Lecture 6 - Improper Integrals | Calculus - Engineering Mathematics PDF Download

 Page 1


Improper Integrals 
 
Institute of Lifelong Learning, University of Delhi                                                     pg. 1 
 
 
 
 
 
 
 
 
 
 
Paper: Calculus 
Lesson: Improper Integrals 
Lesson Developer: Rahul Tomar, Chandra Prakash 
College/ Department: Assistant Professor, Department of 
Mathematics, Shyamlal College, University of Delhi 
 
 
 
 
 
 
 
 
 
 
  
Page 2


Improper Integrals 
 
Institute of Lifelong Learning, University of Delhi                                                     pg. 1 
 
 
 
 
 
 
 
 
 
 
Paper: Calculus 
Lesson: Improper Integrals 
Lesson Developer: Rahul Tomar, Chandra Prakash 
College/ Department: Assistant Professor, Department of 
Mathematics, Shyamlal College, University of Delhi 
 
 
 
 
 
 
 
 
 
 
  
Improper Integrals 
 
Institute of Lifelong Learning, University of Delhi                                                     pg. 2 
 
Table of Contents:    
Chapter : Improper Integrals 
 1 : Learning Outcomes        
 2 : Introduction                                
 3 : Classification of Improper Integrals       
  3.1 : Improper Integrals of Type-I                                                                     
          3.2 : Improper Integrals of Type-II   
 4 :Tests for convergence or divergence of improper  
  4.1 : Direct Comparison Test  
  4.2 : Limit Comparison Test  
  4.3 : Useful Comparison integral(for finite interval)  
 5 : Tests for convergence or divergence at Infinity 
  5.1 : Direct Comparison Test(at Infinity)  
  5.2 : Limit Comparison Test(at Infinity)  
  5.3 : Useful Comparison Integral(for infinite interval)  
 6 : Test for convergence (when integrand changes sign) 
  6.1 : Cauchy’s Test (over finite range of integration)  
  6.2 : Cauchy’s Test (over infinite range of integration) 
  6.3 : Absolute and conditional convergence.  
 7 : Tests for convergence (when integrand is a product of two   
  functions) 
  7.1 : Abel’s Test.  
Page 3


Improper Integrals 
 
Institute of Lifelong Learning, University of Delhi                                                     pg. 1 
 
 
 
 
 
 
 
 
 
 
Paper: Calculus 
Lesson: Improper Integrals 
Lesson Developer: Rahul Tomar, Chandra Prakash 
College/ Department: Assistant Professor, Department of 
Mathematics, Shyamlal College, University of Delhi 
 
 
 
 
 
 
 
 
 
 
  
Improper Integrals 
 
Institute of Lifelong Learning, University of Delhi                                                     pg. 2 
 
Table of Contents:    
Chapter : Improper Integrals 
 1 : Learning Outcomes        
 2 : Introduction                                
 3 : Classification of Improper Integrals       
  3.1 : Improper Integrals of Type-I                                                                     
          3.2 : Improper Integrals of Type-II   
 4 :Tests for convergence or divergence of improper  
  4.1 : Direct Comparison Test  
  4.2 : Limit Comparison Test  
  4.3 : Useful Comparison integral(for finite interval)  
 5 : Tests for convergence or divergence at Infinity 
  5.1 : Direct Comparison Test(at Infinity)  
  5.2 : Limit Comparison Test(at Infinity)  
  5.3 : Useful Comparison Integral(for infinite interval)  
 6 : Test for convergence (when integrand changes sign) 
  6.1 : Cauchy’s Test (over finite range of integration)  
  6.2 : Cauchy’s Test (over infinite range of integration) 
  6.3 : Absolute and conditional convergence.  
 7 : Tests for convergence (when integrand is a product of two   
  functions) 
  7.1 : Abel’s Test.  
Improper Integrals 
 
Institute of Lifelong Learning, University of Delhi                                                     pg. 3 
 
  7.2: Dirichlet’s Test 
 8: Exercises  
 9: References for further readings.                                              
1. Learning Outcome  
After reading this chapter you will be able to understand :  
? the difference between proper and improper integral.    
? the meaning of convergence or divergence of an integral.  
? types of improper integrals.    
? How to integrate a continuous(or, discontinuous) function over finite 
(or, infinite) interval of integration.   
? different tests for the convergence or divergence of improper integrals. 
2. Introduction 
Till now we are familiar with integration of continuous functions over finite 
interval of integration. Immediately the next obvious question in one’s mind 
is “Is it possible to integrate a discontinuous function over unbounded 
interval of integration”. The answer of this relevant question is mathematical 
term “Improper Integral”. Before discussing improper integrals further let’s 
make you familiar with some basic definitions.    
Definition 1.1: Bounded or finite interval: If        such that     then 
any one of the interval                         is said to be finite if     is a 
unique finite number. 
Definition 1.2: Unbounded or infinite interval: An interval from       , such 
that    , is said to be infinite if either                both.  
For example                    .                                                           
Page 4


Improper Integrals 
 
Institute of Lifelong Learning, University of Delhi                                                     pg. 1 
 
 
 
 
 
 
 
 
 
 
Paper: Calculus 
Lesson: Improper Integrals 
Lesson Developer: Rahul Tomar, Chandra Prakash 
College/ Department: Assistant Professor, Department of 
Mathematics, Shyamlal College, University of Delhi 
 
 
 
 
 
 
 
 
 
 
  
Improper Integrals 
 
Institute of Lifelong Learning, University of Delhi                                                     pg. 2 
 
Table of Contents:    
Chapter : Improper Integrals 
 1 : Learning Outcomes        
 2 : Introduction                                
 3 : Classification of Improper Integrals       
  3.1 : Improper Integrals of Type-I                                                                     
          3.2 : Improper Integrals of Type-II   
 4 :Tests for convergence or divergence of improper  
  4.1 : Direct Comparison Test  
  4.2 : Limit Comparison Test  
  4.3 : Useful Comparison integral(for finite interval)  
 5 : Tests for convergence or divergence at Infinity 
  5.1 : Direct Comparison Test(at Infinity)  
  5.2 : Limit Comparison Test(at Infinity)  
  5.3 : Useful Comparison Integral(for infinite interval)  
 6 : Test for convergence (when integrand changes sign) 
  6.1 : Cauchy’s Test (over finite range of integration)  
  6.2 : Cauchy’s Test (over infinite range of integration) 
  6.3 : Absolute and conditional convergence.  
 7 : Tests for convergence (when integrand is a product of two   
  functions) 
  7.1 : Abel’s Test.  
Improper Integrals 
 
Institute of Lifelong Learning, University of Delhi                                                     pg. 3 
 
  7.2: Dirichlet’s Test 
 8: Exercises  
 9: References for further readings.                                              
1. Learning Outcome  
After reading this chapter you will be able to understand :  
? the difference between proper and improper integral.    
? the meaning of convergence or divergence of an integral.  
? types of improper integrals.    
? How to integrate a continuous(or, discontinuous) function over finite 
(or, infinite) interval of integration.   
? different tests for the convergence or divergence of improper integrals. 
2. Introduction 
Till now we are familiar with integration of continuous functions over finite 
interval of integration. Immediately the next obvious question in one’s mind 
is “Is it possible to integrate a discontinuous function over unbounded 
interval of integration”. The answer of this relevant question is mathematical 
term “Improper Integral”. Before discussing improper integrals further let’s 
make you familiar with some basic definitions.    
Definition 1.1: Bounded or finite interval: If        such that     then 
any one of the interval                         is said to be finite if     is a 
unique finite number. 
Definition 1.2: Unbounded or infinite interval: An interval from       , such 
that    , is said to be infinite if either                both.  
For example                    .                                                           
Improper Integrals 
 
Institute of Lifelong Learning, University of Delhi                                                     pg. 4 
 
Definition 1.3: Integrands: An integrand is that function which is to be 
integrated.                                                                                                              
i.e. In         
 
 
         is an integrand.                                                                                   
Definition 1.4: Vertical Asymptote or Infinite discontinuity: An integrand is 
said to have infinite discontinuity at a point if it becomes unbounded at that 
point.  
For example:         
 
 
 has infinite discontinuity at    , and      
 
      
 
has infinite discontinuities at                                                                                                                                           
Definition 1.5: An integral      
 
 
 is said to be proper if it satisfies the 
following two properties:                        
 (i)       is finite, ?     .             
 (ii)      is a finite number.  
For example:   
 
   
   
 
 
,  
     
 
 
 
   are proper integrals.                                  
Definition 1.6: Improper Integral: The integral      
 
 
   is said to be 
improper if it is not proper i.e. it doesn’t satisfies either one or both the 
properties mentioned in the above definition of proper integral.   
For example:  
 
      
 
  
    
 
 
 
 
    
 
    
 
  
   are proper integrals.  
Value addition: 
(i) Proper integrals represent area of a ‘closed region’.    
(ii) Improper integrals represent area of an ‘open region’. 
                                                                                
 
 
Page 5


Improper Integrals 
 
Institute of Lifelong Learning, University of Delhi                                                     pg. 1 
 
 
 
 
 
 
 
 
 
 
Paper: Calculus 
Lesson: Improper Integrals 
Lesson Developer: Rahul Tomar, Chandra Prakash 
College/ Department: Assistant Professor, Department of 
Mathematics, Shyamlal College, University of Delhi 
 
 
 
 
 
 
 
 
 
 
  
Improper Integrals 
 
Institute of Lifelong Learning, University of Delhi                                                     pg. 2 
 
Table of Contents:    
Chapter : Improper Integrals 
 1 : Learning Outcomes        
 2 : Introduction                                
 3 : Classification of Improper Integrals       
  3.1 : Improper Integrals of Type-I                                                                     
          3.2 : Improper Integrals of Type-II   
 4 :Tests for convergence or divergence of improper  
  4.1 : Direct Comparison Test  
  4.2 : Limit Comparison Test  
  4.3 : Useful Comparison integral(for finite interval)  
 5 : Tests for convergence or divergence at Infinity 
  5.1 : Direct Comparison Test(at Infinity)  
  5.2 : Limit Comparison Test(at Infinity)  
  5.3 : Useful Comparison Integral(for infinite interval)  
 6 : Test for convergence (when integrand changes sign) 
  6.1 : Cauchy’s Test (over finite range of integration)  
  6.2 : Cauchy’s Test (over infinite range of integration) 
  6.3 : Absolute and conditional convergence.  
 7 : Tests for convergence (when integrand is a product of two   
  functions) 
  7.1 : Abel’s Test.  
Improper Integrals 
 
Institute of Lifelong Learning, University of Delhi                                                     pg. 3 
 
  7.2: Dirichlet’s Test 
 8: Exercises  
 9: References for further readings.                                              
1. Learning Outcome  
After reading this chapter you will be able to understand :  
? the difference between proper and improper integral.    
? the meaning of convergence or divergence of an integral.  
? types of improper integrals.    
? How to integrate a continuous(or, discontinuous) function over finite 
(or, infinite) interval of integration.   
? different tests for the convergence or divergence of improper integrals. 
2. Introduction 
Till now we are familiar with integration of continuous functions over finite 
interval of integration. Immediately the next obvious question in one’s mind 
is “Is it possible to integrate a discontinuous function over unbounded 
interval of integration”. The answer of this relevant question is mathematical 
term “Improper Integral”. Before discussing improper integrals further let’s 
make you familiar with some basic definitions.    
Definition 1.1: Bounded or finite interval: If        such that     then 
any one of the interval                         is said to be finite if     is a 
unique finite number. 
Definition 1.2: Unbounded or infinite interval: An interval from       , such 
that    , is said to be infinite if either                both.  
For example                    .                                                           
Improper Integrals 
 
Institute of Lifelong Learning, University of Delhi                                                     pg. 4 
 
Definition 1.3: Integrands: An integrand is that function which is to be 
integrated.                                                                                                              
i.e. In         
 
 
         is an integrand.                                                                                   
Definition 1.4: Vertical Asymptote or Infinite discontinuity: An integrand is 
said to have infinite discontinuity at a point if it becomes unbounded at that 
point.  
For example:         
 
 
 has infinite discontinuity at    , and      
 
      
 
has infinite discontinuities at                                                                                                                                           
Definition 1.5: An integral      
 
 
 is said to be proper if it satisfies the 
following two properties:                        
 (i)       is finite, ?     .             
 (ii)      is a finite number.  
For example:   
 
   
   
 
 
,  
     
 
 
 
   are proper integrals.                                  
Definition 1.6: Improper Integral: The integral      
 
 
   is said to be 
improper if it is not proper i.e. it doesn’t satisfies either one or both the 
properties mentioned in the above definition of proper integral.   
For example:  
 
      
 
  
    
 
 
 
 
    
 
    
 
  
   are proper integrals.  
Value addition: 
(i) Proper integrals represent area of a ‘closed region’.    
(ii) Improper integrals represent area of an ‘open region’. 
                                                                                
 
 
Improper Integrals 
 
Institute of Lifelong Learning, University of Delhi                                                     pg. 5 
 
3. Classification of Improper Integrals:      
Based on the behavior (boundedness or unboundedness) of integrand over 
finite or infinite interval of integration we can broadly classify improper 
integrals into two types:                                                                                      
3.1 Improper Integrals of Type-I: 
Type-I Improper integrals are those in which integrand is bounded while 
interval of integration is unbounded. It can have three forms:                                                                                           
(a) If                             , is continuos, then                                           
                               
   
        
 
 
 
 
                                                                                         
(b) If                             , is continuous then                                       
                               
    
        
 
 
 
  
 
(c)                      , is continuos then                           
                              
 
  
         
 
  
        
 
 
 where       .  
                      
 
    
         
 
 
 
   
 
   
       
 
 
 
.     
If the limits on R.H.S of above (a),(b),(c) exists finitely then only the 
corresponding integrals on L.H.S exists and said to be convergent otherwise 
divergent. 
Value addition: 
Convergent improper integral        
 
 
 means that the area under the curve 
    over the given interval of integration is finite while divergent improper 
integral means that the area is infinite or doesn’t exist at all. 
                                 
Example 1: Discuss the convergence or divergence of 
 
  
  
 
 
.                                                 
Solution: Let