IndefiniteIntegral(I)

Page 1

THE INDEfinite INTEGRAL AN INTRODUCTION TO THE INDEFINITE INTEGRAL (PART I) Before you begin this section, I will advise that you review the topic: .ANTIDERIVATIVES So far, we have dealt with definite integrals and we've seen how the Fundamental Theorem relates definite integrals with antiderivatives. Provided that we can find an antiderivative of a function, the definite integral of that function can be easily evaluated. To that effect, we will require a more convenient notation for antiderivatives. This is where the Indefinite Integral comes in. The conventional notation for an indefinite integral has been derived from the fundamental theorem. Let's take a quick flashback:

b a

f(x) dx

=

1

F(b – a)

(Where F is an antiderivative of f, that is, F' = f) If y = f(x), then is

x a

2

f(t) dt

is an antiderivative of F. Based on (1) and (2), we express the indefinite integral as

f(t) dt

Essentially, we are implying that

∫ f(t) dt

=

F(x)

because F'(x) = f(x). Note that there is no interval in the notation for an indefinite integral, which is probably one reason why it's called “indefinite”. That's a personal observation anyway. When we talk about an indefinite integral, what we are really referring to is a GENERAL indefinite integral, which is why we introduce the arbitrary constant C in the evaluation of an indefinite integral. Because the constant C can hold just about any value, we end up with a family of functions; one antiderivative for each constant. Thus, an indefinite integral can also be regarded as a family of functions. When we evaluate a definite integral, we always end up with a NUMBER. My point here is that a definite integral is simply a number, while an indefinite integral is a function/a family of functions. THIS DISTINCTION

IS

CRITICALLY IMPORTANT. There is, however, a connection between the two, which is given by part 2 of the Fundamental Theorem. In the section titled: The Fundamental Theorem – An Introduction, recall that

b

f( x) dx

=

∫ f(t) dt

=

a

F(x)] a

b

3

and now, we are saying that

F(x)

4


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IndefiniteIntegral(I) by Timothy Adu - Issuu