Page 6 - DMTH202_BASIC_MATHEMATICS_II

Richa Nandra, Lovely Professional University Unit 1: Integration

Unit 1: Integration Notes

CONTENTS
Objectives
Introduction

1.1 Integration as an Inverse Process of Differentiation
1.1.1 The Fundamental Theorem of Calculus
1.2 Integration by Substitution

1.2.1 The Guess-and-Check Method
1.2.2 The Method of Substitution
1.2.3 Trigonometric Substitutions
1.2.4 Why does Substitution Work?
1.2.5 More Complex Substitutions

1.3 Summary
1.4 Keywords
1.5 Review Questions

1.6 Further Readings
Objectives

After studying this unit, you will be able to:
 Understand the Integration as inverse process of differentiation
 Illustrate the process of integration by Substitution

Introduction

Differential Calculus is centered on the concept of the derivative. The original motivation for
the derivative was the problem of defining tangent lines to the graphs of functions and calculating
the slope of such lines. Integral Calculus is motivated by the problem of defining and calculating
the area of the region bounded by the graph of the functions. If a function f is differentiable in an
interval I, i.e., its derivative f 2 exists at each point of I, then a natural question arises that given
f 2 at each point of I, can we determine the function? The functions that could possibly have given
function as a derivative are called anti derivatives (or primitive) of the function. Further, the
formula that gives all these anti derivatives is called the indefinite integral of the function and
such process of finding anti derivatives is called integration. Such type of problems arises in
many practical situations. For instance, if we know the instantaneous velocity of an object at any
instant, then there arises a natural question, i.e., can we determine the position of the object at
any instant? There are several such practical and theoretical situations where the process of
integration is involved. The development of integral calculus arises out of the efforts of solving
the problems of the following types: (a) the problem of finding a function whenever its derivative
is given, (b) the problem of finding the area bounded by the graph of a function under certain
conditions. These two problems lead to the two forms of the integrals, e.g., indefinite and
definite integrals, which together constitute the Integral Calculus. There is a connection, known

LOVELY PROFESSIONAL UNIVERSITY 1