Page 303 - DMTH401_REAL ANALYSIS

Unit 25: Mean Value Theorem

What we need to know is that the primitive of one of the two functions should be expressible Notes
in a simple form and that the derivative of the other should also be simple so that the product
of these two is easily integrable. You may note here that the source of the theorem is the
well-known product rule for differentiation.
The Fundamental Theorem of Calculus gives yet another useful technique of integration. This is
known as method by Substitution also named as the change of variable method. In fact this is the
reverse of the well-known chain Rule for differentiation. In other words, we compose the given
function f with another function g so that the composite f o g admits an easy integral. We deduce
this method in the form of the following theorem:

Theorem 2: Let f be a function defined and continuous on the range of a function g. If g' is
continuously differentiable on c,d , then
b d
ò f(x) dx = ò (f o g) (x) g'(x) dx,
a c
where a = g(c) and b = g(d).
b
Proof: Let F(x) = ò f(x) dt be a primitive of the function 1:
a
Note that the function F is defined on the range of g.
Since f is continuous, therefore, by Theorem 2, it follows that F is differentiable and F¢(t) = f(t),
for any t. Denote G(x) = (F o g) (x).
Then, clearly G is defined on [c,d] and it is differentiable there because both F and g are so. By the
Chain Rule of differentiation, it follows that
G¢(x) = (F o g)¢ (x) g¢(x) = (f o g) (x) g¢(x).
Also f og is continuous since both f and g are continuous. Therefore, f o g is integrable.
Since g¢ is integrable, therefore (f o g) g¢ is also integrable. Hence

d d
ò (f o g)(x) g (x) dx = ò G (x) dx
¢
¢
c
= G(d) – G(c) (Why?)
= F(g (d)) – F(g (c))

= F(b) – F(a)
b
= ò f(x) dx.
a
you have seen that the proof of the theorem is based on the Chain Rule for differentiation. In
fact, this theorem is sometimes treated as a Chain Rule for Integration except that it is used
exactly the opposite way from the Chain Rule for differentiation. The Chain Rule for
differentiation tells us how to differentiate a composite function while the Chain Rule for
Integration or the change of variable method tells us how to simplify an integral by rewriting
it as a composite function.
Thus, we are using the equalities in the opposite directions.
We conclude this section by a theorem (without Proof) known as the Second Mean Value Theorem
for Integrals. Only the outlines of the proof are given.

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