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