XPPEDIT 18 0 "int(arctan(x),x) = x*arctan(x)-int(x/(1+x^2),x);" "6
#/-%$intG6$-%'arctanG6#%\"xGF*,&*&F*\"\"\"-F(6#F*F-F--F%6$*&F*F-,&F-F-
*$F*\"\"#F-!\"\"F*F6" }{XPPEDIT 18 0 "`` = x*arctan(x)-1/2;" "6#/%!G,&
*&%\"xG\"\"\"-%'arctanG6#F'F(F(*&F(F(\"\"#!\"\"F." }{TEXT -1 1 " " }
{XPPEDIT 18 0 "ln(1+x^2)+C;" "6#,&-%#lnG6#,&\"\"\"F(*$%\"xG\"\"#F(F(%
\"CGF(" }{TEXT -1 4 "   ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 49 " (The final integration is done by substituting  \+
" }{XPPEDIT 18 0 "u = 1+x^2;" "6#/%\"uG,&\"\"\"F&*$%\"xG\"\"#F&" }
{TEXT -1 2 ", " }{XPPEDIT 18 0 "du = 2*x*dx;" "6#/%#duG*(\"\"#\"\"\"%
\"xGF'%#dxGF'" }{TEXT -1 2 ".)" }}{PARA 0 "" 0 "" {TEXT -1 2 "  " }}
{PARA 0 "" 0 "" {TEXT -1 13 "Verification:" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "testeq(diff(x*arct
an(x)-1/2*ln(1+x^2) + C, x) = arctan(x));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 36 "5. Other In
tegrals Involving  ln(x) " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 
"" 0 "" {TEXT 290 7 "Example" }}{PARA 0 "" 0 "" {TEXT -1 12 "Calculate
   " }{XPPEDIT 18 0 "int(x^p*ln(x),x);" "6#-%$intG6$*&)%\"xG%\"pG\"\"
\"-%#lnG6#F(F*F(" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 291 8 "Solution" }{TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 5 "Set  " }{XPPEDIT 18 0 "u = ln(x);" "6#/%\"uG-%#lnG6#%\"x
G" }{TEXT -1 5 ",    " }{XPPEDIT 18 0 "du = ``;" "6#/%#duG%!G" }{TEXT 
-1 1 " " }{XPPEDIT 18 0 "1/x;" "6#*&\"\"\"F$%\"xG!\"\"" }{TEXT -1 1 " \+
" }{XPPEDIT 18 0 "dx;" "6#%#dxG" }{TEXT -1 8 "   and  " }{XPPEDIT 18 
0 "dv = x^p*dx;" "6#/%#dvG*&)%\"xG%\"pG\"\"\"%#dxGF)" }{TEXT -1 3 ",  \+
" }{XPPEDIT 18 0 "v = 1/(p+1);" "6#/%\"vG*&\"\"\"F&,&%\"pGF&F&F&!\"\"
" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^(p+1);" "6#)%\"xG,&%\"pG\"\"\"F'F'
" }{TEXT -1 6 ". Then" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 20 "                    " }{XPPEDIT 18 0 "int(x^p*ln(x),x)
 = 1/(p+1);" "6#/-%$intG6$*&)%\"xG%\"pG\"\"\"-%#lnG6#F)F+F)*&F+F+,&F*F
+F+F+!\"\"" }{XPPEDIT 18 0 "``*x^(p+1)*ln(x)-1/(p+1);" "6#,&*(%!G\"\"
\")%\"xG,&%\"pGF&F&F&F&-%#lnG6#F(F&F&*&F&F&,&F*F&F&F&!\"\"F0" }{TEXT 
-1 1 " " }{XPPEDIT 18 0 "int(x^(p+1)*1/x,x);" "6#-%$intG6$*()%\"xG,&%
\"pG\"\"\"F+F+F+F+F+F(!\"\"F(" }{TEXT -1 5 "   or" }}{PARA 0 "" 0 "" 
{TEXT -1 19 "                   " }{XPPEDIT 18 0 "int(x^p*ln(x),x) = 1
/(p+1);" "6#/-%$intG6$*&)%\"xG%\"pG\"\"\"-%#lnG6#F)F+F)*&F+F+,&F*F+F+F
+!\"\"" }{XPPEDIT 18 0 "``*x^(p+1)*ln(x)-1/((p+1)^2);" "6#,&*(%!G\"\"
\")%\"xG,&%\"pGF&F&F&F&-%#lnG6#F(F&F&*&F&F&*$,&F*F&F&F&\"\"#!\"\"F2" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "x^(p+1)+C;" "6#,&)%\"xG,&%\"pG\"\"\"F(F
(F(%\"CGF(" }{TEXT -1 5 "     " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Verificat
ion:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 67 "testeq( diff(x^(p+1)*ln(x)/(p+1)-x^(p+1)/(p+1)^2, x) \+
= x^p*ln(x) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 316 7 "Example" }}{PARA 0 "" 
0 "" {TEXT -1 31 "Find a reduction formula for   " }{XPPEDIT 18 0 "int
(ln(x)^p,x);" "6#-%$intG6$)-%#lnG6#%\"xG%\"pGF*" }{TEXT -1 2 " ." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 317 8 "Solution
" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Set  " }{XPPEDIT 18 0 "
u = ln(x)^p;" "6#/%\"uG)-%#lnG6#%\"xG%\"pG" }{TEXT -1 5 ",    " }
{XPPEDIT 18 0 "du = ``;" "6#/%#duG%!G" }{TEXT -1 1 " " }{XPPEDIT 18 0 
"p*ln(x)^(p-1)/x;" "6#*(%\"pG\"\"\")-%#lnG6#%\"xG,&F$F%F%!\"\"F%F*F," 
}{TEXT -1 1 " " }{XPPEDIT 18 0 "dx;" "6#%#dxG" }{TEXT -1 8 "   and  " 
}{XPPEDIT 18 0 "dv = 1*dx;" "6#/%#dvG*&\"\"\"F&%#dxGF&" }{TEXT -1 3 ",
  " }{XPPEDIT 18 0 "v = x;" "6#/%\"vG%\"xG" }{TEXT -1 7 " . Then" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "         \+
            " }{XPPEDIT 18 0 "int(ln(x)^p,x) = x*ln(x)^p-p*int(ln(x)^(
p-1),x);" "6#/-%$intG6$)-%#lnG6#%\"xG%\"pGF+,&*&F+\"\"\")-F)6#F+F,F/F/
*&F,F/-F%6$)-F)6#F+,&F,F/F/!\"\"F+F/F:" }{TEXT -1 2 " ." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
318 7 "Example" }}{PARA 0 "" 0 "" {TEXT -1 12 "Calculate   " }
{XPPEDIT 18 0 "int(ln(x)^3,x);" "6#-%$intG6$*$-%#lnG6#%\"xG\"\"$F*" }
{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 319 8 "Solution" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "U
sing the reduction formula of the preceding example with p = 3, we hav
e" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "    \+
          " }{XPPEDIT 18 0 "int(ln(x)^3,x) = x*ln(x)^3-3*int(ln(x)^2,x
);" "6#/-%$intG6$*$-%#lnG6#%\"xG\"\"$F+,&*&F+\"\"\"*$-F)6#F+F,F/F/*&F,
F/-F%6$*$-F)6#F+\"\"#F+F/!\"\"" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "Using the reduction formu
la of the preceding example with p = 2, we have" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "              " }
{XPPEDIT 18 0 "int(ln(x)^3,x) = x*ln(x)^3-3*(x*ln(x)^2-2*int(ln(x),x))
;" "6#/-%$intG6$*$-%#lnG6#%\"xG\"\"$F+,&*&F+\"\"\"*$-F)6#F+F,F/F/*&F,F
/,&*&F+F/*$-F)6#F+\"\"#F/F/*&F9F/-F%6$-F)6#F+F+F/!\"\"F/F?" }{TEXT -1 
2 " ." }}{PARA 0 "" 0 "" {TEXT -1 5 "Thus," }}{PARA 0 "" 0 "" {TEXT 
-1 13 "             " }{XPPEDIT 18 0 "int(ln(x)^3,x) = x*ln(x)^3-3*(x*
ln(x)^2-2*(x*ln(x)-x))+C;" "6#/-%$intG6$*$-%#lnG6#%\"xG\"\"$F+,(*&F+\"
\"\"*$-F)6#F+F,F/F/*&F,F/,&*&F+F/*$-F)6#F+\"\"#F/F/*&F9F/,&*&F+F/-F)6#
F+F/F/F+!\"\"F/F?F/F?%\"CGF/" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" 
{TEXT -1 2 "or" }}{PARA 0 "" 0 "" {TEXT -1 12 "            " }
{XPPEDIT 18 0 "int(ln(x)^3,x) = x*ln(x)^3-3*x*ln(x)^2+6*x*ln(x)-6*x+C;
" "6#/-%$intG6$*$-%#lnG6#%\"xG\"\"$F+,,*&F+\"\"\"*$-F)6#F+F,F/F/*(F,F/
F+F/-F)6#F+\"\"#!\"\"*(\"\"'F/F+F/-F)6#F+F/F/*&F9F/F+F/F7%\"CGF/" }
{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 13 "Verification:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "test
eq(diff(x*ln(x)^3-3*x*ln(x)^2+6*x*ln(x)-6*x+C,x) = ln(x)^3);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT 277 7 "Example" }}{PARA 0 "" 0 "" {TEXT -1 12 "Calculate   \+
" }{XPPEDIT 18 0 "int(ln((a*x+b)^p),x);" "6#-%$intG6$-%#lnG6#),&*&%\"a
G\"\"\"%\"xGF-F-%\"bGF-%\"pGF." }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 278 8 "Solution" }{TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "First w
e consider   " }{XPPEDIT 18 0 "int(ln(a*x+b),x);" "6#-%$intG6$-%#lnG6#
,&*&%\"aG\"\"\"%\"xGF,F,%\"bGF,F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 5 "Set  " }{XPPEDIT 18 0 "u = ln(a*x+b);" "6#/%\"uG-%#lnG6#,&
*&%\"aG\"\"\"%\"xGF+F+%\"bGF+" }{TEXT -1 5 ",    " }{XPPEDIT 18 0 "du \+
= ``;" "6#/%#duG%!G" }{TEXT -1 1 " " }{XPPEDIT 18 0 "a/(a*x+b);" "6#*&
%\"aG\"\"\",&*&F$F%%\"xGF%F%%\"bGF%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 
18 0 "dx;" "6#%#dxG" }{TEXT -1 8 "   and  " }{XPPEDIT 18 0 "dv = 1*dx;
" "6#/%#dvG*&\"\"\"F&%#dxGF&" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "v = x;
" "6#/%\"vG%\"xG" }{TEXT -1 7 " . Then" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 8 "        " }{XPPEDIT 18 0 "int(ln(a*x+b
),x) = x*ln(a*x+b)-int(a/(a*x+b)*x,x);" "6#/-%$intG6$-%#lnG6#,&*&%\"aG
\"\"\"%\"xGF-F-%\"bGF-F.,&*&F.F--F(6#,&*&F,F-F.F-F-F/F-F-F--F%6$*(F,F-
,&*&F,F-F.F-F-F/F-!\"\"F.F-F.F;" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 279 35 "General \+
rule: Whenever integrating " }{XPPEDIT 280 0 "p(x)/q(x);" "6#*&-%\"pG6
#%\"xG\"\"\"-%\"qG6#F'!\"\"" }{TEXT 281 9 "  where  " }{XPPEDIT 282 0 
"p(x);" "6#-%\"pG6#%\"xG" }{TEXT 283 7 "  and  " }{XPPEDIT 284 0 "q(x)
;" "6#-%\"qG6#%\"xG" }{TEXT 285 22 "  are polynomials and " }{XPPEDIT 
286 0 "degree(q) <= degree(p);" "6#1-%'degreeG6#%\"qG-F%6#%\"pG" }
{TEXT 287 10 ",  divide!" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Since   " }{XPPEDIT 18 0 "a*x/(a*x+
b) = 1-b/(a*x+b);" "6#/*(%\"aG\"\"\"%\"xGF&,&*&F%F&F'F&F&%\"bGF&!\"\",
&F&F&*&F*F&,&*&F%F&F'F&F&F*F&F+F+" }{TEXT -1 12 ",  we have  " }
{XPPEDIT 18 0 "int(a/(a*x+b)*x,x) = x-b/a;" "6#/-%$intG6$*(%\"aG\"\"\"
,&*&F(F)%\"xGF)F)%\"bGF)!\"\"F,F)F,,&F,F)*&F-F)F(F.F." }{TEXT -1 1 " \+
" }{XPPEDIT 18 0 "ln(abs(a*x+b))+C;" "6#,&-%#lnG6#-%$absG6#,&*&%\"aG\"
\"\"%\"xGF-F-%\"bGF-F-%\"CGF-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 5 "Thus," }}{PARA 0 "" 0 "" {TEXT -1 13 "             " }
{XPPEDIT 18 0 "int(ln(a*x+b),x) = x*ln(a*x+b)-x+b/a;" "6#/-%$intG6$-%#
lnG6#,&*&%\"aG\"\"\"%\"xGF-F-%\"bGF-F.,(*&F.F--F(6#,&*&F,F-F.F-F-F/F-F
-F-F.!\"\"*&F/F-F,F6F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(abs(a*x+b))
;" "6#-%#lnG6#-%$absG6#,&*&%\"aG\"\"\"%\"xGF,F,%\"bGF," }{TEXT -1 6 " \+
 + C." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 9 "
 Finally," }}{PARA 0 "" 0 "" {TEXT -1 13 "             " }{XPPEDIT 18 
0 "int(ln((a*x+b)^p),x) = p*int(ln(a*x+b),x);" "6#/-%$intG6$-%#lnG6#),
&*&%\"aG\"\"\"%\"xGF.F.%\"bGF.%\"pGF/*&F1F.-F%6$-F(6#,&*&F-F.F/F.F.F0F
.F/F." }{TEXT -1 3 "   " }{XPPEDIT 18 0 "`` = ` `*p*x*ln(a*x+b)-p*x+p*
b/a;" "6#/%!G,(**%\"~G\"\"\"%\"pGF(%\"xGF(-%#lnG6#,&*&%\"aGF(F*F(F(%\"
bGF(F(F(*&F)F(F*F(!\"\"*(F)F(F1F(F0F3F(" }{TEXT -1 1 " " }{XPPEDIT 18 
0 "ln(abs(a*x+b));" "6#-%#lnG6#-%$absG6#,&*&%\"aG\"\"\"%\"xGF,F,%\"bGF
," }{TEXT -1 6 "  + C." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Verification:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
105 "simplify(diff(p*x*ln(a*x+b) - p*x + p*b/a*ln(abs(a*x+b)) + C, x) \+
- ln((a*x+b)^p)) assuming a>0, b>0, x>0;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT 288 7 "Example" }}{PARA 0 "" 0 "" 
{TEXT -1 12 "Calculate   " }{XPPEDIT 18 0 "int(ln(x^2+7*x+10),x);" "6#
-%$intG6$-%#lnG6#,(*$%\"xG\"\"#\"\"\"*&\"\"(F-F+F-F-\"#5F-F+" }{TEXT 
-1 2 " ." }}{PARA 0 "" 0 "" {TEXT 289 9 "Solution " }{TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 40 "We factor the argument of the logarithm:
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "x^2+7*x+10 = factor( x^2+7*x
+10 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*$)%\"xG\"\"#\"\"\"F)*&\"
\"(F)F'F)F)\"#5F)*&,&F'F)\"\"&F)F),&F'F)F(F)F)" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Thus," }}{PARA 0 "" 0 "" 
{TEXT -1 13 "             " }{XPPEDIT 18 0 "int(ln(x^2+7*x+10),x) = in
t(ln(x+5),x)+int(ln(x+2),x);" "6#/-%$intG6$-%#lnG6#,(*$%\"xG\"\"#\"\"
\"*&\"\"(F.F,F.F.\"#5F.F,,&-F%6$-F(6#,&F,F.\"\"&F.F,F.-F%6$-F(6#,&F,F.
F-F.F,F." }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 85 "\nWe apply t
he result of the preceding example to each integral on the right,  usi
ng  " }{XPPEDIT 18 0 "p = 1;" "6#/%\"pG\"\"\"" }{TEXT -1 7 "  and  " }
{XPPEDIT 18 0 "a = 1;" "6#/%\"aG\"\"\"" }{TEXT -1 16 "  for both and  \+
" }{XPPEDIT 18 0 "b = 5;" "6#/%\"bG\"\"&" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "b = 2;" "6#/%\"bG\"\"#" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 10 "          " }{XPPEDIT 18 0 "int(ln(x^
2+7*x+10),x) = ` `*x*ln(x+5)-x+5;" "6#/-%$intG6$-%#lnG6#,(*$%\"xG\"\"#
\"\"\"*&\"\"(F.F,F.F.\"#5F.F,,(*(%\"~GF.F,F.-F(6#,&F,F.\"\"&F.F.F.F,!
\"\"F8F." }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(abs(x+5));" "6#-%#lnG6#-%
$absG6#,&%\"xG\"\"\"\"\"&F+" }{TEXT -1 5 "  +  " }{XPPEDIT 18 0 "x*ln(
x+2)-x+2;" "6#,(*&%\"xG\"\"\"-%#lnG6#,&F%F&\"\"#F&F&F&F%!\"\"F+F&" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(abs(x+2));" "6#-%#lnG6#-%$absG6#,&%
\"xG\"\"\"\"\"#F+" }{TEXT -1 9 "  + C, or" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "  \+
            " }{XPPEDIT 18 0 "int(ln(x^2+7*x+10),x) = ` `*x*ln(x^2+7*x
+10)-2*x+5;" "6#/-%$intG6$-%#lnG6#,(*$%\"xG\"\"#\"\"\"*&\"\"(F.F,F.F.
\"#5F.F,,(*(%\"~GF.F,F.-F(6#,(*$F,F-F.*&F0F.F,F.F.F1F.F.F.*&F-F.F,F.!
\"\"\"\"&F." }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(abs(x+5));" "6#-%#lnG6
#-%$absG6#,&%\"xG\"\"\"\"\"&F+" }{TEXT -1 5 "  +  " }{XPPEDIT 18 0 "2;
" "6#\"\"#" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(abs(x+2));" "6#-%#lnG6#
-%$absG6#,&%\"xG\"\"\"\"\"#F+" }{TEXT -1 7 "  + C ." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Verification:" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "sim
plify( diff(x*ln(x^2+7*x+10)-2*x+5*ln(abs(x+5))+2*ln(abs(x+2)), x) - l
n( x^2+7*x+10 ))  assuming x > 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 309 44 "Exam
ple (Exercise 35, Page 515 Blank-Krantz)" }}{PARA 0 "" 0 "" {TEXT -1 
11 "Evaluate   " }{XPPEDIT 18 0 "int(ln(1+x^2),x);" "6#-%$intG6$-%#lnG
6#,&\"\"\"F**$%\"xG\"\"#F*F," }{TEXT -1 5 " .  \n" }}{PARA 0 "" 0 "" 
{TEXT 310 10 "\nSolution " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Set  " }{XPPEDIT 
18 0 "u = ln(1+x^2);" "6#/%\"uG-%#lnG6#,&\"\"\"F)*$%\"xG\"\"#F)" }
{TEXT -1 5 ",    " }{XPPEDIT 18 0 "du = ``;" "6#/%#duG%!G" }{TEXT -1 
1 " " }{XPPEDIT 18 0 "2*x/(1+x^2);" "6#*(\"\"#\"\"\"%\"xGF%,&F%F%*$F&F
$F%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dx;" "6#%#dxG" }{TEXT -1 8 "
   and  " }{XPPEDIT 18 0 "dv = 1*dx;" "6#/%#dvG*&\"\"\"F&%#dxGF&" }
{TEXT -1 3 ",  " }{XPPEDIT 18 0 "v = x;" "6#/%\"vG%\"xG" }{TEXT -1 7 "
 . Then" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 
"        " }{XPPEDIT 18 0 "int(ln(1+x^2),x) = x*ln(1+x^2)-int(2*x^2/(1
+x^2),x);" "6#/-%$intG6$-%#lnG6#,&\"\"\"F+*$%\"xG\"\"#F+F-,&*&F-F+-F(6
#,&F+F+*$F-F.F+F+F+-F%6$*(F.F+*$F-F.F+,&F+F+*$F-F.F+!\"\"F-F;" }{TEXT 
-1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
1 " " }{TEXT 311 57 "According to the general rule announced above, we
 divide:" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "quoti
ent := 2*x^2/(1+x^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)quotientG,
$*(\"\"#\"\"\"%\"xGF',&F(F(*$)F)F'F(F(!\"\"F(" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 111 "quotient = quo(numer(quotient), denom(quotient)
, x) + rem(numer(quotient), denom(quotient), x)/denom(quotient);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*(\"\"#\"\"\"%\"xGF&,&F'F'*$)F(F&F'
F'!\"\"F',&F&F'*&F&F'F)F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 4 "Thus" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 8 "        " }{XPPEDIT 18 0 "int(ln(1+x^2),x) = x*ln(
1+x^2)-2*x+int(2/(1+x^2),x);" "6#/-%$intG6$-%#lnG6#,&\"\"\"F+*$%\"xG\"
\"#F+F-,(*&F-F+-F(6#,&F+F+*$F-F.F+F+F+*&F.F+F-F+!\"\"-F%6$*&F.F+,&F+F+
*$F-F.F+F6F-F+" }{TEXT -1 6 "    or" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 8 "        " }{XPPEDIT 18 0 "int(ln(1+x^2),x
) = x*ln(1+x^2)-2*x+2*arctan(x);" "6#/-%$intG6$-%#lnG6#,&\"\"\"F+*$%\"
xG\"\"#F+F-,(*&F-F+-F(6#,&F+F+*$F-F.F+F+F+*&F.F+F-F+!\"\"*&F.F+-%'arct
anG6#F-F+F+" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Verification" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "testeq( diff(x*ln(1+x^2)-2*
x+2*arctan(x), x)  = ln(1+x^2) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%
%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 306 7 "Example" 
}}{PARA 0 "" 0 "" {TEXT -1 10 "Express   " }{XPPEDIT 18 0 "int(ln(a*x^
2+b*x+c),x);" "6#-%$intG6$-%#lnG6#,(*&%\"aG\"\"\"*$%\"xG\"\"#F,F,*&%\"
bGF,F.F,F,%\"cGF,F." }{TEXT -1 39 "  in terms of an integral of the fo
rm  " }{XPPEDIT 18 0 "int((alpha*x+beta)/(a*x^2+b*x+c),x);" "6#-%$intG
6$*&,&*&%&alphaG\"\"\"%\"xGF*F*%%betaGF*F*,(*&%\"aGF**$F+\"\"#F*F**&%
\"bGF*F+F*F*%\"cGF*!\"\"F+" }{TEXT -1 131 ".  \n\n(How we handle the l
atter depends on whether the denominator is an irreducible quadratic o
r the product \nof two linear terms.)" }}{PARA 0 "" 0 "" {TEXT 307 10 
"\nSolution " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Set  " }
{XPPEDIT 18 0 "u = ln(a*x^2+b*x+c);" "6#/%\"uG-%#lnG6#,(*&%\"aG\"\"\"*
$%\"xG\"\"#F+F+*&%\"bGF+F-F+F+%\"cGF+" }{TEXT -1 5 ",    " }{XPPEDIT 
18 0 "du = ``;" "6#/%#duG%!G" }{TEXT -1 1 " " }{XPPEDIT 18 0 "(2*a*x+b
)/(a*x^2+b*x+c);" "6#*&,&*(\"\"#\"\"\"%\"aGF'%\"xGF'F'%\"bGF'F',(*&F(F
'*$F)F&F'F'*&F*F'F)F'F'%\"cGF'!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d
x;" "6#%#dxG" }{TEXT -1 8 "   and  " }{XPPEDIT 18 0 "dv = 1*dx;" "6#/%
#dvG*&\"\"\"F&%#dxGF&" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "v = x;" "6#/%
\"vG%\"xG" }{TEXT -1 7 " . Then" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 8 "        " }{XPPEDIT 18 0 "int(ln(a*x^2+b*x
+c),x) = x*ln(a*x^2+b*x+c)-int((2*a*x^2+b*x)/(a*x^2+b*x+c),x);" "6#/-%
$intG6$-%#lnG6#,(*&%\"aG\"\"\"*$%\"xG\"\"#F-F-*&%\"bGF-F/F-F-%\"cGF-F/
,&*&F/F--F(6#,(*&F,F-*$F/F0F-F-*&F2F-F/F-F-F3F-F-F--F%6$*&,&*(F0F-F,F-
F/F0F-*&F2F-F/F-F-F-,(*&F,F-*$F/F0F-F-*&F2F-F/F-F-F3F-!\"\"F/FF" }
{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 1 " " }{TEXT 308 57 "According to the general rule announced \+
above, we divide:" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
40 "quotient := (2*a*x^2+b*x)/(a*x^2+b*x+c);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%)quotientG*&,&*(\"\"#\"\"\"%\"aGF))%\"xGF(F)F)*&%\"bG
F)F,F)F)F),(*&F*F)F+F)F)F-F)%\"cGF)!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 111 "quotient = quo(numer(quotient), denom(quotient), x) \+
+ rem(numer(quotient), denom(quotient), x)/denom(quotient);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/*&,&*(\"\"#\"\"\"%\"aGF()%\"xGF'F(F(*&%\"bG
F(F+F(F(F(,(*&F)F(F*F(F(F,F(%\"cGF(!\"\",&F'F(*&,&F,F1*&F'F(F0F(F1F(F.
F1F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "T
hus" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "   \+
     " }{XPPEDIT 18 0 "int(ln(a*x^2+b*x+c),x) = x*ln(a*x^2+b*x+c)-2*x+
int((b*x+2*c)/(a*x^2+b*x+c),x);" "6#/-%$intG6$-%#lnG6#,(*&%\"aG\"\"\"*
$%\"xG\"\"#F-F-*&%\"bGF-F/F-F-%\"cGF-F/,(*&F/F--F(6#,(*&F,F-*$F/F0F-F-
*&F2F-F/F-F-F3F-F-F-*&F0F-F/F-!\"\"-F%6$*&,&*&F2F-F/F-F-*&F0F-F3F-F-F-
,(*&F,F-*$F/F0F-F-*&F2F-F/F-F-F3F-F=F/F-" }{TEXT -1 2 " ." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 12 "Verification" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
88 "testeq( diff(x*ln(a*x^2+b*x+c)-2*x, x)  = ln(a*x^2+b*x+c) - ((b*x+
2*c)/(a*x^2+b*x+c)) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT 312 5 "Note:" }{TEXT -1 3 "   " }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 74 "Int((alpha*x+beta)/(a*x^2+b*x+c),x) = int((alpha
*x+beta)/(a*x^2+b*x+c),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG
6$*&,&*&%&alphaG\"\"\"%\"xGF+F+%%betaGF+F+,(*&%\"aGF+)F,\"\"#F+F+*&%\"
bGF+F,F+F+%\"cGF+!\"\"F,,(*&#F+F2F+*(F*F+F0F6-%#lnG6#F.F+F+F+**F2F+,&*
(\"\"%F+F0F+F5F+F+*$)F4F2F+F6#F6F2-%'arctanG6#*&,&*(F2F+F0F+F,F+F+F4F+
F+F?FDF+F-F+F+*,F?FDFEF+F*F+F4F+F0F6F6" }}}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 40 "5. Other Integrals Involvin
g   arctan(x)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 292 7 "Example" }}{PARA 0 "" 0 "" {TEXT -1 12 "Calculate   " }
{XPPEDIT 18 0 "int(x*arctan(x),x);" "6#-%$intG6$*&%\"xG\"\"\"-%'arctan
G6#F'F(F'" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT 293 8 "Solution" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 5 "Set  " }{XPPEDIT 18 0 "u = arctan(x);" "6#/%\"uG-%'arctanG6#%\"x
G" }{TEXT -1 5 ",    " }{XPPEDIT 18 0 "du = ``;" "6#/%#duG%!G" }{TEXT 
-1 1 " " }{XPPEDIT 18 0 "1/(1+x^2);" "6#*&\"\"\"F$,&F$F$*$%\"xG\"\"#F$
!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dx;" "6#%#dxG" }{TEXT -1 8 "   \+
and  " }{XPPEDIT 18 0 "dv = x*dx;" "6#/%#dvG*&%\"xG\"\"\"%#dxGF'" }
{TEXT -1 3 ",  " }{XPPEDIT 18 0 "v = 1/2;" "6#/%\"vG*&\"\"\"F&\"\"#!\"
\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"#" }{TEXT -1 
6 ". Then" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
20 "                    " }{XPPEDIT 18 0 "int(x*arctan(x),x) = 1/2;" "
6#/-%$intG6$*&%\"xG\"\"\"-%'arctanG6#F(F)F(*&F)F)\"\"#!\"\"" }
{XPPEDIT 18 0 "``*x^2*arctan(x)-1/2;" "6#,&*(%!G\"\"\"*$%\"xG\"\"#F&-%
'arctanG6#F(F&F&*&F&F&F)!\"\"F." }{TEXT -1 1 " " }{XPPEDIT 18 0 "int(x
^2/(1+x^2),x);" "6#-%$intG6$*&%\"xG\"\"#,&\"\"\"F**$F'F(F*!\"\"F'" }
{TEXT -1 7 "   or  " }}{PARA 0 "" 0 "" {TEXT -1 13 "             " }
{TEXT 296 51 "Following the general rule mentioned above, since  " }
{XPPEDIT 263 0 "degree(1+x^2) <= degree(x^2);" "6#1-%'degreeG6#,&\"\"
\"F(*$%\"xG\"\"#F(-F%6#*$F*F+" }{TEXT 297 13 ",  we divide:" }}{PARA 
0 "" 0 "" {TEXT -1 18 "                  " }{XPPEDIT 18 0 "int(x*arcta
n(x),x) = 1/2;" "6#/-%$intG6$*&%\"xG\"\"\"-%'arctanG6#F(F)F(*&F)F)\"\"
#!\"\"" }{XPPEDIT 18 0 "``*x^2*arctan(x)-1/2;" "6#,&*(%!G\"\"\"*$%\"xG
\"\"#F&-%'arctanG6#F(F&F&*&F&F&F)!\"\"F." }{TEXT -1 1 " " }{XPPEDIT 
18 0 "int(1-1/(1+x^2),x)+C;" "6#,&-%$intG6$,&\"\"\"F(*&F(F(,&F(F(*$%\"
xG\"\"#F(!\"\"F.F,F(%\"CGF(" }{TEXT -1 5 "     " }}{PARA 0 "" 0 "" 
{TEXT -1 5 "Thus," }}{PARA 0 "" 0 "" {TEXT -1 18 "                  " 
}{XPPEDIT 18 0 "int(x*arctan(x),x) = 1/2;" "6#/-%$intG6$*&%\"xG\"\"\"-
%'arctanG6#F(F)F(*&F)F)\"\"#!\"\"" }{XPPEDIT 18 0 "``*x^2*arctan(x)-1/
2;" "6#,&*(%!G\"\"\"*$%\"xG\"\"#F&-%'arctanG6#F(F&F&*&F&F&F)!\"\"F." }
{TEXT -1 1 " " }{XPPEDIT 18 0 "``*(x-arctan(x))+C;" "6#,&*&%!G\"\"\",&
%\"xGF&-%'arctanG6#F(!\"\"F&F&%\"CGF&" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 13 "Verification:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "testeq( diff
(x^2*arctan(x)/2-x/2+arctan(x)/2, x) = x*arctan(x) );" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT 294 7 "Example" }}{PARA 0 "" 0 "" {TEXT -1 12 "Calculate   \+
" }{XPPEDIT 18 0 "int(x^2*arctan(x),x);" "6#-%$intG6$*&%\"xG\"\"#-%'ar
ctanG6#F'\"\"\"F'" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 295 8 "Solution" }{TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 5 "Set  " }{XPPEDIT 18 0 "u = arctan(x);" "6#/%\"uG-%'arcta
nG6#%\"xG" }{TEXT -1 5 ",    " }{XPPEDIT 18 0 "du = ``;" "6#/%#duG%!G
" }{TEXT -1 1 " " }{XPPEDIT 18 0 "1/(1+x^2);" "6#*&\"\"\"F$,&F$F$*$%\"
xG\"\"#F$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dx;" "6#%#dxG" }{TEXT 
-1 8 "   and  " }{XPPEDIT 18 0 "dv = x^2*dx;" "6#/%#dvG*&%\"xG\"\"#%#d
xG\"\"\"" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "v = 1/3;" "6#/%\"vG*&\"\"
\"F&\"\"$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^3;" "6#*$%\"xG\"\"$
" }{TEXT -1 6 ". Then" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 20 "                    " }{XPPEDIT 18 0 "int(x^2*arctan(x
),x) = 1/3;" "6#/-%$intG6$*&%\"xG\"\"#-%'arctanG6#F(\"\"\"F(*&F-F-\"\"
$!\"\"" }{XPPEDIT 18 0 "``*x^3*arctan(x)-1/3;" "6#,&*(%!G\"\"\"*$%\"xG
\"\"$F&-%'arctanG6#F(F&F&*&F&F&F)!\"\"F." }{TEXT -1 1 " " }{XPPEDIT 
18 0 "int(x^3/(1+x^2),x);" "6#-%$intG6$*&%\"xG\"\"$,&\"\"\"F**$F'\"\"#
F*!\"\"F'" }{TEXT -1 5 "   or" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 2 "  " }{TEXT 298 51 "Following the general r
ule mentioned above, since  " }{XPPEDIT 256 0 "degree(1+x^2) <= degree
(x^3);" "6#1-%'degreeG6#,&\"\"\"F(*$%\"xG\"\"#F(-F%6#*$F*\"\"$" }
{TEXT 299 13 ",  we divide:" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 63 "x^3/(1+x^2) = quo(x^3, 1+x^2, x) + rem(x^3, 1+x^2, x)
/(1+x^2); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"xG\"\"$,&\"\"\"F(*
$)F%\"\"#F(F(!\"\",&F%F(*&F%F(F'F,F," }}}{PARA 0 "" 0 "" {TEXT -1 20 "
    \n               " }{XPPEDIT 18 0 "int(x^2*arctan(x),x) = 1/3;" "6
#/-%$intG6$*&%\"xG\"\"#-%'arctanG6#F(\"\"\"F(*&F-F-\"\"$!\"\"" }
{XPPEDIT 18 0 "``*x^3*arctan(x)-1/3;" "6#,&*(%!G\"\"\"*$%\"xG\"\"$F&-%
'arctanG6#F(F&F&*&F&F&F)!\"\"F." }{TEXT -1 1 " " }{XPPEDIT 18 0 "int(x
-x/(1+x^2),x)+C;" "6#,&-%$intG6$,&%\"xG\"\"\"*&F(F),&F)F)*$F(\"\"#F)!
\"\"F.F(F)%\"CGF)" }{TEXT -1 5 "     " }}{PARA 0 "" 0 "" {TEXT -1 5 "T
hus," }}{PARA 0 "" 0 "" {TEXT -1 18 "                  " }{XPPEDIT 18 
0 "int(x^2*arctan(x),x) = 1/3;" "6#/-%$intG6$*&%\"xG\"\"#-%'arctanG6#F
(\"\"\"F(*&F-F-\"\"$!\"\"" }{XPPEDIT 18 0 "``*x^3*arctan(x)-1/3;" "6#,
&*(%!G\"\"\"*$%\"xG\"\"$F&-%'arctanG6#F(F&F&*&F&F&F)!\"\"F." }{TEXT 
-1 1 " " }{XPPEDIT 18 0 "``*(x^2/2-ln(1+x^2)/2)+C;" "6#,&*&%!G\"\"\",&
*&%\"xG\"\"#F*!\"\"F&*&-%#lnG6#,&F&F&*$F)F*F&F&F*F+F+F&F&%\"CGF&" }}
{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 18 "                  " }{XPPEDIT 18 0 "int(x
^2*arctan(x),x) = 1/3;" "6#/-%$intG6$*&%\"xG\"\"#-%'arctanG6#F(\"\"\"F
(*&F-F-\"\"$!\"\"" }{XPPEDIT 18 0 "``*x^3*arctan(x)-1/6;" "6#,&*(%!G\"
\"\"*$%\"xG\"\"$F&-%'arctanG6#F(F&F&*&F&F&\"\"'!\"\"F/" }{TEXT -1 1 " \+
" }{XPPEDIT 18 0 "x^2+ln(1+x^2)/6+C;" "6#,(*$%\"xG\"\"#\"\"\"*&-%#lnG6
#,&F'F'*$F%F&F'F'\"\"'!\"\"F'%\"CGF'" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Verific
ation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 69 "testeq( diff(x^3*arctan(x)/3-x^2/6+ln(1+x^2)/6, x) = \+
x^2*arctan(x) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 
"" 0 "" {TEXT -1 31 "6. Integrating Powers of sec(x)" }}{PARA 0 "" 0 "
" {TEXT -1 12 "We know that" }}{PARA 0 "" 0 "" {TEXT -1 8 "        " }
}{PARA 0 "" 0 "" {TEXT -1 12 "            " }{XPPEDIT 18 0 "int(sec(x)
,x) = ln(abs(sec(x)+tan(x)))+C;" "6#/-%$intG6$-%$secG6#%\"xGF*,&-%#lnG
6#-%$absG6#,&-F(6#F*\"\"\"-%$tanG6#F*F5F5%\"CGF5" }{TEXT -1 8 "     an
d" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "    \+
        " }{XPPEDIT 18 0 "int(sec(x)^2,x) = tan(x)+C;" "6#/-%$intG6$*$
-%$secG6#%\"xG\"\"#F+,&-%$tanG6#F+\"\"\"%\"CGF1" }{TEXT -1 2 " ." }}
{PARA 0 "" 0 "" {TEXT -1 5 "     " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 300 7 "Example" }}{PARA 0 "" 0 "" {TEXT -1 12 "C
alculate   " }{XPPEDIT 18 0 "int(sec(x)^3,x);" "6#-%$intG6$*$-%$secG6#
%\"xG\"\"$F*" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 301 8 "Solution" }}{PARA 0 "" 0 "" {TEXT -1 5 "S
et  " }{XPPEDIT 18 0 "u = sec(x);" "6#/%\"uG-%$secG6#%\"xG" }{TEXT -1 
5 ",    " }{XPPEDIT 18 0 "du = ``;" "6#/%#duG%!G" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "sec(x)*tan(x);" "6#*&-%$secG6#%\"xG\"\"\"-%$tanG6#F'F(
" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dx;" "6#%#dxG" }{TEXT -1 8 "   and  \+
" }{XPPEDIT 18 0 "dv = sec(x)^2*dx;" "6#/%#dvG*&-%$secG6#%\"xG\"\"#%#d
xG\"\"\"" }{TEXT -1 3 ",  " }{XPPEDIT 18 0 "v = tan(x);" "6#/%\"vG-%$t
anG6#%\"xG" }{TEXT -1 8 " .  Then" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 20 "                    " }{XPPEDIT 18 0 "int
(sec(x)^3,x) = sec(x)*tan(x)-int(sec(x)*tan(x)^2,x);" "6#/-%$intG6$*$-
%$secG6#%\"xG\"\"$F+,&*&-F)6#F+\"\"\"-%$tanG6#F+F1F1-F%6$*&-F)6#F+F1*$
-F36#F+\"\"#F1F+!\"\"" }{TEXT -1 5 "   or" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "                  " }{XPPEDIT 18 
0 "int(sec(x)^3,x) = sec(x)*tan(x)-int(sec(x)*(sec(x)^2-1),x);" "6#/-%
$intG6$*$-%$secG6#%\"xG\"\"$F+,&*&-F)6#F+\"\"\"-%$tanG6#F+F1F1-F%6$*&-
F)6#F+F1,&*$-F)6#F+\"\"#F1F1!\"\"F1F+F?" }{TEXT -1 4 "  or" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "                 \+
  " }{XPPEDIT 18 0 "int(sec(x)^3,x) = sec(x)*tan(x)-int(sec(x)^3,x)+in
t(sec(x),x);" "6#/-%$intG6$*$-%$secG6#%\"xG\"\"$F+,(*&-F)6#F+\"\"\"-%$
tanG6#F+F1F1-F%6$*$-F)6#F+F,F+!\"\"-F%6$-F)6#F+F+F1" }{TEXT -1 3 "  .
" }}{PARA 0 "" 0 "" {TEXT -1 10 "Therefore," }}{PARA 0 "" 0 "" {TEXT 
-1 10 "          " }}{PARA 0 "" 0 "" {TEXT -1 15 "               " }
{XPPEDIT 18 0 "2*int(sec(x)^3,x) = sec(x)*tan(x)+int(sec(x),x);" "6#/*
&\"\"#\"\"\"-%$intG6$*$-%$secG6#%\"xG\"\"$F.F&,&*&-F,6#F.F&-%$tanG6#F.
F&F&-F(6$-F,6#F.F.F&" }}{PARA 0 "" 0 "" {TEXT -1 3 " or" }}{PARA 0 "" 
0 "" {TEXT -1 17 "                 " }{XPPEDIT 18 0 "int(sec(x)^3,x) =
 1/2;" "6#/-%$intG6$*$-%$secG6#%\"xG\"\"$F+*&\"\"\"F.\"\"#!\"\"" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "sec(x)*tan(x);" "6#*&-%$secG6#%\"xG\"\"
\"-%$tanG6#F'F(" }{TEXT -1 6 "  +   " }{XPPEDIT 18 0 "1/2;" "6#*&\"\"
\"F$\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "ln(abs(sec(x)+tan(x)))
;" "6#-%#lnG6#-%$absG6#,&-%$secG6#%\"xG\"\"\"-%$tanG6#F-F." }{TEXT -1 
8 "   +  C." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" 
{TEXT -1 41 "7. Reduction Formula for Powers of sin(x)" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 303 8 "Exercise" }}{PARA 0 
"" 0 "" {TEXT -1 28 "Derive the reduction formula" }}{PARA 0 "" 0 "" 
{TEXT -1 11 "           " }}{PARA 0 "" 0 "" {TEXT -1 13 "             \+
" }{XPPEDIT 18 0 "int(sin(x)^N,x) = -1/N;" "6#/-%$intG6$)-%$sinG6#%\"x
G%\"NGF+,$*&\"\"\"F/F,!\"\"F0" }{TEXT -1 2 "  " }{XPPEDIT 18 0 "sin(x)
^(N-1)*cos(x)+``;" "6#,&*&)-%$sinG6#%\"xG,&%\"NG\"\"\"F,!\"\"F,-%$cosG
6#F)F,F,%!GF," }{TEXT -1 1 " " }{XPPEDIT 18 0 "(N-1)/N;" "6#*&,&%\"NG
\"\"\"F&!\"\"F&F%F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "int(sin(x)^(N-2),
x);" "6#-%$intG6$)-%$sinG6#%\"xG,&%\"NG\"\"\"\"\"#!\"\"F*" }{TEXT -1 
1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT 302 8 "Solution" }}{PARA 0 "" 0 "" {TEXT -1 5 
"Set  " }{XPPEDIT 18 0 "u = sin(x)^(N-1);" "6#/%\"uG)-%$sinG6#%\"xG,&%
\"NG\"\"\"F,!\"\"" }{TEXT -1 5 ",    " }{XPPEDIT 18 0 "du = (N-1)*sin(
x)^(N-2)*cos(x)*dx;" "6#/%#duG**,&%\"NG\"\"\"F(!\"\"F()-%$sinG6#%\"xG,
&F'F(\"\"#F)F(-%$cosG6#F.F(%#dxGF(" }{TEXT -1 9 "    and  " }{XPPEDIT 
18 0 "dv = sin(x)*dx;" "6#/%#dvG*&-%$sinG6#%\"xG\"\"\"%#dxGF*" }{TEXT 
-1 3 ",  " }{XPPEDIT 18 0 "v = -cos(x);" "6#/%\"vG,$-%$cosG6#%\"xG!\"
\"" }{TEXT -1 8 " .  Then" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 20 "                    " }{XPPEDIT 18 0 "int(sin(x)^N
,x) = -sin(x)^(N-1)*cos(x)+(N-1)*int(sin(x)^(N-2)*cos(x)^2,x);" "6#/-%
$intG6$)-%$sinG6#%\"xG%\"NGF+,&*&)-F)6#F+,&F,\"\"\"F3!\"\"F3-%$cosG6#F
+F3F4*&,&F,F3F3F4F3-F%6$*&)-F)6#F+,&F,F3\"\"#F4F3*$-F66#F+FAF3F+F3F3" 
}{TEXT -1 5 "   or" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 19 "                   " }{XPPEDIT 18 0 "int(sin(x)^N,x) = -s
in(x)^(N-1)*cos(x)+(N-1)*int(sin(x)^(N-2)*(1-sin(x)^2),x);" "6#/-%$int
G6$)-%$sinG6#%\"xG%\"NGF+,&*&)-F)6#F+,&F,\"\"\"F3!\"\"F3-%$cosG6#F+F3F
4*&,&F,F3F3F4F3-F%6$*&)-F)6#F+,&F,F3\"\"#F4F3,&F3F3*$-F)6#F+FAF4F3F+F3
F3" }{TEXT -1 4 "  or" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 19 "                   " }{XPPEDIT 18 0 "int(sin(x)^N,x) =
 -sin(x)^(N-1)*cos(x)+(N-1)*int(sin(x)^(N-2),x)-(N-1)*int(sin(x)^N,x);
" "6#/-%$intG6$)-%$sinG6#%\"xG%\"NGF+,(*&)-F)6#F+,&F,\"\"\"F3!\"\"F3-%
$cosG6#F+F3F4*&,&F,F3F3F4F3-F%6$)-F)6#F+,&F,F3\"\"#F4F+F3F3*&,&F,F3F3F
4F3-F%6$)-F)6#F+F,F+F3F4" }{TEXT -1 3 "  ." }}{PARA 0 "" 0 "" {TEXT 
-1 10 "Therefore," }}{PARA 0 "" 0 "" {TEXT -1 10 "          " }}{PARA 
0 "" 0 "" {TEXT -1 15 "               " }{XPPEDIT 18 0 "(1+N-1)*int(si
n(x)^N,x) = -sin(x)^(N-1)*cos(x)+(N-1)*int(sin(x)^(N-2),x);" "6#/*&,(
\"\"\"F&%\"NGF&F&!\"\"F&-%$intG6$)-%$sinG6#%\"xGF'F0F&,&*&)-F.6#F0,&F'
F&F&F(F&-%$cosG6#F0F&F(*&,&F'F&F&F(F&-F*6$)-F.6#F0,&F'F&\"\"#F(F0F&F&
" }}{PARA 0 "" 0 "" {TEXT -1 3 " or" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 18 "                  " }{XPPEDIT 18 0 "int(
sin(x)^N,x) = -1/N;" "6#/-%$intG6$)-%$sinG6#%\"xG%\"NGF+,$*&\"\"\"F/F,
!\"\"F0" }{TEXT -1 2 "  " }{XPPEDIT 18 0 "sin(x)^(N-1)*cos(x)+``;" "6#
,&*&)-%$sinG6#%\"xG,&%\"NG\"\"\"F,!\"\"F,-%$cosG6#F)F,F,%!GF," }{TEXT 
-1 1 " " }{XPPEDIT 18 0 "(N-1)/N;" "6#*&,&%\"NG\"\"\"F&!\"\"F&F%F'" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "int(sin(x)^(N-2),x);" "6#-%$intG6$)-%$s
inG6#%\"xG,&%\"NG\"\"\"\"\"#!\"\"F*" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "
" {TEXT -1 17 "                 " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 41 "8. Reduction Formula for Powers of cos(x)
" }}{PARA 0 "" 0 "" {TEXT 305 8 "Exercise" }}{PARA 0 "" 0 "" {TEXT -1 
28 "Derive the reduction formula" }}{PARA 0 "" 0 "" {TEXT -1 11 "     \+
      " }}{PARA 0 "" 0 "" {TEXT -1 13 "             " }{XPPEDIT 18 0 "
int(cos(x)^N,x) = 1/N;" "6#/-%$intG6$)-%$cosG6#%\"xG%\"NGF+*&\"\"\"F.F
,!\"\"" }{TEXT -1 2 "  " }{XPPEDIT 18 0 "sin(x)*cos(x)^(N-1)+``;" "6#,
&*&-%$sinG6#%\"xG\"\"\")-%$cosG6#F(,&%\"NGF)F)!\"\"F)F)%!GF)" }{TEXT 
-1 1 " " }{XPPEDIT 18 0 "(N-1)/N;" "6#*&,&%\"NG\"\"\"F&!\"\"F&F%F'" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "int(cos(x)^(N-2),x);" "6#-%$intG6$)-%$c
osG6#%\"xG,&%\"NG\"\"\"\"\"#!\"\"F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 304 8 "Solution" }}{PARA 0 "" 0 "" {TEXT -1 5 "Set  " }{XPPEDIT 
18 0 "u = cos(x)^(N-1);" "6#/%\"uG)-%$cosG6#%\"xG,&%\"NG\"\"\"F,!\"\"
" }{TEXT -1 5 ",    " }{XPPEDIT 18 0 "du = -(N-1)*cos(x)^(N-2)*sin(x)*
dx;" "6#/%#duG,$**,&%\"NG\"\"\"F)!\"\"F))-%$cosG6#%\"xG,&F(F)\"\"#F*F)
-%$sinG6#F/F)%#dxGF)F*" }{TEXT -1 9 "    and  " }{XPPEDIT 18 0 "dv = c
os(x)*dx;" "6#/%#dvG*&-%$cosG6#%\"xG\"\"\"%#dxGF*" }{TEXT -1 3 ",  " }
{XPPEDIT 18 0 "v = sin(x);" "6#/%\"vG-%$sinG6#%\"xG" }{TEXT -1 8 " .  \+
Then" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 " \+
                   " }{XPPEDIT 18 0 "int(cos(x)^N,x) = sin(x)*cos(x)^(
N-1)+(N-1)*int(cos(x)^(N-2)*sin(x)^2,x);" "6#/-%$intG6$)-%$cosG6#%\"xG
%\"NGF+,&*&-%$sinG6#F+\"\"\")-F)6#F+,&F,F2F2!\"\"F2F2*&,&F,F2F2F7F2-F%
6$*&)-F)6#F+,&F,F2\"\"#F7F2*$-F06#F+FAF2F+F2F2" }{TEXT -1 5 "   or" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "         \+
          " }{XPPEDIT 18 0 "int(cos(x)^N,x) = sin(x)*cos(x)^(N-1)+(N-1
)*int(cos(x)^(N-2)*(1-cos(x)^2),x);" "6#/-%$intG6$)-%$cosG6#%\"xG%\"NG
F+,&*&-%$sinG6#F+\"\"\")-F)6#F+,&F,F2F2!\"\"F2F2*&,&F,F2F2F7F2-F%6$*&)
-F)6#F+,&F,F2\"\"#F7F2,&F2F2*$-F)6#F+FAF7F2F+F2F2" }{TEXT -1 4 "  or" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "       \+
           " }{XPPEDIT 18 0 "int(cos(x)^N,x) = cos(x)^(N-1)*sin(x)+(N-
1)*int(cos(x)^(N-2),x)-(N-1)*int(cos(x)^N,x);" "6#/-%$intG6$)-%$cosG6#
%\"xG%\"NGF+,(*&)-F)6#F+,&F,\"\"\"F3!\"\"F3-%$sinG6#F+F3F3*&,&F,F3F3F4
F3-F%6$)-F)6#F+,&F,F3\"\"#F4F+F3F3*&,&F,F3F3F4F3-F%6$)-F)6#F+F,F+F3F4
" }{TEXT -1 5 "  .  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 10 "Therefore," }}{PARA 0 "" 0 "" {TEXT -1 10 "          " 
}}{PARA 0 "" 0 "" {TEXT -1 15 "               " }{XPPEDIT 18 0 "(1+N-1
)*int(cos(x)^N,x) = sin(x)*cos(x)^(N-1)+(N-1)*int(cos(x)^(N-2),x);" "6
#/*&,(\"\"\"F&%\"NGF&F&!\"\"F&-%$intG6$)-%$cosG6#%\"xGF'F0F&,&*&-%$sin
G6#F0F&)-F.6#F0,&F'F&F&F(F&F&*&,&F'F&F&F(F&-F*6$)-F.6#F0,&F'F&\"\"#F(F
0F&F&" }}{PARA 0 "" 0 "" {TEXT -1 3 " or" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "                  " }{XPPEDIT 18 
0 "int(cos(x)^N,x) = 1/N;" "6#/-%$intG6$)-%$cosG6#%\"xG%\"NGF+*&\"\"\"
F.F,!\"\"" }{TEXT -1 2 "  " }{XPPEDIT 18 0 "sin(x)*cos(x)^(N-1)+``;" "
6#,&*&-%$sinG6#%\"xG\"\"\")-%$cosG6#F(,&%\"NGF)F)!\"\"F)F)%!GF)" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "(N-1)/N;" "6#*&,&%\"NG\"\"\"F&!\"\"F&F%
F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "int(cos(x)^(N-2),x);" "6#-%$intG6$
)-%$cosG6#%\"xG,&%\"NG\"\"\"\"\"#!\"\"F*" }{TEXT -1 2 " ." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 
3 "" 0 "" {TEXT -1 17 "9. Other Examples" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT 313 31 "Example (Exercise 39, Page 515)" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Calcula
te    " }{XPPEDIT 18 0 "int(x/sqrt(x+3),x = -1 .. 1);" "6#-%$intG6$*&%
\"xG\"\"\"-%%sqrtG6#,&F'F(\"\"$F(!\"\"/F';,$F(F.F(" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 315 10 "Solution
 1" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 34 "J := Int(x/sqrt(x+3),x = -1 .. 1);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%$IntG6$*&%\"xG\"\"\",&F)F*\"\"$F*#!
\"\"\"\"#/F);F.F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "K := s
tudent[intparts](J, x); #This carries out integration by parts with u \+
= x" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG,(*&\"\"#\"\"\"\"\"%#F(F'
F(*&F'F(F'F*F(-%$IntG6$,$*&F'F(,&%\"xGF(\"\"$F(F*F(/F2;!\"\"F(F6" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(K);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,(*&\"\"#\"\"\"\"\"%#F&F%F&*(\"#9F&\"\"$!\"\"F%F(F&#
\"#KF+F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT 314 10 "Solution 2" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "J := Int(x/s
qrt(x+3),x = -1 .. 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%$Int
G6$*&%\"xG\"\"\",&F)F*\"\"$F*#!\"\"\"\"#/F);F.F*" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 40 "K := student[changevar](y = x+3, J, y); " }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG-%$IntG6$*&,&\"\"$!\"\"%\"yG\"\"
\"F-F,#F+\"\"#/F,;F/\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
114 "Int(-3*y^(-1/2), y = 2 .. 4) + Int(y^(1/2), y = 2 .. 4) = int(-3*
y^(-1/2), y = 2 .. 4) + int(y^(1/2), y = 2 .. 4);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/,&-%$IntG6$,$*&\"\"$\"\"\"%\"yG#!\"\"\"\"#F./F,;F/\"\"
%F+-F&6$*$F,#F+F/F0F+,&#\"#?F*F.*(\"#9F+F*F.F/F6F+" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
320 7 "Example" }}{PARA 0 "" 0 "" {TEXT -1 14 "Suppose that  " }
{XPPEDIT 18 0 "a <> b;" "6#0%\"aG%\"bG" }{TEXT -1 16 ".    Evaluate   \+
" }{XPPEDIT 18 0 "int(cos(a*x)*sin(b*x),x);" "6#-%$intG6$*&-%$cosG6#*&
%\"aG\"\"\"%\"xGF,F,-%$sinG6#*&%\"bGF,F-F,F,F-" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 321 8 "Solution
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "J := Int(cos(a*x)*sin(b*
x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%$IntG6$*&-%$cosG6#*&
%\"aG\"\"\"%\"xGF.F.-%$sinG6#*&%\"bGF.F/F.F.F/" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 67 "J = student[intparts](J, cos(a*x)); #int by pa
rts with u = cos(a*x)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&-
%$cosG6#*&%\"aG\"\"\"%\"xGF-F--%$sinG6#*&%\"bGF-F.F-F-F.,&*(F(F-F3!\"
\"-F)F1F-F6-F%6$**-F0F*F-F,F-F3F6F7F-F.F6" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 80 "A := -cos(a*x)/b*cos(b*x);\nK := Int(sin(a*x)*cos(b
*x),x);\neqn := J = A -a/b*K;  " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"AG,$*(-%$cosG6#*&%\"aG\"\"\"%\"xGF,F,%\"bG!\"\"-F(6#*&F.F,F-F,F,F/" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KG-%$IntG6$*&-%$sinG6#*&%\"aG\"
\"\"%\"xGF.F.-%$cosG6#*&%\"bGF.F/F.F.F/" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%$eqnG/-%$IntG6$*&-%$cosG6#*&%\"aG\"\"\"%\"xGF/F/-%$sinG6#*&%\"
bGF/F0F/F/F0,&*(F*F/F5!\"\"-F+F3F/F8,$-F'6$*&-F2F,F/F9F/F0*&F.F/F5F8F8
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
75 "eqn2 := K = student[intparts](K, sin(a*x)); #int by parts with u =
 sin(a*x)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/-%$IntG6$*&-%$si
nG6#*&%\"aG\"\"\"%\"xGF/F/-%$cosG6#*&%\"bGF/F0F/F/F0,&*(F*F/-F+F3F/F5!
\"\"F/-F'6$**-F2F,F/F.F/F8F/F5F9F0F9" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 46 "B := sin(a*x)*sin(b*x)/b;\neqn3 := K = B-a/b*J;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG*(-%$sinG6#*&%\"aG\"\"\"%\"xGF+F
+-F'6#*&%\"bGF+F,F+F+F0!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn
3G/-%$IntG6$*&-%$sinG6#*&%\"aG\"\"\"%\"xGF/F/-%$cosG6#*&%\"bGF/F0F/F/F
0,&*(F*F/-F+F3F/F5!\"\"F/,$-F'6$*&-F2F,F/F8F/F0*&F.F/F5F9F9" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "eqn4 := subs(eqn3, eqn);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn4G/-%$IntG6$*&-%$cosG6#*&%\"aG\"
\"\"%\"xGF/F/-%$sinG6#*&%\"bGF/F0F/F/F0,&*(F*F/F5!\"\"-F+F3F/F8*(F.F/F
5F8,&*(-F2F,F/F1F/F5F8F/,$F&*&F.F/F5F8F8F/F8" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 50 "eqn5 := J = solve(eqn4, Int(cos(a*x)*sin(b*x),x)
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn5G/-%$IntG6$*&-%$cosG6#*&%
\"aG\"\"\"%\"xGF/F/-%$sinG6#*&%\"bGF/F0F/F/F0*&,&*(F*F/-F+F3F/F5F/F/*(
F.F/-F2F,F/F1F/F/F/,&*$)F5\"\"#F/!\"\"*$)F.F?F/F/F@" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 46 "testeq(diff(rhs(eqn5),x) = cos(a*x)*sin(b*x));" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 32 "Copyright and Author Informa
tion" }}{PARA 0 "" 0 "" {TEXT -1 100 "\nWorksheet Title: BlankKrantz-0
7_1R8.mws     A Maple  Release 8 worksheet.\n\nAuthor:  Brian E. Blank
 " }}{PARA 0 "" 0 "" {TEXT -1 30 "Date Created:  26 January 2000" }}
{PARA 0 "" 0 "" {TEXT -1 485 "Date Last Revised: 29 August 2007\n\nThi
s document may not be distributed by any medium,\nincluding print, dis
k, and electronic transfer, without\nprior written permission of the a
uthor.\n\nFor more information,  please contact the author:\n    \n   \+
  Department of Mathematics, \n     Washington University in St. Louis
\n     St. Louis, MO   63130\n   \n     Telephone:    (314) 935-6763\n
             e-mail:    brian@math.wustl.edu\n\nCopyright:  \251 2000-
2007 Brian E. Blank,  All Rights Reserved.\n" }}{PARA 3 "" 0 "" {TEXT 
-1 0 "" }}}}{MARK "5 100 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }