The Basic Properties of the Integral
When we compute the derivative of a “complicated” function, like
x
2
+ sin
x
, we usually
use diFerentiation rules, like
d
d
x
[
f
(
x
) +
g
(
x
)] =
d
d
x
f
(
x
) +
d
d
x
g
(
x
), to reduce the computation
to that of ±nding the derivatives of the simple parts, like
x
2
and sin
x
, of the complicated
function. Exactly the same technique is used in computing integrals. Here are a bunch of
integration rules.
Let
a, b
and
A, B, C
be real numbers. Let the functions
f
(
x
) and
g
(
x
) be integrable
on an interval that contains
a
and
b
. Then
(a)
i
b
a
[
f
(
x
) +
g
(
x
)]
dx
=
i
b
a
f
(
x
)
dx
+
i
b
a
g
(
x
)
dx
(b)
i
b
a
[
f
(
x
)

g
(
x
)]
dx
=
i
b
a
f
(
x
)
dx

i
b
a
g
(
x
)
dx
(c)
i
b
a
[
Cf
(
x
)]
dx
=
C
i
b
a
f
(
x
)
dx
(d)
i
b
a
[
Af
(
x
) +
Bg
(
x
)]
dx
=
A
i
b
a
f
(
x
)
dx
+
B
i
b
a
g
(
x
)
dx
That is, integrals depend linearly on the integrand.
(e)
i
b
a
dx
=
b

a
Theorem 1
(Arithmetic of Integration)
.
Proof.
The ±rst three formulae are all special cases of formula (d). ²or example, formula (a)
is just formula (d) with
A
=
B
= 1. So to get the ±rst four formulae it’s good enough to
just prove formula (d), which we’ll do by just comparing the de±nitions of the left and right
hand sides. Let’s introduce the notation
R
n
(
h, a, b
) =
n
s
i
=1
h
p
a
+ (
i

1
2
)
b

a
n
P
b

a
n
(1)
for the midpoint Riemann sum approximation to
I
b
a
h
(
x
)
dx
with
n
subintervals. Then,
by de±nition, the left hand side, namely
I
b
a
[
Af
(
x
) +
(
x
)]
dx
, is the limit as
n
→ ∞
of
R
n
(
Af
+
Bg, a, b
) while the right hand side, namely
A
I
b
a
f
(
x
)
dx
+
B
I
b
a
g
(
x
)
dx
, is the limit
as
n
→ ∞
of
AR
n
(
f, a, b
) +
BR
n
(
g, a, b
). But
R
n
(
Af
+
Bg, a, b
) =
n
s
i
=1
b
Af
p
a
+ (
i

1
2
)
b

a
n
P
+
p
a
+ (
i

1
2
)
b

a
n
PB
b

a
n
=
n
s
i
=1
b
Af
p
a
+ (
i

1
2
)
b

a
n
P
b

a
n
+
p
a
+ (
i

1
2
)
b

a
n
P
b

a
n
B
1
January 13, 2014
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n
s
i
=1
Af
p
a
+ (
i

1
2
)
b

a
n
P
b

a
n
+
n
s
i
=1
Bg
p
a
+ (
i

1
2
)
b

a
n
P
b

a
n
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 Spring '08
 Broughton
 Calculus, Derivative, dx, Riemann, 1 1 g, Arithmetic of Integration

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