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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 11 "Calculus II" }}{PARA 264 
"" 0 "" {TEXT -1 47 "Lesson 12:  Integration of Trigonometric Powers" 
}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 12 "Introduction" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 131 "One of the most important uses of reduction formulas is to eva
luate anti-derivatives of powers of trigonometric functions, such as \+
" }{XPPEDIT 18 0 "Int(sin(x)^n,x);" "6#-%$IntG6$)-%$sinG6#%\"xG%\"nGF*
" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "Int(sin(x)^n*cos(x)^m,x);" "6#-%$In
tG6$*&)-%$sinG6#%\"xG%\"nG\"\"\")-%$cosG6#F+%\"mGF-F+" }{TEXT -1 9 " o
r even " }{XPPEDIT 18 0 "Int(tan(x)^n*sec(x)^m,x);" "6#-%$IntG6$*&)-%$
tanG6#%\"xG%\"nG\"\"\")-%$secG6#F+%\"mGF-F+" }{TEXT -1 52 " .  In this
 worksheet, we will see how this is done." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "Of course, you already know some s
imple integrals of this type:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 35 "Int(sin(x),x) = int(sin(x), x) + C;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%$IntG6$-%$sinG6#%\"xGF*,&-%$cosGF)!\"\"%\"CG\"\"\"" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Int(cos(x), x) = int(cos(
x), x) + C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-%$cosG6#%\"x
GF*,&-%$sinGF)\"\"\"%\"CGF." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
39 "Int(sec(x)^2,x) = int(sec(x)^2, x) + C;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%$IntG6$*$)-%$secG6#%\"xG\"\"#\"\"\"F,,&*&-%$sinGF+F.
-%$cosGF+!\"\"F.%\"CGF." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "(It is
 hard to get " }{TEXT 256 5 "Maple" }{TEXT -1 10 " to write " }
{XPPEDIT 18 0 "tan(x);" "6#-%$tanG6#%\"xG" }{TEXT -1 79 " in this last
 expression--it seems to prefer to work with sines and cosines.)  " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "Some othe
r integrals involving low powers can be done quickly with a substituti
on:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "p1 := Int(sin(x)^5 * cos(x),
 x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p1G-%$IntG6$*&)-%$sinG6#%\"
xG\"\"&\"\"\"-%$cosGF,F/F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
33 "p2 := changevar(u=sin(x), p1, u);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%#p2G-%$IntG6$*$)%\"uG\"\"&\"\"\"F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "p3 := value(p2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%#p3G,$*$)%\"uG\"\"'\"\"\"#F*F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 25 "p4 := subs(u=sin(x), p3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%#p4G,$*$)-%$sinG6#%\"xG\"\"'\"\"\"#F-F," }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "p1 = p4 + C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$I
ntG6$*&)-%$sinG6#%\"xG\"\"&\"\"\"-%$cosGF+F.F,,&*$)F)\"\"'F.#F.F4%\"CG
F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "(Of course, we should check
 this answer by differentiating it.)  Integrals of the form " }
{XPPEDIT 18 0 "Int(cos(x)^n*sin(x),x);" "6#-%$IntG6$*&)-%$cosG6#%\"xG%
\"nG\"\"\"-%$sinG6#F+F-F+" }{TEXT -1 40 " can be done similarly, by su
bstituting " }{XPPEDIT 18 0 "u = cos(x);" "6#/%\"uG-%$cosG6#%\"xG" }
{TEXT -1 46 ".  On the other hand, since the derivative of " }
{XPPEDIT 18 0 "tan(x);" "6#-%$tanG6#%\"xG" }{TEXT -1 4 " is " }
{XPPEDIT 18 0 "sec(x)^2;" "6#*$-%$secG6#%\"xG\"\"#" }{TEXT -1 69 ", fo
r a tangent substitution to work we need an integral of the form " }
{XPPEDIT 18 0 "Int(tan(x)^n*sec(x)^2,x);" "6#-%$IntG6$*&)-%$tanG6#%\"x
G%\"nG\"\"\"*$-%$secG6#F+\"\"#F-F+" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 214 "Since we know, or know
 how to find, anti-derivatives of certain small powers of trigonometri
c functions, it is natural to ask whether examples involving higher po
wers can be turned into these by reduction formulas." }}}{SECT 0 
{PARA 4 "" 0 "" {TEXT -1 10 "Question 1" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 74 "Find the following anti-derivatives by performing a suita
ble substitution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT 268 10 "Solutions." }{TEXT -1 31 "  (Be sure you \+
have loaded the " }{TEXT 269 7 "student" }{TEXT -1 43 " package before
 executing these solutions.)" }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }
{XPPEDIT 18 0 "Int(cos(x)^3*sin(x),x);" "6#-%$IntG6$*&-%$cosG6#%\"xG\"
\"$-%$sinG6#F*\"\"\"F*" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "p1 := Int( cos(x)^3 * sin(x), x);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#p1G-%$IntG6$*&)-%$cosG6#%\"xG\"\"$\"\"\"-%$sinGF,F
/F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "p2 := changevar(u=co
s(x), p1, u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p2G-%$IntG6$,$*$)%
\"uG\"\"$\"\"\"!\"\"F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "p
3 := simplify(p2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p3G,$-%$IntG6
$*$)%\"uG\"\"$\"\"\"F+!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
16 "p4 := value(p3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p4G,$*$)%\"
uG\"\"%\"\"\"#!\"\"F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "p5
 := subs(u=cos(x), p4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p5G,$*$)
-%$cosG6#%\"xG\"\"%\"\"\"#!\"\"F," }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "p1 = p5 + C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$I
ntG6$*&)-%$cosG6#%\"xG\"\"$\"\"\"-%$sinGF+F.F,,&*$)F)\"\"%F.#!\"\"F4%
\"CGF." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 4 "(b) " }{XPPEDIT 18 0 "Int(cos(x)^9*sin(x),x);" "
6#-%$IntG6$*&-%$cosG6#%\"xG\"\"*-%$sinG6#F*\"\"\"F*" }{TEXT -1 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "p1 := Int( cos(x)^9 * sin(x
), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p1G-%$IntG6$*&)-%$cosG6#%
\"xG\"\"*\"\"\"-%$sinGF,F/F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 33 "p2 := changevar(u=cos(x), p1, u);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#p2G-%$IntG6$,$*$)%\"uG\"\"*\"\"\"!\"\"F+" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 19 "p3 := simplify(p2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#p3G,$-%$IntG6$*$)%\"uG\"\"*\"\"\"F+!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "p4 := value(p3);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#p4G,$*$)%\"uG\"#5\"\"\"#!\"\"F)" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 25 "p5 := subs(u=cos(x), p4);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#p5G,$*$)-%$cosG6#%\"xG\"#5\"\"\"#!\"\"F," }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "p1 = p5 + C;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/-%$IntG6$*&)-%$cosG6#%\"xG\"\"*\"\"\"-%$sinGF+F.F
,,&*$)F)\"#5F.#!\"\"F4%\"CGF." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "(c) " }{XPPEDIT 18 0 "Int
(tan(x)^4*sec(x)^2,x);" "6#-%$IntG6$*&-%$tanG6#%\"xG\"\"%-%$secG6#F*\"
\"#F*" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "p1 \+
:= Int( tan(x)^4 * sec(x)^2 , x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%#p1G-%$IntG6$*&)-%$tanG6#%\"xG\"\"%\"\"\")-%$secGF,\"\"#F/F-" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "p2 := changevar( u=tan(x), p
1, u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p2G-%$IntG6$*$)%\"uG\"\"%
\"\"\"F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "p3 := value(p2)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p3G,$*$)%\"uG\"\"&\"\"\"#F*F)
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "p4 := subs( u=tan(x), p
3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p4G,$*$)-%$tanG6#%\"xG\"\"&
\"\"\"#F-F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "p1 = p4 + C;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&)-%$tanG6#%\"xG\"\"%\"
\"\")-%$secGF+\"\"#F.F,,&*$)F)\"\"&F.#F.F6%\"CGF." }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "Question 2" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 77 "Find each of the following anti-derivativ
es by using a suitable substitution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 270 10 "Solutions." }{TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }{XPPEDIT 18 0 "Int(sin(x)^n*
cos(x),x);" "6#-%$IntG6$*&)-%$sinG6#%\"xG%\"nG\"\"\"-%$cosG6#F+F-F+" }
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "p1 := Int( \+
sin(x)^n * cos(x), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p1G-%$Int
G6$*&)-%$sinG6#%\"xG%\"nG\"\"\"-%$cosGF,F/F-" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 34 "p2 := changevar( u=sin(x), p1, u);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#p2G-%$IntG6$)%\"uG%\"nGF)" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 16 "p3 := value(p2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#p3G*&)%\"uG,&%\"nG\"\"\"F*F*F*F(!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "p4 := subs( u=sin(x), p3);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p4G*&)-%$sinG6#%\"xG,&%\"nG\"\"\"F-
F-F-F+!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "p1 = p4 + C;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&)-%$sinG6#%\"xG%\"nG\"
\"\"-%$cosGF+F.F,,&*&)F),&F-F.F.F.F.F4!\"\"F.%\"CGF." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "(b
) " }{XPPEDIT 18 0 "Int(cos(x)^n*sin(x),x);" "6#-%$IntG6$*&)-%$cosG6#%
\"xG%\"nG\"\"\"-%$sinG6#F+F-F+" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 32 "p1 := Int(cos(x)^n * sin(x), x);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#p1G-%$IntG6$*&)-%$cosG6#%\"xG%\"nG\"\"\"-%$sinG
F,F/F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "p2 := changevar( \+
u=cos(x), p1, u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p2G-%$IntG6$,$
)%\"uG%\"nG!\"\"F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "p3 :=
 simplify(p2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p3G,$-%$IntG6$)%
\"uG%\"nGF*!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "p4 := v
alue(p3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p4G,$*&)%\"uG,&%\"nG\"
\"\"F+F+F+F)!\"\"F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "p5 :
= subs( u=cos(x), p4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p5G,$*&)-
%$cosG6#%\"xG,&%\"nG\"\"\"F.F.F.F,!\"\"F/" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 12 "p1 = p5 + C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%
$IntG6$*&)-%$cosG6#%\"xG%\"nG\"\"\"-%$sinGF+F.F,,&*&)F),&F-F.F.F.F.F4!
\"\"F5%\"CGF." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "(c) " }{XPPEDIT 18 0 "Int(tan(x)^n*
sec(x)^2,x);" "6#-%$IntG6$*&)-%$tanG6#%\"xG%\"nG\"\"\"*$-%$secG6#F+\"
\"#F-F+" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "p
1 := Int( tan(x)^n * sec(x)^2 , x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%#p1G-%$IntG6$*&)-%$tanG6#%\"xG%\"nG\"\"\")-%$secGF,\"\"#F/F-" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "p2 := changevar( u=tan(x), p
1, u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p2G-%$IntG6$)%\"uG%\"nGF)
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "p3 := value(p2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p3G*&)%\"uG,&%\"nG\"\"\"F*F*F*F(!\"
\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "p4 := subs( u=tan(x),
 p3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p4G*&)-%$tanG6#%\"xG,&%\"n
G\"\"\"F-F-F-F+!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "p1 \+
= p4 + C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&)-%$tanG6#%\"
xG%\"nG\"\"\")-%$secGF+\"\"#F.F,,&*&)F),&F-F.F.F.F.F6!\"\"F.%\"CGF." }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 4 "(d) " }{XPPEDIT 18 0 "Int(cot(x)^n*csc(x)^2,x);" "6#-%$I
ntG6$*&)-%$cotG6#%\"xG%\"nG\"\"\"*$-%$cscG6#F+\"\"#F-F+" }{TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "p1 := Int( cot(x)^n * c
sc(x)^2 , x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p1G-%$IntG6$*&)-%$
cotG6#%\"xG%\"nG\"\"\")-%$cscGF,\"\"#F/F-" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 34 "p2 := changevar( u=cot(x), p1, u);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#p2G-%$IntG6$,$)%\"uG%\"nG!\"\"F*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "p3 := simplify(p2);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#p3G,$-%$IntG6$)%\"uG%\"nGF*!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "p4 := value(p3);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#p4G,$*&)%\"uG,&%\"nG\"\"\"F+F+F+F)!\"\"F," }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "p5 := subs( u=cot(x), p4);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p5G,$*&)-%$cotG6#%\"xG,&%\"nG\"\"
\"F.F.F.F,!\"\"F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "p1 = p
5 + C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&)-%$cotG6#%\"xG%
\"nG\"\"\")-%$cscGF+\"\"#F.F,,&*&)F),&F-F.F.F.F.F6!\"\"F7%\"CGF." }}}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "
" {TEXT -1 22 "Integrals of the form " }{XPPEDIT 18 0 "Int(sin(x)^n*co
s(x)^m,x);" "6#-%$IntG6$*&)-%$sinG6#%\"xG%\"nG\"\"\")-%$cosG6#F+%\"mGF
-F+" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student):" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "We will begin with a special case:
 " }{XPPEDIT 18 0 "m = 0;" "6#/%\"mG\"\"!" }{TEXT -1 23 ".  Our integr
al is then" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "p1 := Int(sin
(x)^n , x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p1G-%$IntG6$)-%$sinG
6#%\"xG%\"nGF," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "There is no pro
blem if " }{XPPEDIT 18 0 "n = 0;" "6#/%\"nG\"\"!" }{TEXT -1 4 " or " }
{XPPEDIT 18 0 "n = 1;" "6#/%\"nG\"\"\"" }{TEXT -1 19 ", so we can assu
me " }{XPPEDIT 18 0 "2 <= n;" "6#1\"\"#%\"nG" }{TEXT -1 73 ".  In this
 case, as discussed in class, the relevant reduction formula is" }}
{PARA 256 "" 0 "" {XPPEDIT 18 0 "Int(sin(x)^n,x);" "6#-%$IntG6$)-%$sin
G6#%\"xG%\"nGF*" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "-1/n;" "6#,$*&\"\"
\"F%%\"nG!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sin(x)^(n-1)*cos(x);
" "6#*&)-%$sinG6#%\"xG,&%\"nG\"\"\"F+!\"\"F+-%$cosG6#F(F+" }{TEXT -1 
5 "  +  " }{XPPEDIT 18 0 "(n-1)/n;" "6#*&,&%\"nG\"\"\"F&!\"\"F&F%F'" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(sin(x)^(n-2),x);" "6#-%$IntG6$)-%$s
inG6#%\"xG,&%\"nG\"\"\"\"\"#!\"\"F*" }{TEXT -1 2 " ." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 36 "
Derivation of the formula (optional)" }}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 41 "The trick is to separate off a factor of " }{XPPEDIT 18 0 "sin(
x)^2;" "6#*$-%$sinG6#%\"xG\"\"#" }{TEXT -1 18 ", and write it as " }
{XPPEDIT 18 0 "1-cos(x)^2;" "6#,&\"\"\"F$*$-%$cosG6#%\"xG\"\"#!\"\"" }
{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "p2 := Int(
 sin(x)^2*sin(x)^(n-2), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p2G-
%$IntG6$*&)-%$sinG6#%\"xG\"\"#\"\"\")F*,&%\"nGF/F.!\"\"F/F-" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "p3 := subs(sin(x)^2 = (1-cos
(x)^2), p2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p3G-%$IntG6$*&,&\"
\"\"F**$)-%$cosG6#%\"xG\"\"#F*!\"\"F*)-%$sinGF/,&%\"nGF*F1F2F*F0" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "p4 := expand(p3, n-2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p4G,&-%$IntG6$)-%$sinG6#%\"xG,&%\"n
G\"\"\"\"\"#!\"\"F-F0-F'6$*&F)F0)-%$cosGF,F1F0F-F2" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 23 "(The optional argument " }{TEXT 256 3 "n-2" }
{TEXT -1 8 " in the " }{TEXT 257 6 "expand" }{TEXT -1 18 " command pre
vents " }{TEXT 258 5 "Maple" }{TEXT -1 29 " from expanding the exponen
t " }{XPPEDIT 18 0 "n-2;" "6#,&%\"nG\"\"\"\"\"#!\"\"" }{TEXT -1 73 ".)
  We now want to integrate the second integral by parts, writing it as
 " }{XPPEDIT 18 0 "Int(sin(x)^(n-2)*cos(x)*cos(x),x);" "6#-%$IntG6$*()
-%$sinG6#%\"xG,&%\"nG\"\"\"\"\"#!\"\"F.-%$cosG6#F+F.-F26#F+F.F+" }
{TEXT -1 43 ", and differentiating the second factor of " }{XPPEDIT 
18 0 "cos(x);" "6#-%$cosG6#%\"xG" }{TEXT -1 53 ".  This requires us to
 anti-differentiate the factor " }{XPPEDIT 18 0 "sin(x)^(n-2)*cos(x);
" "6#*&)-%$sinG6#%\"xG,&%\"nG\"\"\"\"\"#!\"\"F+-%$cosG6#F(F+" }{TEXT 
-1 52 " , but we saw how to do this earlier.  When getting " }{TEXT 
264 5 "Maple" }{TEXT -1 66 " to do this calculation, it is convenient \+
to use a command called " }{TEXT 259 2 "op" }{TEXT -1 13 " .  The name
 " }{TEXT 261 2 "op" }{TEXT -1 15 "  is short for " }{TEXT 260 7 "oper
and" }{TEXT -1 10 ", and the " }{TEXT 262 2 "op" }{TEXT -1 324 " comma
nd refers to a particular operand, or piece, of an expression.  For ex
ample, in the expression p4 you see two integrals: two operands.  (Eac
h of these is itself built up from lower-level pieces, but that need n
ot concern us here.)  Each of these integrals can be referred to by pi
cking out the correct operand from p4:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "op(1, p4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG
6$)-%$sinG6#%\"xG,&%\"nG\"\"\"\"\"#!\"\"F*" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 10 "op(2, p4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%$
IntG6$*&)-%$sinG6#%\"xG,&%\"nG\"\"\"\"\"#!\"\"F/)-%$cosGF+F0F/F,F1" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "With the " }{TEXT 263 2 "op" }
{TEXT -1 68 " command, we can integrate just one of these two integral
s by parts:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "p5 := op(1, \+
p4) + intparts(op(2, p4), cos(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%#p5G,(-%$IntG6$)-%$sinG6#%\"xG,&%\"nG\"\"\"\"\"#!\"\"F-F0*&*&-%$cosG
F,F0)F*,&F/F0F0F2F0F0F8F2F2-F'6$,$*&*&F*F0F7F0F0F8F2F2F-F0" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "p6 := op(1, p5) + op(2, p5) + simpl
ify(op(3, p5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p6G,(-%$IntG6$)-
%$sinG6#%\"xG,&%\"nG\"\"\"\"\"#!\"\"F-F0*&*&-%$cosGF,F0)F*,&F/F0F0F2F0
F0F8F2F2*&-F'6$)F*F/F-F0F8F2F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
331 "We are almost done.  This last expression, p6, is equal to our or
iginal integral p1.  The first term is what we want: an integral like \+
p1, but with a lower power.  Unfortunately, the original integral p1 h
as shown up again in the last term.  We can get around this problem, t
hough, by solving the resulting equation for p1.  Since " }{TEXT 265 
5 "Maple" }{TEXT -1 193 " thinks that p1 already has a value, it will \+
probably refuse to solve an equation for it, so we first replace p1 in
 the equation by an unassigned expression, and then solve for that exp
ression." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "eqn1 := p1 = p6
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/-%$IntG6$)-%$sinG6#%\"xG
%\"nGF-,(-F'6$)F*,&F.\"\"\"\"\"#!\"\"F-F4*&*&-%$cosGF,F4)F*,&F.F4F4F6F
4F4F<F6F6*&F&F4F<F6F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "eq
n2 := subs(p1=a, eqn1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/%
\"aG,(-%$IntG6$)-%$sinG6#%\"xG,&%\"nG\"\"\"\"\"#!\"\"F/F2*&*&-%$cosGF.
F2)F,,&F1F2F2F4F2F2F:F4F4*&F&F2F:F4F4" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "p7 := solve(eqn2, a);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#p7G*&,(*&-%$IntG6$)-%$sinG6#%\"xG,&%\"nG\"\"\"\"\"#!\"\"F/F2F
1F2F2F(F4*&-%$cosGF.F2)F,,&F1F2F2F4F2F4F2F1F4" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 12 "p1 = p7 + C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
/-%$IntG6$)-%$sinG6#%\"xG%\"nGF+,&*&,(*&-F%6$)F(,&F,\"\"\"\"\"#!\"\"F+
F5F,F5F5F1F7*&-%$cosGF*F5)F(,&F,F5F5F7F5F7F5F,F7F5%\"CGF5" }}}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 
-1 10 "Question 3" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Use the reduc
tion formula for " }{XPPEDIT 18 0 "Int(sin(x)^n,x);" "6#-%$IntG6$)-%$s
inG6#%\"xG%\"nGF*" }{TEXT -1 30 " to find anti-derivatives for " }
{XPPEDIT 18 0 "Int(sin(x)^5,x);" "6#-%$IntG6$*$-%$sinG6#%\"xG\"\"&F*" 
}{TEXT -1 3 " , " }{XPPEDIT 18 0 "Int(sin(x)^4,x);" "6#-%$IntG6$*$-%$s
inG6#%\"xG\"\"%F*" }{TEXT -1 4 " and" }{XPPEDIT 18 0 "Int(sin(x)^19,x)
;" "6#-%$IntG6$*$-%$sinG6#%\"xG\"#>F*" }{TEXT -1 2 " ." }}{PARA 0 "" 
0 "" {TEXT -1 77 "You can do this painfully by hand, or you could try \+
to be clever and write a " }{TEXT 266 5 "Maple" }{TEXT -1 109 " proced
ure that works out the formula.  For example, you could write a proced
ure whose input is the exponent " }{XPPEDIT 18 0 "n;" "6#%\"nG" }
{TEXT -1 88 " and whose output is the right-hand side of the reduction
 formula.  Even better: with a " }{TEXT 267 3 "for" }{TEXT -1 72 " loo
p, or a recursive program, you could write a procedure that accepts " 
}{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 48 " as input and evaluates th
e integral completely." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }{TEXT 271 10 "Solutions." }{TEXT -1 82 "  We will \+
do the first two anti-derivatives by writing a procedure whose input i
s " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 79 ", and whose output is \+
the right-hand side of the formula.  Note the use of the " }{TEXT 272 
2 "op" }{TEXT -1 101 " command (explained in the optional section abov
e) to separate off terms in the intermediate answers." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "sinred := proc(n) (-1/n)*sin(x)^(n-
1) * cos(x)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "            + (n-1)/
n * Int( sin(x)^(n-2), x);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "     \+
     end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Here is a solution f
or " }{XPPEDIT 18 0 "Int(sin(x)^5,x);" "6#-%$IntG6$*$-%$sinG6#%\"xG\"
\"&F*" }{TEXT -1 2 " :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "p
1 := Int( sin(x)^5 , x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p1G-%$I
ntG6$*$)-%$sinG6#%\"xG\"\"&\"\"\"F-" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "p2 := sinred(5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%#p2G,&*&)-%$sinG6#%\"xG\"\"%\"\"\"-%$cosGF*F-#!\"\"\"\"&*&#F,F2F--%$I
ntG6$*$)F(\"\"$F-F+F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "
p3 := op(1, p2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p3G,$*&)-%$sinG
6#%\"xG\"\"%\"\"\"-%$cosGF*F-#!\"\"\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "p4 := (4/5)*sinred(3);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#p4G,&*&)-%$sinG6#%\"xG\"\"#\"\"\"-%$cosGF*F-#!\"%\"#:*&#\"\")
F2F--%$IntG6$F(F+F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "p5
 := p3 + value(p4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p5G,(*&)-%$s
inG6#%\"xG\"\"%\"\"\"-%$cosGF*F-#!\"\"\"\"&*&#F,\"#:F-*&)F(\"\"#F-F.F-
F-F1*&#\"\")F5F-F.F-F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "p
1 = p5 + C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*$)-%$sinG6#%
\"xG\"\"&\"\"\"F,,**&)F)\"\"%F.-%$cosGF+F.#!\"\"F-*&#F2\"#:F.*&)F)\"\"
#F.F3F.F.F6*&#\"\")F9F.F3F.F6%\"CGF." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 8 "Here is " }{XPPEDIT 18 0 "Int(sin(x)^4,x);" "6#-%$IntG6$*$-%$sin
G6#%\"xG\"\"%F*" }{TEXT -1 34 " , organised slightly differently:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "q1 := Int( sin(x)^4 , x);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q1G-%$IntG6$*$)-%$sinG6#%\"xG\"\"%
\"\"\"F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "q2 := sinred(4)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q2G,&*&)-%$sinG6#%\"xG\"\"$\"
\"\"-%$cosGF*F-#!\"\"\"\"%*&#F,F2F--%$IntG6$*$)F(\"\"#F-F+F-F-" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "q3 := (3/4)*sinred(2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q3G,&*&-%$sinG6#%\"xG\"\"\"-%$cosGF
)F+#!\"$\"\")*&#\"\"$F0F+-%$IntG6$F+F*F+F+" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 16 "q4 := value(q3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%#q4G,&*&-%$sinG6#%\"xG\"\"\"-%$cosGF)F+#!\"$\"\")*&#\"\"$F0F+F*F+F+
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "q5 := op(1, q2) + q4;" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q5G,(*&)-%$sinG6#%\"xG\"\"$\"\"\"
-%$cosGF*F-#!\"\"\"\"%*&#F,\"\")F-*&F(F-F.F-F-F1*&#F,F5F-F+F-F-" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "q1 = q5 + C;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/-%$IntG6$*$)-%$sinG6#%\"xG\"\"%\"\"\"F,,**&)F)\"
\"$F.-%$cosGF+F.#!\"\"F-*&#F2\"\")F.*&F)F.F3F.F.F6*&#F2F9F.F,F.F.%\"CG
F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "The anti-derivative " }
{XPPEDIT 18 0 "Int(sin(x)^19,x);" "6#-%$IntG6$*$-%$sinG6#%\"xG\"#>F*" 
}{TEXT -1 85 " could be done the same way, but it would obviously be l
ong.  Instead, let's write a " }{TEXT 273 9 "recursive" }{TEXT -1 246 
" procedure ( a procedure which calls itself repeatedly) to do the red
uction formula as many times as needed.  The procedure below is very b
asic: a more sophisticated version would have checks to make sure that
 you didn't try to use it with (say) " }{XPPEDIT 18 0 "n = 1/2;" "6#/%
\"nG*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "n = -3;" 
"6#/%\"nG,$\"\"$!\"\"" }{TEXT -1 37 ".  The program checks to see whet
her " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 111 " is at least 2; if \+
so, it applies the reduction formula, which involves calling itself wi
th a smaller value of " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 10 "; \+
if not, " }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 63 " is presumably e
ither 0 or 1 and the integral can be evaluated." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 22 "sinpowerint := proc(n)" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 88 "                 if(n >= 2) then (-1/n)*sin(x)^(n-1) \+
* cos(x)                           " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
51 "                   + (n-1)/n * sinpowerint(n-2);   " }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 46 "                 else value(Int(sin(x)^n, x));
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "                 fi;" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 19 "               end:" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 82 "We can quickly check our new procedure by evaluati
ng the two cases we did earlier:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "sinpowerint(5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(
*&)-%$sinG6#%\"xG\"\"%\"\"\"-%$cosGF(F+#!\"\"\"\"&*&#F*\"#:F+*&)F&\"\"
#F+F,F+F+F/*&#\"\")F3F+F,F+F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 15 "sinpowerint(4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&)-%$sinG6
#%\"xG\"\"$\"\"\"-%$cosGF(F+#!\"\"\"\"%*&#F*\"\")F+*&F&F+F,F+F+F/*&#F*
F3F+F)F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "These answers agree
 with what we got earlier, so we can go for the big kahuna:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Int( sin(x)^19, x) = sinpowe
rint(19) + C;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*$)-%$sinG6
#%\"xG\"#>\"\"\"F,,8*&)F)\"#=F.-%$cosGF+F.#!\"\"F-*&#F2\"$B$F.*&)F)\"#
;F.F3F.F.F6*&#\"#'*\"%:;F.*&)F)\"#9F.F3F.F.F6*&#\"%W8\"&&*4#F.*&)F)\"#
7F.F3F.F.F6*&#\"&Gh\"\"'X4BF.*&)F)\"#5F.F3F.F.F6*&#\"%%e$\"&*=YF.*&)F)
\"\")F.F3F.F.F6*&#\"%'4%FUF.*&)F)\"\"'F.F3F.F.F6*&#\"&wX#FNF.*&)F)\"\"
%F.F3F.F.F6*&#\"&oF$FNF.*&)F)\"\"#F.F3F.F.F6*&#\"&Ob'FNF.F3F.F6%\"CGF.
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Congratulations: you have jus
t re-written a small part of " }{TEXT 274 5 "Maple" }{TEXT -1 3 "'s " 
}{TEXT 275 3 "int" }{TEXT -1 9 " command." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "The next special case is the integ
ral " }{XPPEDIT 18 0 "Int(cos(x)^m,x);" "6#-%$IntG6$)-%$cosG6#%\"xG%\"
mGF*" }{TEXT -1 64 " .  This type of integral can be done with the red
uction formula" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "Int(cos(x)^m,x) = 1
/m;" "6#/-%$IntG6$)-%$cosG6#%\"xG%\"mGF+*&\"\"\"F.F,!\"\"" }{TEXT -1 
1 " " }{XPPEDIT 18 0 "cos(x)^(m-1)*sin(x);" "6#*&)-%$cosG6#%\"xG,&%\"m
G\"\"\"F+!\"\"F+-%$sinG6#F(F+" }{TEXT -1 5 "  +  " }{XPPEDIT 18 0 "(m-
1)/m;" "6#*&,&%\"mG\"\"\"F&!\"\"F&F%F'" }{TEXT -1 1 " " }{XPPEDIT 18 
0 "Int(cos(x)^(m-2),x);" "6#-%$IntG6$)-%$cosG6#%\"xG,&%\"mG\"\"\"\"\"#
!\"\"F*" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 54 "Finally, for the general case, there are two formu
las:" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "Int(sin(x)^n*cos(x)^m,x);" "6
#-%$IntG6$*&)-%$sinG6#%\"xG%\"nG\"\"\")-%$cosG6#F+%\"mGF-F+" }{TEXT 
-1 3 " = " }{XPPEDIT 18 0 "-sin(x)^(n-1)*cos(x)^(m+1)/(n+m);" "6#,$*()
-%$sinG6#%\"xG,&%\"nG\"\"\"F,!\"\"F,)-%$cosG6#F),&%\"mGF,F,F,F,,&F+F,F
3F,F-F-" }{TEXT -1 5 "  +  " }{XPPEDIT 18 0 "(n-1)/(n+m);" "6#*&,&%\"n
G\"\"\"F&!\"\"F&,&F%F&%\"mGF&F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(s
in(x)^(n-2)*cos(x)^m,x);" "6#-%$IntG6$*&)-%$sinG6#%\"xG,&%\"nG\"\"\"\"
\"#!\"\"F.)-%$cosG6#F+%\"mGF.F+" }{TEXT -1 2 " ," }}{PARA 0 "" 0 "" 
{TEXT -1 3 "and" }}{PARA 259 "" 0 "" {XPPEDIT 18 0 "Int(sin(x)^n*cos(x
)^m,x);" "6#-%$IntG6$*&)-%$sinG6#%\"xG%\"nG\"\"\")-%$cosG6#F+%\"mGF-F+
" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "sin(x)^(n+1)*cos(x)^(m-1)/(n+m);" 
"6#*()-%$sinG6#%\"xG,&%\"nG\"\"\"F+F+F+)-%$cosG6#F(,&%\"mGF+F+!\"\"F+,
&F*F+F1F+F2" }{TEXT -1 5 "  +  " }{XPPEDIT 18 0 "(m-1)/(n+m);" "6#*&,&
%\"mG\"\"\"F&!\"\"F&,&%\"nGF&F%F&F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "I
nt(sin(x)^n*cos(x)^(m-2),x);" "6#-%$IntG6$*&)-%$sinG6#%\"xG%\"nG\"\"\"
)-%$cosG6#F+,&%\"mGF-\"\"#!\"\"F-F+" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "
" {TEXT -1 114 "Repeated use of one or other of these will reduce the \+
problem to one of the special cases we have already studied." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 "" 
{TEXT -1 10 "Question 4" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Find " }
{XPPEDIT 18 0 "Int(sin(x)^7*cos(x)^5,x);" "6#-%$IntG6$*&-%$sinG6#%\"xG
\"\"(-%$cosG6#F*\"\"&F*" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "Int(sin(x)^4
*cos(x)^7,x);" "6#-%$IntG6$*&-%$sinG6#%\"xG\"\"%-%$cosG6#F*\"\"(F*" }
{TEXT -1 5 " and " }{XPPEDIT 18 0 "Int(sin(x)^6*cos(x)^4,x);" "6#-%$In
tG6$*&-%$sinG6#%\"xG\"\"'-%$cosG6#F*\"\"%F*" }{TEXT -1 1 "." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 276 10 
"Solutions." }{TEXT -1 223 "  For the first two, we will lower the pow
er of cosine.  In the first example, it probably desn't make much diff
erence which way we go, but the cosine has the lower power; in the sec
ond, it will be much quicker to lower the" }}{PARA 0 "" 0 "" {TEXT -1 
134 "odd power than the even one.  To save typing, we will write a pro
cedure which will lower a cosine power in an integral of this type.  \+
" }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 17 " is the power of " }
{XPPEDIT 18 0 "sin(x);" "6#-%$sinG6#%\"xG" }{TEXT -1 2 "; " }{XPPEDIT 
18 0 "m;" "6#%\"mG" }{TEXT -1 17 " is the power of " }{XPPEDIT 18 0 "c
os(x);" "6#-%$cosG6#%\"xG" }{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 54 "lowercos := proc(n,m) sin(x)^(n+1)*cos(x)^(m-1)/(n+
m) " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "              + (m-1)/(n+m)*
Int(sin(x)^n * cos(x)^(m-2) , x);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
16 "            end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "p1 \+
:= Int(sin(x)^7 * cos(x)^5 , x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
#p1G-%$IntG6$*&)-%$sinG6#%\"xG\"\"(\"\"\")-%$cosGF,\"\"&F/F-" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "p2 := lowercos(7,5);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p2G,&*&)-%$sinG6#%\"xG\"\")\"\"\")-
%$cosGF*\"\"%F-#F-\"#7*&#F-\"\"$F--%$IntG6$*&)F(\"\"(F-)F/F6F-F+F-F-" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "p3 := (1/3)*lowercos(7,3)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p3G,&*&)-%$sinG6#%\"xG\"\")\"
\"\")-%$cosGF*\"\"#F-#F-\"#I*&#F-\"#:F--%$IntG6$*&)F(\"\"(F-F/F-F+F-F-
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "p4 := value(p3);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p4G,&*&)-%$sinG6#%\"xG\"\")\"\"\")-
%$cosGF*\"\"#F-#F-\"#I*&#F-\"$?\"F-F'F-F-" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 23 "p1 = op(1,p2) + p4 + C;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%$IntG6$*&)-%$sinG6#%\"xG\"\"(\"\"\")-%$cosGF+\"\"&F.
F,,**&)F)\"\")F.)F0\"\"%F.#F.\"#7*(#F.\"#IF.F5F.)F0\"\"#F.F.*&#F.\"$?
\"F.F5F.F.%\"CGF." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "(It is inter
esting to try and check this result by differentiation!)" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 155 "Here is the secon
d one.  Notice that by choosing to reduce the odd power, we end up wit
h an easy anti-derivative, and never need to reduce the other power." 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "q1 := Int( sin(x)^4 * cos
(x)^7 , x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q1G-%$IntG6$*&)-%$si
nG6#%\"xG\"\"%\"\"\")-%$cosGF,\"\"(F/F-" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "q2 := lowercos(4,7);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%#q2G,&*&)-%$sinG6#%\"xG\"\"&\"\"\")-%$cosGF*\"\"'F-#F-\"#6*&#F1F3
F--%$IntG6$*&)F(\"\"%F-)F/F,F-F+F-F-" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "q3 := (6/11)*lowercos(4,5);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#q3G,&*&)-%$sinG6#%\"xG\"\"&\"\"\")-%$cosGF*\"\"%F-#
\"\"#\"#L*&#\"\")F4F--%$IntG6$*&)F(F1F-)F/\"\"$F-F+F-F-" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "q4 := (8/33)*lowercos(4,3);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q4G,&*&)-%$sinG6#%\"xG\"\"&\"\"\")-
%$cosGF*\"\"#F-#\"\")\"$J#*&#\"#;F4F--%$IntG6$*&)F(\"\"%F-F/F-F+F-F-" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "The last anti-derivative can be
 done by substituting " }{XPPEDIT 18 0 "u = sin(x);" "6#/%\"uG-%$sinG6
#%\"xG" }{TEXT -1 26 "; we will \"cheat\" and get " }{TEXT 277 5 "Mapl
e" }{TEXT -1 28 " to evaluate it in one step." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 16 "q5 := value(q4);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#q5G,&*&)-%$sinG6#%\"xG\"\"&\"\"\")-%$cosGF*\"\"#F-#\"\")\"$J#
*&#\"#;\"%b6F-F'F-F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Now we ne
ed to put all the pieces together:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 36 "q1 = op(1, q2) + op(1, q3) + q5 + C;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/-%$IntG6$*&)-%$sinG6#%\"xG\"\"%\"\"\")-%$cosGF+\"
\"(F.F,,,*&)F)\"\"&F.)F0\"\"'F.#F.\"#6*(#\"\"#\"#LF.F5F.)F0F-F.F.*(#\"
\")\"$J#F.F5F.)F0F=F.F.*&#\"#;\"%b6F.F5F.F.%\"CGF." }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 377 "For the last example, since both powers are even
, it probably doesn't make much difference which way we go, but it can
't hurt to lower the smaller power.  There is another advantage to doi
ng that here: we already have a procedure which will lower a power of \+
cosine for us, as well as one which will evaluate the integral that re
mains when the powers of cosine have disappeared." }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 35 "r1 := Int( sin(x)^6 * cos(x)^4, x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r1G-%$IntG6$*&)-%$sinG6#%\"xG\"\"'
\"\"\")-%$cosGF,\"\"%F/F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
20 "r2 := lowercos(6,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r2G,&*&
)-%$sinG6#%\"xG\"\"(\"\"\")-%$cosGF*\"\"$F-#F-\"#5*&#F1F3F--%$IntG6$*&
)F(\"\"'F-)F/\"\"#F-F+F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
27 "r3 := (3/10)*lowercos(6,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#
r3G,&*&)-%$sinG6#%\"xG\"\"(\"\"\"-%$cosGF*F-#\"\"$\"#!)*&F0F--%$IntG6$
*$)F(\"\"'F-F+F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "r4 :=
 (3/80)*sinpowerint(6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r4G,**&)
-%$sinG6#%\"xG\"\"&\"\"\"-%$cosGF*F-#!\"\"\"$g\"*&#F-\"$G\"F-*&)F(\"\"
$F-F.F-F-F1*&#F8\"$c#F-*&F(F-F.F-F-F1*&#F8F;F-F+F-F-" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 34 "r1 = op(1,r2) + op(1,r3) + r4 + C;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&)-%$sinG6#%\"xG\"\"'\"\"\"
)-%$cosGF+\"\"%F.F,,0*&)F)\"\"(F.)F0\"\"$F.#F.\"#5*(#F8\"#!)F.F5F.F0F.
F.*&#F.\"$g\"F.*&)F)\"\"&F.F0F.F.!\"\"*&#F.\"$G\"F.*&)F)F8F.F0F.F.FD*&
#F8\"$c#F.*&F)F.F0F.F.FD*&#F8FLF.F,F.F.%\"CGF." }}}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 22 "Inte
grals of the form " }{XPPEDIT 18 0 "Int(tan(x)^n*sec(x)^m,x);" "6#-%$I
ntG6$*&)-%$tanG6#%\"xG%\"nG\"\"\")-%$secG6#F+%\"mGF-F+" }{TEXT -1 0 "
" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "Integrals of this type can be
 handled in a very similar maner to those of the previous section.  Th
e special cases " }{XPPEDIT 18 0 "m = 0;" "6#/%\"mG\"\"!" }{TEXT -1 4 
" or " }{XPPEDIT 18 0 "n = 0;" "6#/%\"nG\"\"!" }{TEXT -1 41 " can be h
andled by the reduction formulas" }}{PARA 260 "" 0 "" {XPPEDIT 18 0 "I
nt(tan(x)^n,x);" "6#-%$IntG6$)-%$tanG6#%\"xG%\"nGF*" }{TEXT -1 5 "  = \+
 " }{XPPEDIT 18 0 "1/(n-1);" "6#*&\"\"\"F$,&%\"nGF$F$!\"\"F'" }{TEXT 
-1 1 " " }{XPPEDIT 18 0 "tan(x)^(n-1);" "6#)-%$tanG6#%\"xG,&%\"nG\"\"
\"F*!\"\"" }{TEXT -1 5 "   - " }{XPPEDIT 18 0 "Int(tan(x)^(n-2),x);" "
6#-%$IntG6$)-%$tanG6#%\"xG,&%\"nG\"\"\"\"\"#!\"\"F*" }{TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 261 "" 0 "" {XPPEDIT 18 0 "In
t(sec(x)^m,x);" "6#-%$IntG6$)-%$secG6#%\"xG%\"mGF*" }{TEXT -1 5 "  =  \+
" }{XPPEDIT 18 0 "1/(m-1);" "6#*&\"\"\"F$,&%\"mGF$F$!\"\"F'" }{TEXT 
-1 1 " " }{XPPEDIT 18 0 "tan(x)*sec(x)^(m-2);" "6#*&-%$tanG6#%\"xG\"\"
\")-%$secG6#F',&%\"mGF(\"\"#!\"\"F(" }{TEXT -1 5 "  +  " }{XPPEDIT 18 
0 "(m-2)/(m-1);" "6#*&,&%\"mG\"\"\"\"\"#!\"\"F&,&F%F&F&F(F(" }{TEXT 
-1 1 " " }{XPPEDIT 18 0 "Int(sec(x)^(m-2),x);" "6#-%$IntG6$)-%$secG6#%
\"xG,&%\"mG\"\"\"\"\"#!\"\"F*" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 82 "respectively.  The general case can then be done with one or ot
her of the formulas" }}{PARA 262 "" 0 "" {XPPEDIT 18 0 "Int(tan(x)^n*s
ec(x)^m,x);" "6#-%$IntG6$*&)-%$tanG6#%\"xG%\"nG\"\"\")-%$secG6#F+%\"mG
F-F+" }{TEXT -1 5 "  =  " }{XPPEDIT 18 0 "1/(n+1);" "6#*&\"\"\"F$,&%\"
nGF$F$F$!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "tan(x)^(n+1)*sec(x)^(m-
2);" "6#*&)-%$tanG6#%\"xG,&%\"nG\"\"\"F+F+F+)-%$secG6#F(,&%\"mGF+\"\"#
!\"\"F+" }{TEXT -1 5 "  -  " }{XPPEDIT 18 0 "(m-2)/(n+1);" "6#*&,&%\"m
G\"\"\"\"\"#!\"\"F&,&%\"nGF&F&F&F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "In
t(tan(x)^(n+2)*sec(x)^(m-2),x);" "6#-%$IntG6$*&)-%$tanG6#%\"xG,&%\"nG
\"\"\"\"\"#F.F.)-%$secG6#F+,&%\"mGF.F/!\"\"F.F+" }{TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 2 "or" }}{PARA 263 "" 0 "" {XPPEDIT 18 0 "Int
(tan(x)^n*sec(x)^m,x);" "6#-%$IntG6$*&)-%$tanG6#%\"xG%\"nG\"\"\")-%$se
cG6#F+%\"mGF-F+" }{TEXT -1 5 "  =  " }{XPPEDIT 18 0 "1/m;" "6#*&\"\"\"
F$%\"mG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "tan(x)^(n-1)*sec(x)^m;" 
"6#*&)-%$tanG6#%\"xG,&%\"nG\"\"\"F+!\"\"F+)-%$secG6#F(%\"mGF+" }{TEXT 
-1 5 "  -  " }{XPPEDIT 18 0 "(n-1)/m;" "6#*&,&%\"nG\"\"\"F&!\"\"F&%\"m
GF'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(tan(x)^(n-2)*sec(x)^(m+2),x);
" "6#-%$IntG6$*&)-%$tanG6#%\"xG,&%\"nG\"\"\"\"\"#!\"\"F.)-%$secG6#F+,&
%\"mGF.F/F.F.F+" }{TEXT -1 2 " ." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 
10 "Question 5" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Find the anti-de
rivatives " }{XPPEDIT 18 0 "Int(tan(x)^4,x);" "6#-%$IntG6$*$-%$tanG6#%
\"xG\"\"%F*" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "Int(sec(x)^5,x);" "6#-%$
IntG6$*$-%$secG6#%\"xG\"\"&F*" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "Int(t
an(x)^4*sec(x)^4,x);" "6#-%$IntG6$*&-%$tanG6#%\"xG\"\"%-%$secG6#F*F+F*
" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Int(tan(x)^4*sec(x)^5,x);" "6#-%
$IntG6$*&-%$tanG6#%\"xG\"\"%-%$secG6#F*\"\"&F*" }{TEXT -1 2 " ." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 
278 10 "Solutions." }{TEXT -1 87 "  We will do the first two by writin
g procedures to reduce powers of tangent or secant." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 68 "tanred := proc(n) 1/(n-1) * tan(x)^(n-1) \+
- Int(tan(x)^(n-2), x) end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
47 "secred := proc(m) 1/(m-1) * tan(x)*sec(x)^(m-2)" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 54 "            + (m-2)/(m-1) * Int(sec(x)^(m-2) , x) \+
end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "p1 := Int( tan(x)^4
, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p1G-%$IntG6$*$)-%$tanG6#%
\"xG\"\"%\"\"\"F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "p2 := \+
tanred(4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p2G,&*$)-%$tanG6#%\"x
G\"\"$\"\"\"#F-F,-%$IntG6$*$)F(\"\"#F-F+!\"\"" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 17 "p3 := -tanred(2);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#p3G,&-%$tanG6#%\"xG!\"\"-%$IntG6$\"\"\"F)F." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 16 "p4 := value(p3);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#p4G,&-%$tanG6#%\"xG!\"\"F)\"\"\"" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 24 "p1 = op(1, p2) + p4 + C;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/-%$IntG6$*$)-%$tanG6#%\"xG\"\"%\"\"\"F,,**$)F)\"\"$F
.#F.F2F)!\"\"F,F.%\"CGF." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "q1 := Int( sec(x)^5, x)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q1G-%$IntG6$*$)-%$secG6#%\"xG
\"\"&\"\"\"F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "q2 := secr
ed(5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q2G,&*&-%$tanG6#%\"xG\"\"
\")-%$secGF)\"\"$F+#F+\"\"%*&#F/F1F+-%$IntG6$*$F,F+F*F+F+" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "q3 := (3/4)*secred(3);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%#q3G,&*&-%$tanG6#%\"xG\"\"\"-%$secGF)F+#\"\"$
\"\")*&F.F+-%$IntG6$F,F*F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "T
he reduction formula is of no further help, but you are supposed to kn
ow this anti-derivative." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 
"q4 := value(q3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#q4G,&*&-%$tanG
6#%\"xG\"\"\"-%$secGF)F+#\"\"$\"\")*&F.F+-%#lnG6#,&F,F+F'F+F+F+" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "q1 = op(1, q2) + q4 + C;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*$)-%$secG6#%\"xG\"\"&\"\"\"
F,,**&-%$tanGF+F.)F)\"\"$F.#F.\"\"%*(#F4\"\")F.F1F.F)F.F.*&F8F.-%#lnG6
#,&F)F.F1F.F.F.%\"CGF." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 416 "For the last two examples, not
ice that if we start off by reducing the power of tangent each time, w
e will (after two applications of the appropriate reduction formula) e
nd up with an integral which involves only secant.  The clever way to \+
do these two would therefore be to write two recursive programs: one t
o reduce powers of tangent, and one to evaluate the anti-derivative of
 a power of secant.  Here they are (" }{XPPEDIT 18 0 "n;" "6#%\"nG" }
{TEXT -1 26 " is the power of tangent; " }{XPPEDIT 18 0 "m;" "6#%\"mG
" }{TEXT -1 25 " is the power of secant)." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 21 "tanred2 := proc(n,m) " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 57 "             if(n >= 2) then (1/m)*tan(x)^(n-1)*sec(x
)^m " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "               - (n-1)/m * \+
tanred2(n-2,m+2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "             e
lse Int(tan(x)^n * sec(x)^m, x);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 
"             fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "           end
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "secpowerint := proc(m)
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "                 if(m >= 2) the
n 1/(m-1)*tan(x)*sec(x)^(m-2)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "  \+
                 + (m-2)/(m-1) * secpowerint(m-2);" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 46 "                 else value(Int(sec(x)^m, x));" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "                 fi;" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 19 "               end:" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 42 "We can now evaluate both anti-derivatives." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "r1 := Int(tan(x)^4 * sec(x)^4 , x);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r1G-%$IntG6$*&)-%$tanG6#%\"xG\"
\"%\"\"\")-%$secGF,F.F/F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
19 "r2 := tanred2(4,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r2G,(*&)
-%$tanG6#%\"xG\"\"$\"\"\")-%$secGF*\"\"%F-#F-F1*&#F-\"\")F-*&F(F-)F/\"
\"'F-F-!\"\"*&#F-F5F--%$IntG6$*$)F/F5F-F+F-F-" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 27 "r3 := (3/8)*secpowerint(8);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#r3G,**&-%$tanG6#%\"xG\"\"\")-%$secGF)\"\"'F+#\"\"$\"
#c*(#\"\"*\"$S\"F+F'F+)F-\"\"%F+F+*(#F1\"#NF+F'F+)F-\"\"#F+F+*&#F/F;F+
F'F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "r1 = op(1,r2) + o
p(2,r2) + r3 + C;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&)-%$t
anG6#%\"xG\"\"%\"\"\")-%$secGF+F-F.F,,.*&)F)\"\"$F.F/F.#F.F-*&#F.\"#9F
.*&F)F.)F0\"\"'F.F.!\"\"*(#\"\"*\"$S\"F.F)F.F/F.F.*(#F5\"#NF.F)F.)F0\"
\"#F.F.*&#F<FDF.F)F.F.%\"CGF." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "s1 := Int( tan(x)^
4 * sec(x)^5 , x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#s1G-%$IntG6$*
&)-%$tanG6#%\"xG\"\"%\"\"\")-%$secGF,\"\"&F/F-" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 19 "s2 := tanred2(4,5);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#s2G,(*&)-%$tanG6#%\"xG\"\"$\"\"\")-%$secGF*\"\"&F-#F
-F1*&#F,\"#NF-*&F(F-)F/\"\"(F-F-!\"\"*&#F,F5F--%$IntG6$*$)F/\"\"*F-F+F
-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "s3 := (3/35)*secpowe
rint(9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#s3G,,*&-%$tanG6#%\"xG\"
\"\")-%$secGF)\"\"(F+#\"\"$\"$!G*(#F+\"#!)F+F'F+)F-\"\"&F+F+*(#F+\"#kF
+F'F+)F-F1F+F+*(#F1\"$G\"F+F'F+F-F+F+*&F=F+-%#lnG6#,&F-F+F'F+F+F+" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "s1 = op(1,s2) + op(2,s2) + s
3 + C;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$*&)-%$tanG6#%\"xG
\"\"%\"\"\")-%$secGF+\"\"&F.F,,0*&)F)\"\"$F.F/F.#F.F2*&#F6\"#SF.*&F)F.
)F0\"\"(F.F.!\"\"*(#F.\"#!)F.F)F.F/F.F.*(#F.\"#kF.F)F.)F0F6F.F.*(#F6\"
$G\"F.F)F.F0F.F.*&FGF.-%#lnG6#,&F0F.F)F.F.F.%\"CGF." }}}}}}{MARK "0 1 \+
0" 9 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }