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Sachin Kaushal, Lovely Professional University Unit 3: Differentiation of an Integral

Unit 3: Differentiation of an Integral Notes

CONTENTS
Objectives
Introduction

3.1 Differentiation of an Integral
3.2 Summary
3.3 Keyword

3.4 Review Questions
3.5 Further Readings

Objectives

After studying this unit, you will be able to:
 Define differentiation of an integral.

 Solve problems related to it.

Introduction

If f is an integrable function on [a, b], we define its indefinite integral to be the function F defined
on [a, b] by

x
F (x) = f(t) dt
a
Here, it is shown that the derivative of the indefinite integral of an integrable function is equal
to the integrand almost everywhere. We begin by establishing some lemmas.

3.1 Differentiation of an Integral

If f is an integrable function on [a, b] then f is integrable on any interval [a, x] [a, b]. The
function F given by
x
F (x) = f(t) dt c ,
a
where c is a constant, called the indefinite integral of f.
Lemma 1: If f is integrable on [a, b] then the indefinite integral of f namely the function F on
x
[a, b] given by F (x) = f(t) is a continuous function of bounded variation on [a, b].
a
Proof: Let x be any point of [a, b].
o

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