Calculus Of One Real Variable – By Pheng Kim Ving
Chapter 5: Applications Of The Deriivative Part 1 – Section 5.7: Antiderivatives And Indefinite Integrals

 

5.7
Antiderivatives And Indefinite Integrals

 

 

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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)
for all
x in I.

 

 

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) =
F(x) + C on I, where C is some constant.

 

 

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

 

Remarks 2.1

 

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 F2(x) is an AD of f(x). So
     
F2(x) = F(x) + C2 for some constant C2. Then G(x) = F(x) + C = F2(x) – C2 + C = F2(x) + K, where K = – C2 +
     
C is a constant. This proves our affirmative answer.

 

v.  Any two ADs of f(x) differ by a constant. To see why, suppose G1(x) = F(x) + C1 and G2(x) = F(x) + C2. Then
    
G1(x) – G2(x) = (F(x) + C1) – (F(x) + C2) = C1C2.

 

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.

 

Graphs Of Antiderivatives

 

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) = x2 – 3x + 2.

 

Solution

 

EOS

 

The answer can be verified by simply differentiating it.

 

Antidifferentiation

 

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.

 

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Problems And Solutions

 

1.  Find the general antiderivative of each of the following functions.
     a.  2.
     b. 
x3.

 

Solution

 

 

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2.  Find each of the following indefinite integrals.

 

 

Solution

 

 

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3.  Find the general antiderivative of each of the following functions.

 

   

 

Solution

 

 

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4.  Prove that:

 

   

 

     where a and b are non-0 constants.

 

Solution

 

Let F(x) and G(x) be antiderivatives of f(x) and g(x) respectively. We have:

 

 

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5.  Here we give an example as promised in Remarks 2.1 vi. Let:

 

      f(x) = 0 for all x,
    
F1(x) = 2 for all x, and:

 

   

 

Solution

 

 

Notes

 

 

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