Lecture 8 Antiderivative. Indefinite integral and its properties. Table of integrals. Main methods of integration
1. Antiderivative and indefinite integral
Concept of antiderivative and indefinite integral
Definition 1. A
function
is an antiderivative
of
on an interval
if
for all
in
.
Theorem 1. If
is an antiderivative of
on an interval
,
then
is an antiderivative of
on the interval
if and only if
is of the form
,
for all
in
where
is a constant.
The constant is called the constant of integration.
The family of functions represented by is the general antiderivative of
is the general
solution of the differential equation
.
Notation for antiderivatives
When solving a differential equation of the form
,
we solve for
,
giving us the equivalent differential form
.
The operation of finding all
solutions of this equation is called antidifferentiation
or indefinite
integration and is denoted by an
integral sign
.
The general solution is denoted by
Definition 2.
Suppose f
is a function defined on an interval I
and suppose further that f
has an antiderivative on the interval I.
The family of all antiderivatives of f
is called the indefinite integral of
f and is
denoted by the symbol
(read the integral of
f with respect to x).
In this notation the function is called the integrand
of the indefinite integral. The process of finding the indefinite
integral is called integration
or, sometimes anti-differentiation.
More specifically, given a function f,
the expression “integrating f”
means “finding the indefinite integral of f”.
There is a difference between an antiderivative
and the indefinite integral. An antiderivative is a member of the
family of functions represented by the indefinite integral. In the
above example,
,
where C is
a constant. The functions
,
and
are
all members of the family
and
are all antiderivatives of the function
.
It is also correct to write
,
where K is
a constant. The difference is the choice of constants, where in this
case
.
Notice that all these curves are “parallel” in
the sense that they never cross each other since they are translates
of the function
.
To get a particular antiderivative one needs to know a specific point
the antiderivative passes through. For example, if we want the curve
that is a member of the family
that
passes through the point
,
we have
and
.
So
and
it follows that
.
The process of integration involves finding one
antiderivative of the given function so that the indefinite integral
is that one antiderivative plus the constant of integration. This can
be easy for certain functions because we know the differentiation
formulae. For example, because
,
it follows that
,
C a
constant. Here are the integration formulae that follow directly from
the appropriate differentiation formulae: The symbol C
is a constant (called the constant
of integration). For any
differentiable function
it
is always the case that
meaning
that the function
is
a member of the family
.
for
any constant
.
Notice that when
,
the
formula
reduces to
or
just
.
In fact every
differentiation formula provides an
integration formula. Consider the differentiation formula
.
This is a statement that two derivatives are equal. So the families
of antiderivatives are also the same families. Therefore
.
In the same manner,
.
For reference, number these formulae as
11.
12.
These two formulae and those above can be combined to produce integrals of more complicated functions as shown in the examples below.
The table of indefinite integrals
1 |
|
|
2 |
|
|
3 |
|
|
4 |
|
|
5 |
|
|
6 |
|
|
7 |
|
|
8 |
|
|
9 |
|
|
1 |
|
|
1 |
|
|
1 |
|
|
Examples
Example 1:
Evaluate
Solution:
(Formula
12, applied twice)
(Formula
11)
(Formula
2)
(Arithmetic)
Example 2:
Evaluate
Solution:
(Formula
12)
(Formula
11)
(Algebra)
(Formula
2)
(Arithmetic)
Example 3: Find
all the antiderivatives of
.
Solution: This
question is just another way to ask that
be
evaluated.
(Formula
12)
(Formula
11)
(Formulae
2 and 7)
(Arithmetic)