Calculus Of One Real
Variable – By Pheng Kim Ving 
5.7 
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1.
Antiderivatives 
Definition 1.1
An antiderivative (or a primitive)
of a function f on an interval I is a function F whose
derivative on I is f, ie, F '(x) = f(x) 
The Abbreviation AD
For the remainder of this section, the abbreviation “AD” stands for “antiderivative”.
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2.
Relationships Among Antiderivatives 
Suppose F(x) is an AD of f(x). Clearly F(x), F(x) + 1, F(x) – 2/5, F(x) – 100, and F(x) + 1000 are ADs
of f(x),
because their derivatives are all equal to f(x). Recall that the
derivative of a constant is 0; eg, (d/dx) (F(x) + 1) =
(d/dx) F(x) + 0 = f(x). An AD isn't unique. Indeed, for any constant C, the function F(x) + C is an AD of the
function
f(x). This means that
if G(x) is such that G(x) = F(x) + C for some constant C, then G(x) is an AD of f(x). The
following theorem states that the converse is also true, ie, if G(x) is an AD of f(x), then G(x) is of the form G(x) =
F(x) + C for some constant C.
Theorem 2.1
Suppose F(x) is an
antiderivative of f (x) on an interval I. Then every
antiderivative G(x) of f (x) on I is of the form G(x) = 
Proof
Let H(x) = G(x) – F(x). Then H '(x) = G '(x) – F '(x) = f(x) – f(x) = 0. So H(x) = C on I, where C is some constant.
Thus, G(x) – F(x) = C, hence G(x) = F(x) + C, on I.
EOP
i. The phrase “on I ” means “for all x in I ”.
ii. All
ADs of f(x) on I are each of the form F(x) + C for some constant
C. All of them are obtained in this way. There
are no others.
iii. What about F(x) itself ? Well, F(x) = F(x) + 0.
iv. Can
the function F(x) be any AD of f(x), not just some
particular one? Yes. Suppose F_{2}(x) is an AD of f(x). So
F_{2}(x) = F(x) + C_{2} for some constant
C_{2}. Then G(x) = F(x) + C = F_{2}(x) – C_{2} + C = F_{2}(x) + K, where K = – C_{2} +
C is a constant.
This proves our affirmative answer.
v. Any
two ADs of f(x) differ by a constant. To see why, suppose G_{1}(x) = F(x) + C_{1} and G_{2}(x) = F(x) + C_{2}. Then
G_{1}(x) – G_{2}(x) = (F(x) + C_{1}) – (F(x) + C_{2}) = C_{1} – C_{2}.
vi.
It's essential that the set on which f(x) and its ADs are
considered is an interval. If that set isn't an interval, there may
be ADs that aren't of the form F(x) + C for any constant C. We'll give an
example of this situation in
Problem
& Solution 5.
Theorem 1 implies that the graphs of various
ADs of a function on an interval are vertically displaced versions of the same
curve, as shown in Fig. 2.1. That is, the vertical distance between the
graphs of any 2 ADs of a function on an interval is a
constant. This is also implied by Remarks 2.1 v above.

Fig. 2.1 Graphs Of Some Antiderivatives Of A Function. 
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3. General
Antiderivatives 
Let F(x) be an AD of f(x). Then, by theorem 2.1, all ADs of f(x) are each of the
form F(x) + C for some constant C. We
use the function F(x) + C, where C is an arbitrary constant, to represent every AD of f(x), and call it the
general
antiderivative of f(x).
Definition 3.1
Let F(x) be an antiderivative of the function f (x) on an interval I. Then the function F(x) + C, where C is an arbitrary constant, is called the general antiderivative of f (x) on I. 
The Abbreviation GAD
For the remainder of this section, the abbreviation “ GAD” stands for “ general antiderivative”.
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4. Indefinite
Integrals 
The GAD of f(x) is also called
the indefinite integral of f(x). The reason for
this will be known in Section
9.4 Indefinite
Integrals.
Definition 4.1
The general antiderivative of a function f on an interval I is also called
the indefinite integral of f on I, which is denoted by:
where C is an arbitrary
constant. 
Remarks 4.1
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5. Finding
General Antiderivatives 
Example 5.1
Find the general antiderivative of f(x) = x^{2} – 3x + 2.
Solution
EOS
The answer can be verified by simply differentiating it.
Recall that the finding of the derivative of
a function is called differentiation. Finding the general antiderivative of a
function is called antidifferentiation.
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6. General
Antiderivatives Of Integer Powers Of The Variable 

The case where n = –1, which
causes the denominator n + 1 to become 0, will be treated in Section
7.1. There, we'll find an
AD of x^{–1} = 1/x.
Problems And Solutions 
1. Find the general antiderivative of each of
the following functions.
a. 2.
b. x^{3}.
Solution
2. Find each of the following indefinite
integrals.
Solution
3. Find the general antiderivative of each of the following functions.
Solution
4.
Prove that:
where a and b are non0 constants.
Solution
Let F(x) and G(x) be antiderivatives of f(x) and g(x) respectively. We have:
5. Here we give an example as promised in Remarks 2.1 vi. Let:
f(x) = 0 for all x,
F_{1}(x) = 2 for all x, and:
Solution
Notes
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