Page 297 - DMTH401_REAL ANALYSIS

Unit 24: Fundamental Theorem of Calculus

You may note that any suitable primitive will serve the purpose because when the primitive is Notes
known, then the process described by the Fundamental Theorem is much simpler than other
methods. However, it is just possible that the primitive may not exist while you may keep on
trying to find it. It is, therefore, essential to formulate some conditions which can ensure the
existence of a primitive. Thus now the next step is to find the conditions on the integral, (function
to be integrated) which will ensure the existence of a primitive. One such condition is provided
by the theorem.
According to theorem 2 if f is continuous in [a, b], then the function F given by

x
Î
F(x) = ò f(t) dt x [a,b] is differentiable in [a, b] and F'(x) = f(x) " x [a,b]
Î
a
i.e. F is the primitive of f in [a, b]
The following observations are obvious from the theorems 1 and 2:

(i) If f is integrable on [a, b], then there is a function F which is associated with f through the
process of integration and the domain of F is the same as the interval [a, b] over which f is
integrated.
(ii) F is continuous. In other words, the process of integration generates continuous function.

(iii) If the function f is continuous on [a, b], then F is differentiable on [a, b]. Thus, the process
of integration generates differentiable functions.
(iv) At any point of continuity of f, we will have f(c) = f(c) for c  [a, b]. This means that if f is
continuous on the whole of [a, b], then F will be a member of the family of primitives of
f on [a, b].
In the case of continuous functions, this leads us to the notion

ò f(x) dx

for the family of primitives of f. Such an integral, as you know, is called an Indefinite integral of
f. It does not simply denote one function, but it denotes a family of functions. Thus, a member of
the indefinite integral of f will always be an antiderivative for f.
Theorem 3 gives US a condition on the function to be integrated which ensures the existence of
a primitive. But how to obtain the primitives, once this condition is satisfied. In the next section,
we look for the two most important techniques for finding the primitives. Before we do so, we
need to study two important mean-values theorems of integrability.

Self Assessment

Fill in the blanks:

1. This function is not the derivative of any function F : [–1, 1]  R. Indeed if f is the derivative
of a function F : [–1, 1]  R then f must have the ................................... .
2. Let f : [a, b]  R be a continuous function. Let F : [a, b]  R be a function defined by
..................................... .
3. If f and F are functions defined on [a, b] such that F’(x) = f(x) for x Î [a, b] then F is called a
'primitive' or an '...............................' off on [a, b].

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