This paper considers a special integral
(
I
)
∫
a
b
(
f
d
g
+
H
)
(I)\smallint _a^b(fdg + H)
which is a subdivision-refinement-type limit of the approximating sum
\[
∑
1
n
{
f
(
t
i
)
[
g
(
x
i
)
−
g
(
x
i
−
1
)
]
+
H
(
x
i
−
1
,
x
i
)
}
,
\sum \limits _1^n {\{ f({t_i})[g({x_i}) - g({x_{i - 1}})] + H({x_{i - 1}},{x_i})\} ,}
\]
where
x
i
−
1
>
t
i
>
x
i
{x_{i - 1}} > {t_i} > {x_i}
. The author shows, with appropriate restrictions, that
(
I
)
∫
a
b
(
f
d
g
+
H
)
(I)\smallint _a^b(fdg + H)
exists if and only if
\[
(
R
)
∫
x
y
(
f
d
g
+
H
−
A
−
)
=
(
L
)
∫
x
y
(
f
d
g
+
H
+
A
+
)
(R)\smallint _x^y(fdg + H - {A^ - }) = (L)\smallint _x^y(fdg + H + {A^ + })
\]
for
a
⩽
x
>
y
⩽
b
a \leqslant x > y \leqslant b
, where
A
(
p
,
q
)
=
[
f
(
q
)
−
f
(
p
)
]
[
g
(
q
)
−
g
(
p
)
]
,
A
−
(
p
,
q
)
=
A
(
q
−
,
q
)
A(p,q) = [f(q) - f(p)][g(q) - g(p)],{A^ - }(p,q) = A({q^ - },q)
and
A
+
(
p
,
q
)
=
A
(
p
,
p
+
)
{A^ + }(p,q) = A(p,{p^ + })
. Furthermore, if either of the equivalent statements is true, then all the integrals are equal. These equivalent statements are used to prove an integration-by-parts theorem and to solve a Gronwall inequality involving this special integral. Product integrals are used in the solution of the Gronwall inequality.