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File: Indefinite Integral Pdf 171683 | Unit 3
unit 3 integral calculus integral calculus structure 3 1 introduction objectives 3 2 antiderivatives 3 3 basic definitions 3 3 1 standard integrals 3 3 2 algebra of integrals 3 ...

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                  UNIT 3  INTEGRAL CALCULUS                                                                                Integral Calculus 
                  Structure 
                         3.1    Introduction 
                                Objectives 
                         3.2    Antiderivatives 
                         3.3    Basic Definitions 
                                3.3.1   Standard Integrals 
                                3.3.2   Algebra of Integrals 
                         3.4    Methods of Integration 
                                3.4.1   Integration by Substitution 
                                3.4.2   Integration by Parts 
                         3.5    Integration of Rational Functions 
                                3.5.1   Some Simple Rational Functions 
                                3.5.2   Partial Fraction Decomposition 
                         3.6    Integration of Irrational Functions 
                         3.7    Integration of Trigonometric Functions 
                         3.8    Definite Integrals 
                         3.9    Fundamental Theorem of Calculus 
                                3.9.1   Area Function 
                                3.9.2   First Fundamental Theorem of Integral Calculus 
                                3.9.3   Second Fundamental Theorem of Integral Calculus 
                                3.9.4   Evaluation of a Definite Integral by Substitution 
                         3.10  Properties of Definite Integrals 
                         3.11  Applications 
                         3.12  Summary 
                         3.13  Answers to SAQs 
                  3.1  INTRODUCTION 
                  In this unit, we shall introduce the notions of antiderivatives, indefinite integral and various 
                  methods and techniques of integration. The unit will also cover definite integrals which 
                  can be evaluated using these methods. 
                  We have seen in Unit 2 that one of the problems which motivated the concept of a 
                  derivative was a geometrical one – that of finding a tangent to a curve at a point. The 
                  concept of integration was also similarly motivated by a geometrical problem – that of 
                  finding the areas of plane regions enclosed by curves. Some recently discovered Egyptian 
                  manuscripts reveal that the formulas for finding the areas of triangles and rectangles were 
                  known even in 1800 BC. Using these formulas, one could also find the area of any figure 
                  bounded by straight line segments. But no method for finding the area of figures bounded 
                  by curves had evolved till much later. 
                  In the third century BC, Archimedes was successful in rigorously proving the formula for 
                  the area of a circle. His solution contained the seeds of the present day integral calculus. 
                  But it was only later, in the seventeenth century, that Newton and Leibniz were able to                              119 
                                               
                                               
               Calculus : Basic Concepts      generalize. Archimedes’ method and also to establish the link between differential and 
                                              integral calculus. The definition of the definite integral of a function, which we shall give 
                                              in this unit, was first given by Riemann in 1854. We will also acquaint you with various 
                                              application of integration. 
                                              Objectives 
                                              After studying this unit, you should be able to 
                                                     •      compute the antiderivative of a given function, 
                                                     •      define the indefinite integral of a function, 
                                                     •      evaluate certain standard integrals by finding the antiderivatives of the 
                                                            integrals, 
                                                     •      compute integrals of various elementary and trigonometric functions, 
                                                     •      integrate rational functions of a variable by using the method of partial 
                                                            fractions, 
                                                     •      evaluate the integrals of some specified types of irrational functions, 
                                                     •      define the definite integral of a given function as a limit of a sum, 
                                                     •      state the fundamental theorems of calculus, 
                                                     •      learn the different properties of definite integral, 
                                                     •      use the fundamental theorems to calculate the definite integral of an 
                                                            integrable function, and 
                                                     •      use the definite integrals to evaluate areas of figures bounded by curves. 
                                              3.2  ANTIDERIVATIVES 
                                              In Unit 1, we have been occupied with the problem of finding the derivative of a given 
                                              function. Some of the important applications of the calculus lead to the inverse problem, 
                                              namely, given the derivative of a function, is it possible to find the function? This process 
                                              is called antidifferentiation and the result of antidifferentiation is called an 
                                              antiderivative. The importance of the antiderivative results partly from the fact that 
                                              scientific laws often specify the rates of change of quantities. The quantities themselves 
                                              are then found by antidifferentiation. 
                                              To get started, suppose we are given that f ′ (x) = 9, can we find f (x)? It is easy to see 
                                              that one such function f is given by f (x) = 9x, since the derivative of 9x is 9. 
                                              Before making any definite decision, consider the functions 
                                                                         9x + 4, 9x − 10, 9x +     3 
                                              Each of these functions has 9 as its derivative. Thus, not only can f (x) be 9x, but it can 
                                              also be  9x + 4 or 9x − 10, 9x +      3. Not enough information is given to help us 
                                              determine which is the correct answer. 
                                              Let us look at each of these possible functions a bit more carefully. We notice that each 
                                              of these functions differs from another only by a constant. Therefore, we can say that if 
                                              f ′ (x) = 9, then f (x) must be of the form f (x) = 9x + c, where c is a constant. We call 
                                              9x + c the antiderivative of 9. 
                                              More generally, we have the following definition. 
                                              Definition 
                 120 
                                                   
                           
                                                                                                                                                                                           
                                   Suppose f is a given function. Then a function F is called an antiderivative of                                                              Integral Calculus 
                                   f, if F′(x) = f (x) ∀x . 
                          We now state an important theorem without giving its proof. 
                          Theorem 1 
                                   If F  and F  are two antiderivatives of the same function, then F  and F  
                                         1           2                                                                                        1          2
                                   differ by a constant, that is 
                                                               F (x)= F (x) +c 
                                                                  1            2
                          Remark 
                                   From above Theorem, it follows that we can find all the antiderivatives of a given 
                                   function, once we know one antiderivative of it. For instance, in the above example, 
                                   since one antiderivative of 9 is 9x, all antiderivative of 9 have the form 9x + c, 
                                   where c is a constant. Let us do one example. 
                          Example 3.1 
                                   Find all the antiderivatives of 4x. 
                          Solution 
                                   We have to look for a function F such that  F′(x) = 4x. Now, an antiderivative of 
                                               2                                                                                             2
                                   4x is 2x . Thus, by Theorem 1, all antiderivatives of 4x are given by 2x  + c, where 
                                   c is a constant. 
                          SAQ 1 
                                   Find all the antiderivatives of each of the following function 
                                   (i)       f (x) = 10x 
                                                             10
                                   (ii)      f (x) = 11x  
                                   (iii)     f (x) = − 5x 
                           
                           
                           
                           
                           
                           
                          3.3  BASIC DEFINITIONS 
                          We have seen, that the antiderivative of a function is not unique. More precisely, we have 
                          seen that if a function F is an antiderivative of a function f, then F + c is also an 
                          antiderivative of f, where c is any arbitrary constant. Now we shall introduce a notation 
                          here : we shall use the symbol  ∫ f (x) dx to denote the class of all antiderivatives of f. 
                          We call it the indefinite integral or just the integral of f. Thus, if F (x) is an antiderivative 
                          of f (x), then we can write  ∫ f (x) dx = F (x) + c . 
                          Here c is called the constant of integration. The function f (x) is called the integrand, 
                          f (x) dx is called the element of integration and the symbol ∫ stands for the integral sign. 
                                                                                                                                                                                                 121 
                                               
                                               
               Calculus : Basic Concepts      The indefinite integral  ∫ f (x) dx is a class of functions which differ from one another by 
                
                                              constant. It is not a definite number; it is not even a definite function. We say that the 
                                              indefinite integral is unique up to an arbitrary constant. 
                                              Thus, having defined an indefinite integral, let us get acquainted with the various 
                                              techniques for evaluating integrals. 
                                              3.3.1  Standard Integrals 
                                              We give below some elementary standard integrals which can be obtained directly from 
                                              our knowledge of derivatives. 
                                                                                         Table 3.1 
                                                           Sl. No.             Function                          Integral 
                                                                                     n                        n+1
                                                               1                   x                         x     + c, n ≠ − 1 
                                                                                                             n+1
                                                               2                  sin x                         − cos x + c 
                                                               3                  cos x                          sin x + c 
                                                                                     2
                                                               4                 sec  x                          tan x + c 
                                                                                      2
                                                               5                cosec  x                        − cot x + c 
                                                               6               sec x tan x                       sec x + c 
                                                               7             cosec x cot x                    − cosec x + c 
                                                               8                    1                 sin−1 x + c or − cos−1 x + c  
                                                                                  1− x2
                                                               9                    1                     −1                 −1        
                                                                                                      tan    x + c or − cot     x +c
                                                                                 1+x2
                                                              10                    1                sec−1 x + c or − cosec−1 x + c  
                                                                               x   x2 −1
                                                              11                    1                           ln | x | + c 
                                                                                    x
                                                                                     x                              x
                                                              12                   e                               e  + c 
                                                                                     x                              x
                                                              13                   a                              a    + c  
                                                                                                                 ln |a|
                                              Now let us see how to evaluate some functions which are linear combination of the 
                                              functions listed in Table 3.1. 
                                              3.3.2  Algebra of Integrals 
                                              You are familiar with the rule for differential of sum of functions, which says 
                                                                 d [a f (x) + bg(x)] = a d [ f (x)] + b d [g(x)] 
                                                                 dx                        dx              dx
                                              There is a similar rule for integration : 
                                              Rule 1 
                 122 
                                                   
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...Unit integral calculus structure introduction objectives antiderivatives basic definitions standard integrals algebra of methods integration by substitution parts rational functions some simple partial fraction decomposition irrational trigonometric definite fundamental theorem area function first second evaluation a properties applications summary answers to saqs in this we shall introduce the notions indefinite and various techniques will also cover which can be evaluated using these have seen that one problems motivated concept derivative was geometrical finding tangent curve at point similarly problem areas plane regions enclosed curves recently discovered egyptian manuscripts reveal formulas for triangles rectangles were known even bc could find any figure bounded straight line segments but no method figures had evolved till much later third century archimedes successful rigorously proving formula circle his solution contained seeds present day it only seventeenth newton leibniz a...

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