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Basic Mathematics-II Richa Nandra, Lovely Professional University

Notes Unit 3: Integration by Parts

CONTENTS
Objectives
Introduction

3.1 Integration by Parts
3.1.1 Usage
3.2 The Substitution z = tan(x/2)

3.3 Summary
3.4 Keywords
3.5 Review Questions
3.6 Further Readings

Objectives

After studying this unit, you will be able to:

 Understand the concept of integration by parts
 Recognize the usage of integration by parts
 Discuss the substitution with respect to integration by parts

Introduction

As we know, there is a power rule intended for derivatives; there is a power rule intended for
integrals. There is a chain rule intended for derivatives; there is a chain rule intended for
integrals. There is a product rule intended for derivatives; and now what do you think will be
there for integrals? There is integration by parts for integrals. In this unit, you will understand
the concept of integration by parts with their examples as well.

3.1 Integration by Parts

This is a method depending on the product rule for differentiation, for articulating one integral
in provisions of another. It is mostly functional for integrating functions that are products of
two types of functions: like power times an exponent, and functions including logarithms.

d

g
x
x
x
f
x

If f(x) and g(x) be two given functions of x we know that  ( ). ( )f x g x  f ( ). ( ) g ( ). ( )
dx
Hence, by definition
1
x
x
x
g
x

x
x
g
f ( ) ( )   f ( ). ( )dx   f 1 ( ). ( )d
or f ( . 1( )dx  f ( ). ( )   f 1( ). ( )dx
g

x
g
x
x
x
x
g
x
Notes To apply this formula, the integrand should be expressible as the product of two
functions such that one of them can be easily integrated. This is taken as the second function.
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