The document discusses antiderivatives and indefinite integrals. It defines an antiderivative as a function whose derivative is equal to the given function. It provides examples of finding antiderivatives using properties like power rules. The power rule states the antiderivative of x^r is x^(r+1)/(r+1) for any rational number r not equal to -1. It also discusses the linearity property which allows breaking up integrals of sums into sums of integrals. The generalized power rule extends the power rule to functions of x raised to some power.

1. 1|Antiderivative
Chapter 1
Antiderivatives
Most of the mathematical operations that we work with come in inverse pairs:
addition and subraction, multiplication and division, exponentiation and root taking.
We have studied how to find the derivative of a function. However, many problems
require that we recover a function from its known derivative (from its known rate of
change). For instance, we may know the velocity function of an object falling from
an initial height and need to know its height at any time over some period. More
generally, we want to find a function F from its derivative ƒ. If such a function F
exists, it is called an anti-derivative of ƒ.
DEFINITION 1.1 Antiderivative
A function F is an antiderivative of ƒ on an interval I if
The process of recovering a function
for all x in I.
F(x) from its derivative ƒ(x) is called
antidifferentiation. We use capital letters such as F to represent an antiderivative of a
function ƒ, G to represent an antiderivative of g, and so forth.
EXAMPLE 1.1
Find an antiderivative of the function
on
SOLUTION
We seek a function F satisfying
with differentiation, we know that
for all real x. From our experience
is one such function.
A moment’s thought will suggest other solution to Example 1. The function
also satisfies
In fact,
(see Figure 1).
, it too is an antiderivative of
, where C is any constant, is an antiderivative of
.
on

2. 2|Antiderivative
Figure 1
EXAMPLE 1.2
Find the general antiderivative of
on
SOLUTION
,
which
antiderivative is
satisfies
.
However,
the
general
.
More generally, we have the following result.
If F is an antiderivative of an interval I, then the most general antiderivative of f on I
is
, where C is an arbitrary constant.
Notation for Antiderivatives
Let a function F is an antiderivative of ƒ on an interval I. The process of find an
antiderivative of f on an interval I called indefinite integral of function f , we wrote
where C be a constant.
Theorem 1.1 Power Rule
If r is any rational number except -1, then
Proof
The derivative of the right side is

3. 3|Antiderivative
We make two comments about Theorem 1.1. First, it is meant to include the
case r = 0; that is,
Second, since no interval I is specified, the conclusion is understood to be valid only
on interval on which
is defined. In particular, we must exclude any interval
containing the origin if r < 0.
EXAMPLE 1.3
Find the general antiderivative of
SOLUTION
Theorem 1.2
and
Proof
Simply note that
and
Theorem 1.3 Indefinite Integral is a Linier Operator
Let f and g have antiderivatives (indefinite integrals) and let k be a constant.
Then
(i)
(ii)
(iii)

4. 4|Antiderivative
Proof
To show (i) and (ii), we simply differentiate the right side and observe that we get the
integrand of the left side.
Property (iii) follows from (i) and (ii).
EXAMPLE 1. 4
Using the linearity of ∫, evaluate
(a)
(b)
(c)
SOLUTION
(a)
Two arbitrary constants
dan
appeared, but they were combined into one
constant, C, a practice we consistently follow.
(b)
(c)
+

5. 5|Antiderivative
Theorem 1.4 Generalized Power Rule
Let g be a differentiable function and r is a rational number different from -1.
Then
EXAMPLE 1.5 Evaluate
(a)
(b)
SOLUTION
(a) Let
(b) Let
; then
. Thus, by Theorem 1.4
. Thus
+C=

6. 6|Antiderivative
Exercise 1
Find the general antiderivative
1.
for each of the following.
+3
2.
3.
In Problem 4 – 10, evaluate the indicated indefinite integrals.
4.
5.
6.
7.
8.
9.
10.