1
Antiderivatives and Inde°nite Integral
De°nition 1.1
Let
I
denote an interval in
R
and
f
:
I
!
R
be a function.
We say that a function
F
:
I
!
R
is an antiderivative of
f
(
x
)
on
I
if
F
0
(
x
) =
f
(
x
)
for all
x
2
I;
or equivalently, using the notion of the di/erential
dF
(
x
) =
f
(
x
)
dx
for all
x
2
I:
Proposition 1.2
Let
F
1
:
I
!
R
and
F
2
:
I
!
R
be two antiderivatives of
f
:
I
!
R
. Then, there exists
a constant
C
2
R
such that
F
1
(
x
) =
F
2
(
x
) +
C
for all
x
2
I
.
Proof.
Since the function
’
(
x
) =
F
1
(
x
)
°
F
2
(
x
)
; x
2
I;
has the derivative
’
0
(
x
) = 0
, it follows that
’
(
x
)
is a constant function, i.e. there exists
C
2
R
such that
’
(
x
) =
F
1
(
x
)
°
F
2
(
x
) =
C
and the conclusion follows.
It is clear, by Proposition 1.2, that in order to °nd all antiderivatives of a given function
f
(
x
)
on an
interval
I
, it is su¢ cient to °nd just one antiderivative
F
(
x
)
and any other antiderivative will be equal
to
F
(
x
) +
C
for some constant
C
. The expression
F
(
x
) +
C
, where
C
denotes an arbitrary constant, is
called the
inde°nite integral
of
f
(
x
)
and is denoted by
Z
f
(
x
)
dx:
It is convenient to think about the inde°nite integral
Z
f
(
x
)
dx
=
F
(
x
) +
C
as a parameterized family (where
C
is a parameter) of all antiderivatives of the function
f
(
x
)
. However,
this explanation is correct only when the function
f
(
x
)
is de°ned on an interval. For example the function
F
(
x
) =
°
1
2
x
2
+
C
is the inde°nite integral of
f
(
x
) =
1
x
3
;
but as we can see the function
G
(
x
) =
(
°
1
2
x
2
+ 1
for
x <
0;
°
1
2
x
2
°
1
for
x >
0
;
is also an antiderivative of
f
(
x
)
which can not be expressed as
F
(
x
)+
C
with one value of the constant
C
.
Therefore, we must ±improve²this explanation of the de°nite integral by adding that
R
f
(
x
)
dx
denotes
the family of all antiderivatives on intervals contained in the domain of
f
(
x
)
. That means, if we know
the formula of
R
f
(
x
)
dx
=
F
(
x
) +
C
, then on every interval
I
contained in
Dom
(
f
)
\
Dom
(
F
)
, every
antiderivative of
f
(
x
)
can be expressed in a form
F
(
x
) +
C
for a certain value of constant
C
.
The symbol
R
can be easily explained in a context of de°nite integrals (assuming that we know
Fundamental Theorem of Calculus). The symbol
dx
, which is associated with the notion of di/erential
1

indicates that
x
is the independent variable of the function
f
(
x
)
and that the derivative of
F
(
x
)
with
respect to this variable should be equal to
f
(
x
)
. On the other hand,
R
can be considered as an operator³
called
integral
or
inde°nite integral
which is the inverse operator of the operator of di/erentiation
d
dx
,
i.e.
d
dx
Z
f
(
x
)
dx
=
f
(
x
)
:
However, if we think formally, the operator
R
should be considered rather as the inverse operator
d
associated with the operation of °nding a di/erential. Indeed, we have that
d
°Z
f
(
x
)
dx
±
=
f
(
x
)
dx
and
Z
df
(
x
)
dx
=
f
(
x
) +
C;
so indeed, in a formal way the symbols
d
and
R
cancel each other out. The process of °nding the inde°nite
integral of a certain function
f
(
x
)
is called
integration
and the function
f
(
x
)
is called the
integrant
. There
is a fundamental question related to the above de°ned operation of integration. Which exactly are the
functions
f
(
x
)
for which an antiderivative
F
(
x
)
exists, i.e. the inde°nite integral