Functions are from
R
×
R
R \times R
to R, where R represents the set of real numbers. If c is a number and either (1)
∫
a
b
G
2
\smallint _a^b{G^2}
exists and
∫
a
b
G
\smallint _a^bG
exists, (2)
∫
a
b
G
\smallint _a^bG
exists and
a
Π
b
(
1
+
G
)
_a{{\mathbf {\Pi }}^b}(1 + G)
exists and is not zero or (3) each of
a
Π
b
(
1
+
G
)
_a{{\mathbf {\Pi }}^b}(1 + G)
and
a
Π
b
(
1
−
G
)
_a{\Pi ^b}(1 - G)
exists and is not zero, then
∫
a
b
c
G
\smallint _a^bcG
exists,
∫
a
b
|
c
G
−
∫
c
G
|
=
0
,
x
Π
y
(
1
+
c
G
)
\smallint _a^b|cG - \smallint cG| = 0{,_x}{{\mathbf {\Pi }}^y}(1 + cG)
exists for
a
≤
x
>
y
≤
b
a \leq x > y \leq b
and
∫
a
b
|
1
+
c
G
−
Π
(
1
+
c
G
)
|
=
0
\smallint _a^b|1 + cG - {\mathbf {\Pi }}(1 + cG)| = 0
. Furthermore, if H is a function such that
lim
x
→
p
−
H
(
x
,
p
)
,
lim
x
→
p
+
H
(
p
,
x
)
,
lim
x
,
y
→
p
−
H
(
x
,
y
)
{\lim _{x \to {p^ - }}}H(x,p),{\lim _{x \to {p^ + }}}H(p,x),{\lim _{x,y \to {p^ - }}}H(x,y)
and
lim
x
,
y
→
p
+
H
(
x
,
y
)
{\lim _{x,y \to {p^ + }}}H(x,y)
exist for each
p
∈
[
a
,
b
]
,
n
≥
2
p \in [a,b],n \geq 2
is an integer, and G satisfies either (1), (2) or (3) of the above, then
∫
a
b
H
G
n
\smallint _a^bH{G^n}
exists,
∫
a
b
|
H
G
n
−
∫
H
G
n
|
=
0
,
x
Π
y
(
1
+
H
G
n
)
\smallint _a^b|H{G^n} - \smallint H{G^n}| = 0{,_x}{{\mathbf {\Pi }}^y}(1 + H{G^n})
exists for
a
≤
x
>
y
≤
b
a \leq x > y \leq b
and
∫
a
b
|
1
+
H
G
n
−
Π
(
1
+
H
G
n
)
|
=
0
\smallint _a^b|1 + H{G^n} - {\mathbf {\Pi }}(1 + H{G^n})| = 0
.