0.1 Indefinite Integrals
Integration is a formalization of the process of antidifferentiation. When we differentiate a function,
we lose some information about the function - although we know how it is changing we don’t know
where it started from. Thus, when we go through the reverse process of differentiation, we end up
with a family of functions rather than just a single function. Because of this we need an initial
value to find a unique solution to a differential equation. We define the indefinite integral in order
to encompass the fact that we have a family of functions.
Definition 0.1.1 (Indefinite Integral).We call the set of all antiderivatives of fthe indefinite
integral of f, denoted by Zf(x)dx.
The symbol Ris an integral sign. The function fis the called the integrand and xis the variable
of integration. We say that dx is a differential of x.
If we know any antiderivative Fto a given function f, then we simply find the indefinite integral
as Zf(x)dx =F(x) + c,
because the expression on the right-hand side represents all possible antiderivatives of f(x), if we
let cbe an arbitrary constant.
When we find the indefinite integral of a function f(x) we say that we integrate the integrand
f(x). Thus, the process of evaluating an integral is referred to as integration. Unfortunately, the
notation we have introduced for the indefinite integral probably looks rather arcane. As we move
further into the study of integration, the reason for the notation will become more clear. For the
moment let us note that this notation is based on the initial developments of calculus, where the
notion of infinitesimal and differential were prevalent. The resemblance of Rand an elongated
Sis not accidental. This notation reflects the relationship between integration and summation.
Integration is essentially the summation of the area underneath the integrand f(x) over intervals
of infinitesimal length. The differential dx represents an infinitesimal change in x, which represents
the intervals of infinitesimal length over which the summation involved in integration occurs. When
it will not lead to confusion we will refer to the indefinite integral as simply the integral. The term
indefinite integral is used to distinguish the process of indefinite and definite integration (which we
will consider soon). Although both of these concepts are related to the area underneath a function,
the indefinite integral is a function, whereas the definite integral is a constant, which is given by
the area underneath a function over a set interval (defined by the limits of integration, which are
not present in an indefinite integral).
Since the process of (indefinite) integration is an inverse to differentiation, we can derive many
rules for integration using rules we already know for differentiation. For instance
d
dxxn+1 = (n+ 1)xn
so we see that d
dxxn+1
n+ 1=xnforn6=−1.
This leads us to the product rule for integrals.
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