6.2 Properties of the Riemann Integral
89
Putting this minor result together with the previous two shows that
D
(
I
(
f
)) =
f,
but
I
(
D
(
f
)) =
f
+
c,
so integration and diﬀerentiation are
almost
inverse operations.
Theorem 6.2.6
(Integration by parts)
.
If
f,g
∈
C
1
[
a,b
]
, then
Z
b
a
f
(
x
)
g
0
(
x
)
dx
=
f
(
b
)
g
(
b
)

f
(
a
)
g
(
a
)

Z
b
a
f
0
(
x
)
g
(
x
)
dx.
Proof.
Put
h
(
x
) =
f
(
x
)
g
(
x
), so
h,h
0
∈ R
by Integral properties thm. Use the Integration
of derivative thm on
h
0
.
NOTE: IBP is product rule in reverse, just like CoV is chain rule in reverse.
Theorem 6.2.7
(Change of variable)
.
If
g
∈
C
1
[
a,b
]
,
g
is increasing, and
f
∈ R
[
g
(
a
)
,g
(
b
)]
,
then
f
◦
g
∈ R
[
a,b
]
and
Z
b
a
f
(
g
(
x
))
g
0
(
x
)
dx
=
Z
g
(
b
)
g
(
a
)
f
(
y
)
dy.
Proof.
First, show
f
◦
g
∈ R
. To each partition
P
=
{
x
0
,...,x
n
}
of [
a,b
], there corresponds
a partition
Q
=
{
y
0
,...,y
n
}
of [
g
(
a
)
,g
(
b
)], so that
g
(
x
i
) =
y
i
. Since the range of
f
on
[
y
i

1
,y