BENCZE MIHÁLY
FLORENTIN SMARANDACHE
One Application of Wallis
Theorem
In Florentin Smarandache: “Collected Papers”, vol. III.
Oradea (Romania): Abaddaba, 2000.
COLLECTED PAPERS, vol. III
One Application Of Wallis Theorem l
Theorem 1. (Wallis, 1616-1703)
nl2
f sin
Jf/2
'n+' xdx
f cos
=
'n+i
2.4. (2 n )
lL(2n+l)
xdx
Proof. Using integration by parts, we obtain
It,'
2
It
12
In= fSin2n+lxdx = fsin2nxsinxdx =-cosx.sin2nxl~2+
~Il 0
0
+ 2n fsin In-I x(l- sin 1 x}:u- = 2nl n-I
from where
2nl n
-
2n
In = 2n + l i n - I '
By multiplication we obtain the statement. We prove in the same
way f~~~;r:m 2.
'r
'n
sin
y cos ~.
xdx = •
o
:..::.3~...:..... ~(2=.;.:...n_-~I..L)
xdx =.:...1
2.4 ... (2 n)
0
Proof. Same as the first theorem.
Theorem 3. If
f (x ) = f a " x ", then
f f(sin
o
."
f f(eos~
• 12
x)dx =
0
, ..
x)dx
I 3
lr
.2 .
(2 k _ I)
= !!..a o +!!.. I au ..... (
r
).
2.4 .... 2k
a" x 2' function we substitute x by sill x
2k~'
2
Proof. In the f (x ) = I
and then integrate from 0 t6 nl2, and we use the second theorem.
Theorem 4. If
g (x ) = . . f a x 11 +,
then
~
2k·..!
~
H
!2.
H
f g(sln
f
•
~
~0
k
2.4 ..... (2 k )
r
t;,
x}dx =
g(co: x}dx = a, +
a,,+'1.3 ..... (2k + 1)
Theorem 5. If h (x ) = I a I x I . then
Hf"h(' \.J_ H'2h(
\.J_ lr
'0 u ~(lr 1.3 .... ~2k -I)
2.4 .... ~2k) \
SlnXJ<U= f eosxJ<U=-a,,+a,+~ -a21
' ) +a 21 +,
~,
)"
U
./
2
hi
2
2.4 .... ~2k
1." .... ~2k + I
." Application 1. • " .
f Sin (sin x}dx = f Sin (cos x}dx = Lx (- I Y , ' ( I
)' '
o
0
Proof. We use that
Application 2.
f ' 'cos (SIO x}dx =
"
sin x
x
=
2 A.
lr
(cos
x) =
2
l
• •
f sh
(Sin
Jsh (cos
=u
i.
,h
x=
(_
.
I
Y
L
. ( ,)' ,
I." 4 k.
_
1
I,/1-3·.2k+l' , (
r
::co
x)it =
:c
Proof. We use that
x
!
I (- 1Y-(~.
) "
k
frl-:!,
x)dx =
1
r: I;
x
x =
~
I,~O (- I Y(x
)
2k + I
,
fo cos
Proof. We use that cos
Application 3.
fr
1·3-",2k+1
<00
:z.
\_ ~A-. i
~/ (2~
+ I) /
:Together with Mihaly Bcnczc.
119
FLORENTIN SMARANDACHE
IZ
APP lication4·'f h ( .
o
I
"fl' h (
C
0
x
Proof. We usc that
Application 5.
x
\,,- _
Sin X JU-' -
C
ch
\,,__
x JU-' -
COS
2k
!!.... ~
~
2ko04
t (
I
')'
k.
I-x- .
to() (2k) !
X=
;r 2
I2=.
Hk
6
_
f-
1.3 ..... (2k-I)X 2 k+1
Proof. In the arcsin X - X + ~
expres t d 2.4 ..... 2k
2k+1
sion we substitute x by sin x, and use theorem 4. It results that
<
Jr '
/
I',0 (2k
8 =
we get
f
1
! I k' =
+ 1
(X
X!
x
)'.
Because
L !Ik
)
2
=
*01
L (2k+! )'
bO
!
+-
4
I6 .
;r
k =1
'2
Application 6.
12
B
J
sinx ctg{sinx}dt= Jcosx ctg{cosx}dt= Jr - JrI( ,~
o
2 2
k.,
fC,
fC
'"
0
k=1
where Bk is the k-th Bernoulli type number (see[l]).
Proof. We use that xctg
Application 7.
,./2
I arctg
~) I +
r·
x = I-
f. ~ x"
bl
(2k)
I
,
(sin x}iT =
:t (-
I)
arctg (cos x}iT =
2.4 ..... (2 k )
,
1.3 ..... (2k-I)(2k+IY
'01
2'(: ... 1
x
Proof. We use that arctg x= L(-I} _x_
Application 8.
to()
2k + I
,I,
x
24 (2k)
!
I argth{sinx)dx = fargth{cosx}dT =I + L'011.3····
...(.2k-1)(2k+I,
,12
\2
0
'J
Proof. We use that
Application 9.
argth x =
;r,2
X2k+1
r
L-.
k~ 2k+1
I
1f/2
x
0
kdJ
far~~sinx)dx= f args~cosx)dx= IHY ()2·
2k+1
o
Proof. We use that
args
h
*"
Application 10.
,12.
_ ,</2
0
Proof. We use that
"
tg
-I)B.
f·
13 .. 2k - I k
x ik(4' -I)B
~ 2'<-1(4'
_
ftg(Slnx)dx- ftg(cosx)dx-I
o
X=
2'
L
H
120
IV. IJ. ...(2k-I~r+l
24····(2kX2k+I)
X= L.)- ,
Hi
~
(2k)!
k X,·-I
x
L !Ik
*01
2
COLLECTED PAPERS, vol. III
Application 11.
KI' . sin x x) dx = KI' . cos(cosx x) dx
o
Sin (Sin
=
~ + 1f
Proof. We use that
f=,
x
2
Sin
0
(2 ":' - I~B,
2"'(k')"
.f (2" , - I )B
x
(2 k) ,
Application 12.
fr!2·
J
'2
Jr
J
smx d.x=
o sh(sinx)
(
sh(cosx)
0
2
'"'
\n
2/..-1
,-IF'
2"(k!j
_x_ = 1+ 2::t (-1 j
sh x
k~'
Proof. We use that
12
dt=~+Jr!2
COSX
"
-'x'
--=1+2L.,sin x
, ~,
(2
2k
-1 )BkX2t
(2k) !
·'
Application 13.
1(!
2
7f
.f
E,
x)cix = 2+
Jr 7-:, 2H"(k!Y
where Ek is the k-th Euler type number (see([ I D.
It
f sec (sin
x)cix
f sec (cos
=
Proof. We use that
sec x
= I +
E
I --'
-x'k
~
'"1
(2 k)
Application 14.
,,/2
,,/2
r
0
*=1
!
E
f sech{sinx}ix= fsech(cosx}ix=!::+JrL(-lj
u"( ,y
o
2
2 k.
Proof. We use that
r
E",
-(
) ,.e
ser:h x=l+ ~J-Ir
*"1
2k.
References:
[I] Octav Mayer: Teoria functiilor de 0 variabila complexa, Ed.
Academica, Bucure~ti, 1981.
[2] Mihaly Bencze, About Taylor-formula (manuscript).
["Octogon", Vo1.6, No.2, 117-20, 1998.]
121