Page 1 INTEGRALS 287 vJust as a mountaineer climbs a mountain â€“ because it is there, so a good mathematics student studies new material because it is there. â€” JAMES B. BRISTOL v 7.1 Introduction Differential Calculus is centred 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 ' exists at each point of I, then a natural question arises that given f '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 arise 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. Chapter 7 INTEGRALS G .W. Leibnitz (1646 -1716) Page 2 INTEGRALS 287 vJust as a mountaineer climbs a mountain â€“ because it is there, so a good mathematics student studies new material because it is there. â€” JAMES B. BRISTOL v 7.1 Introduction Differential Calculus is centred 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 ' exists at each point of I, then a natural question arises that given f '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 arise 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. Chapter 7 INTEGRALS G .W. Leibnitz (1646 -1716) 288 MATHEMATICS There is a connection, known as the Fundamental Theorem of Calculus, between indefinite integral and definite integral which makes the definite integral as a practical tool for science and engineering. The definite integral is also used to solve many interesting problems from various disciplines like economics, finance and probability. In this Chapter, we shall confine ourselves to the study of indefinite and definite integrals and their elementary properties including some techniques of integration. 7.2 Integration as an Inverse Process of Differentiation Integration is the inverse process of differentiation. Instead of differentiating a function, we are given the derivative of a function and asked to find its primitive, i.e., the original function. Such a process is called integration or anti differentiation. Let us consider the following examples: We know that (sin ) d x dx = cos x ... (1) 3 ( ) 3 d x dx = x 2 ... (2) and ( ) x d e dx = e x ... (3) W e observe that in (1), the function cos x is the derived function of sin x. We say that sin x is an anti derivative (or an integral) of cos x. Similarly, in (2) and (3), 3 3 x and e x are the anti derivatives (or integrals) of x 2 and e x , respectively. Again, we note that for any real number C, treated as constant function, its derivative is zero and hence, we can write (1), (2) and (3) as follows : (sin + C) cos = d x x dx , 3 2 ( + C) 3 = d x x dx and ( + C) = x x d e e dx Thus, anti derivatives (or integrals) of the above cited functions are not unique. Actually, there exist infinitely many anti derivatives of each of these functions which can be obtained by choosing C arbitrarily from the set of real numbers. For this reason C is customarily referred to as arbitrary constant. In fact, C is the parameter by varying which one gets different anti derivatives (or integrals) of the given function. More generally, if there is a function F such that F ( ) = ( ) d x f x dx , ? x ? I (interval), then for any arbitrary real number C, (also called constant of integration) [ ] F ( ) + C d x dx = f (x), x ? I Page 3 INTEGRALS 287 vJust as a mountaineer climbs a mountain â€“ because it is there, so a good mathematics student studies new material because it is there. â€” JAMES B. BRISTOL v 7.1 Introduction Differential Calculus is centred 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 ' exists at each point of I, then a natural question arises that given f '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 arise 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. Chapter 7 INTEGRALS G .W. Leibnitz (1646 -1716) 288 MATHEMATICS There is a connection, known as the Fundamental Theorem of Calculus, between indefinite integral and definite integral which makes the definite integral as a practical tool for science and engineering. The definite integral is also used to solve many interesting problems from various disciplines like economics, finance and probability. In this Chapter, we shall confine ourselves to the study of indefinite and definite integrals and their elementary properties including some techniques of integration. 7.2 Integration as an Inverse Process of Differentiation Integration is the inverse process of differentiation. Instead of differentiating a function, we are given the derivative of a function and asked to find its primitive, i.e., the original function. Such a process is called integration or anti differentiation. Let us consider the following examples: We know that (sin ) d x dx = cos x ... (1) 3 ( ) 3 d x dx = x 2 ... (2) and ( ) x d e dx = e x ... (3) W e observe that in (1), the function cos x is the derived function of sin x. We say that sin x is an anti derivative (or an integral) of cos x. Similarly, in (2) and (3), 3 3 x and e x are the anti derivatives (or integrals) of x 2 and e x , respectively. Again, we note that for any real number C, treated as constant function, its derivative is zero and hence, we can write (1), (2) and (3) as follows : (sin + C) cos = d x x dx , 3 2 ( + C) 3 = d x x dx and ( + C) = x x d e e dx Thus, anti derivatives (or integrals) of the above cited functions are not unique. Actually, there exist infinitely many anti derivatives of each of these functions which can be obtained by choosing C arbitrarily from the set of real numbers. For this reason C is customarily referred to as arbitrary constant. In fact, C is the parameter by varying which one gets different anti derivatives (or integrals) of the given function. More generally, if there is a function F such that F ( ) = ( ) d x f x dx , ? x ? I (interval), then for any arbitrary real number C, (also called constant of integration) [ ] F ( ) + C d x dx = f (x), x ? I INTEGRALS 289 Thus, {F + C, C ? R} denotes a family of anti derivatives of f. Remark Functions with same derivatives differ by a constant. To show this, let g and h be two functions having the same derivatives on an interval I. Consider the function f = g â€“ h defined by f (x) = g(x) â€“ h(x), ? x ? I Then df dx = f' = g ' â€“ h' giving f' (x) = g' (x) â€“ h' (x) ? x ? I or f' (x) = 0, ? x ? I by hypothesis, i.e., the rate of change of f with respect to x is zero on I and hence f is constant. In view of the above remark, it is justified to infer that the family {F + C, C ? R} provides all possible anti derivatives of f. We introduce a new symbol, namely, ( ) f x dx ? which will represent the entire class of anti derivatives read as the indefinite integral of f with respect to x. Symbolically, we write ( ) = F ( ) + C f x dx x ? . Notation Given that ( ) dy f x dx = , we write y = ( ) f x dx ? . For the sake of convenience, we mention below the following symbols/terms/phrases with their meanings as given in the Table (7.1). T able 7.1 Symbols/Terms/Phrases Meaning ( ) f x dx ? Integral of f with respect to x f (x) in ( ) f x dx ? Integrand x in ( ) f x dx ? Variable of integration Integrate Find the integral An integral of f A function F such that F'(x) = f (x) Integration The process of finding the integral Constant of Integration Any real number C, considered as constant function Page 4 INTEGRALS 287 vJust as a mountaineer climbs a mountain â€“ because it is there, so a good mathematics student studies new material because it is there. â€” JAMES B. BRISTOL v 7.1 Introduction Differential Calculus is centred 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 ' exists at each point of I, then a natural question arises that given f '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 arise 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. Chapter 7 INTEGRALS G .W. Leibnitz (1646 -1716) 288 MATHEMATICS There is a connection, known as the Fundamental Theorem of Calculus, between indefinite integral and definite integral which makes the definite integral as a practical tool for science and engineering. The definite integral is also used to solve many interesting problems from various disciplines like economics, finance and probability. In this Chapter, we shall confine ourselves to the study of indefinite and definite integrals and their elementary properties including some techniques of integration. 7.2 Integration as an Inverse Process of Differentiation Integration is the inverse process of differentiation. Instead of differentiating a function, we are given the derivative of a function and asked to find its primitive, i.e., the original function. Such a process is called integration or anti differentiation. Let us consider the following examples: We know that (sin ) d x dx = cos x ... (1) 3 ( ) 3 d x dx = x 2 ... (2) and ( ) x d e dx = e x ... (3) W e observe that in (1), the function cos x is the derived function of sin x. We say that sin x is an anti derivative (or an integral) of cos x. Similarly, in (2) and (3), 3 3 x and e x are the anti derivatives (or integrals) of x 2 and e x , respectively. Again, we note that for any real number C, treated as constant function, its derivative is zero and hence, we can write (1), (2) and (3) as follows : (sin + C) cos = d x x dx , 3 2 ( + C) 3 = d x x dx and ( + C) = x x d e e dx Thus, anti derivatives (or integrals) of the above cited functions are not unique. Actually, there exist infinitely many anti derivatives of each of these functions which can be obtained by choosing C arbitrarily from the set of real numbers. For this reason C is customarily referred to as arbitrary constant. In fact, C is the parameter by varying which one gets different anti derivatives (or integrals) of the given function. More generally, if there is a function F such that F ( ) = ( ) d x f x dx , ? x ? I (interval), then for any arbitrary real number C, (also called constant of integration) [ ] F ( ) + C d x dx = f (x), x ? I INTEGRALS 289 Thus, {F + C, C ? R} denotes a family of anti derivatives of f. Remark Functions with same derivatives differ by a constant. To show this, let g and h be two functions having the same derivatives on an interval I. Consider the function f = g â€“ h defined by f (x) = g(x) â€“ h(x), ? x ? I Then df dx = f' = g ' â€“ h' giving f' (x) = g' (x) â€“ h' (x) ? x ? I or f' (x) = 0, ? x ? I by hypothesis, i.e., the rate of change of f with respect to x is zero on I and hence f is constant. In view of the above remark, it is justified to infer that the family {F + C, C ? R} provides all possible anti derivatives of f. We introduce a new symbol, namely, ( ) f x dx ? which will represent the entire class of anti derivatives read as the indefinite integral of f with respect to x. Symbolically, we write ( ) = F ( ) + C f x dx x ? . Notation Given that ( ) dy f x dx = , we write y = ( ) f x dx ? . For the sake of convenience, we mention below the following symbols/terms/phrases with their meanings as given in the Table (7.1). T able 7.1 Symbols/Terms/Phrases Meaning ( ) f x dx ? Integral of f with respect to x f (x) in ( ) f x dx ? Integrand x in ( ) f x dx ? Variable of integration Integrate Find the integral An integral of f A function F such that F'(x) = f (x) Integration The process of finding the integral Constant of Integration Any real number C, considered as constant function 290 MATHEMATICS We already know the formulae for the derivatives of many important functions. From these formulae, we can write down immediately the corresponding formulae (referred to as standard formulae) for the integrals of these functions, as listed below which will be used to find integrals of other functions. Derivatives Integrals (Anti derivatives) (i) 1 1 n n d x x dx n + ? ? = ? ? + ? ? ; 1 C 1 n n x x dx n + = + + ? , n ? â€“1 Particularly, we note that ( ) 1 d x dx = ; C dx x = + ? (ii) ( ) sin cos d x x dx = ; cos sin C x dx x = + ? (iii) ( ) â€“ cos sin d x x dx = ; sin cos C x dx â€“ x = + ? (iv) ( ) 2 tan sec d x x dx = ; 2 sec tan C x dx x = + ? (v) ( ) 2 â€“ cot cosec d x x dx = ; 2 cosec cot C x dx â€“ x = + ? (vi) ( ) sec sec tan d x x x dx = ; sec tan sec C x x dx x = + ? (vii) ( ) â€“ cosec cosec cot d x x x dx = ; cosec cot â€“ cosec C x x dx x = + ? (viii) ( ) â€“ 1 2 1 sin 1 d x dx â€“ x = ; â€“ 1 2 sin C 1 dx x â€“ x = + ? (ix) ( ) â€“ 1 2 1 â€“ cos 1 d x dx â€“ x = ; â€“ 1 2 cos C 1 dx â€“ x â€“ x = + ? (x) ( ) â€“ 1 2 1 tan 1 d x dx x = + ; â€“ 1 2 tan C 1 dx x x = + + ? (xi) ( ) â€“ 1 2 1 â€“ cot 1 d x dx x = + ; â€“ 1 2 cot C 1 dx â€“ x x = + + ? Page 5 INTEGRALS 287 vJust as a mountaineer climbs a mountain â€“ because it is there, so a good mathematics student studies new material because it is there. â€” JAMES B. BRISTOL v 7.1 Introduction Differential Calculus is centred 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 ' exists at each point of I, then a natural question arises that given f '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 arise 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. Chapter 7 INTEGRALS G .W. Leibnitz (1646 -1716) 288 MATHEMATICS There is a connection, known as the Fundamental Theorem of Calculus, between indefinite integral and definite integral which makes the definite integral as a practical tool for science and engineering. The definite integral is also used to solve many interesting problems from various disciplines like economics, finance and probability. In this Chapter, we shall confine ourselves to the study of indefinite and definite integrals and their elementary properties including some techniques of integration. 7.2 Integration as an Inverse Process of Differentiation Integration is the inverse process of differentiation. Instead of differentiating a function, we are given the derivative of a function and asked to find its primitive, i.e., the original function. Such a process is called integration or anti differentiation. Let us consider the following examples: We know that (sin ) d x dx = cos x ... (1) 3 ( ) 3 d x dx = x 2 ... (2) and ( ) x d e dx = e x ... (3) W e observe that in (1), the function cos x is the derived function of sin x. We say that sin x is an anti derivative (or an integral) of cos x. Similarly, in (2) and (3), 3 3 x and e x are the anti derivatives (or integrals) of x 2 and e x , respectively. Again, we note that for any real number C, treated as constant function, its derivative is zero and hence, we can write (1), (2) and (3) as follows : (sin + C) cos = d x x dx , 3 2 ( + C) 3 = d x x dx and ( + C) = x x d e e dx Thus, anti derivatives (or integrals) of the above cited functions are not unique. Actually, there exist infinitely many anti derivatives of each of these functions which can be obtained by choosing C arbitrarily from the set of real numbers. For this reason C is customarily referred to as arbitrary constant. In fact, C is the parameter by varying which one gets different anti derivatives (or integrals) of the given function. More generally, if there is a function F such that F ( ) = ( ) d x f x dx , ? x ? I (interval), then for any arbitrary real number C, (also called constant of integration) [ ] F ( ) + C d x dx = f (x), x ? I INTEGRALS 289 Thus, {F + C, C ? R} denotes a family of anti derivatives of f. Remark Functions with same derivatives differ by a constant. To show this, let g and h be two functions having the same derivatives on an interval I. Consider the function f = g â€“ h defined by f (x) = g(x) â€“ h(x), ? x ? I Then df dx = f' = g ' â€“ h' giving f' (x) = g' (x) â€“ h' (x) ? x ? I or f' (x) = 0, ? x ? I by hypothesis, i.e., the rate of change of f with respect to x is zero on I and hence f is constant. In view of the above remark, it is justified to infer that the family {F + C, C ? R} provides all possible anti derivatives of f. We introduce a new symbol, namely, ( ) f x dx ? which will represent the entire class of anti derivatives read as the indefinite integral of f with respect to x. Symbolically, we write ( ) = F ( ) + C f x dx x ? . Notation Given that ( ) dy f x dx = , we write y = ( ) f x dx ? . For the sake of convenience, we mention below the following symbols/terms/phrases with their meanings as given in the Table (7.1). T able 7.1 Symbols/Terms/Phrases Meaning ( ) f x dx ? Integral of f with respect to x f (x) in ( ) f x dx ? Integrand x in ( ) f x dx ? Variable of integration Integrate Find the integral An integral of f A function F such that F'(x) = f (x) Integration The process of finding the integral Constant of Integration Any real number C, considered as constant function 290 MATHEMATICS We already know the formulae for the derivatives of many important functions. From these formulae, we can write down immediately the corresponding formulae (referred to as standard formulae) for the integrals of these functions, as listed below which will be used to find integrals of other functions. Derivatives Integrals (Anti derivatives) (i) 1 1 n n d x x dx n + ? ? = ? ? + ? ? ; 1 C 1 n n x x dx n + = + + ? , n ? â€“1 Particularly, we note that ( ) 1 d x dx = ; C dx x = + ? (ii) ( ) sin cos d x x dx = ; cos sin C x dx x = + ? (iii) ( ) â€“ cos sin d x x dx = ; sin cos C x dx â€“ x = + ? (iv) ( ) 2 tan sec d x x dx = ; 2 sec tan C x dx x = + ? (v) ( ) 2 â€“ cot cosec d x x dx = ; 2 cosec cot C x dx â€“ x = + ? (vi) ( ) sec sec tan d x x x dx = ; sec tan sec C x x dx x = + ? (vii) ( ) â€“ cosec cosec cot d x x x dx = ; cosec cot â€“ cosec C x x dx x = + ? (viii) ( ) â€“ 1 2 1 sin 1 d x dx â€“ x = ; â€“ 1 2 sin C 1 dx x â€“ x = + ? (ix) ( ) â€“ 1 2 1 â€“ cos 1 d x dx â€“ x = ; â€“ 1 2 cos C 1 dx â€“ x â€“ x = + ? (x) ( ) â€“ 1 2 1 tan 1 d x dx x = + ; â€“ 1 2 tan C 1 dx x x = + + ? (xi) ( ) â€“ 1 2 1 â€“ cot 1 d x dx x = + ; â€“ 1 2 cot C 1 dx â€“ x x = + + ? INTEGRALS 291 (xii) ( ) â€“ 1 2 1 sec 1 d x dx x x â€“ = ; â€“ 1 2 sec C 1 dx x x x â€“ = + ? (xiii) ( ) â€“ 1 2 1 â€“ cosec 1 d x dx x x â€“ = ; â€“ 1 2 cosec C 1 dx â€“ x x x â€“ = + ? (xiv) ( ) x x d e e dx = ; C x x e dx e = + ? (xv) ( ) 1 log| | d x dx x = ; 1 log| | C dx x x = + ? (xvi) x x d a a dx log a ? ? = ? ? ? ? ; C x x a a dx log a = + ? A Note In practice, we normally do not mention the interval over which the various functions are defined. However, in any specific problem one has to keep it in mind. 7.2.1 Geometrical interpretation of indefinite integral Let f (x) = 2x. Then 2 ( ) C f x dx x = + ? . For different values of C, we get different integrals. But these integrals are very similar geometrically. Thus, y = x 2 + C, where C is arbitrary constant, represents a family of integrals. By assigning different values to C, we get dif ferent members of the family. These together constitute the indefinite integral. In this case, each integral represents a parabola with its axis along y-axis. Clearly, for C = 0, we obtain y = x 2 , a parabola with its vertex on the origin. The curve y = x 2 + 1 for C = 1 is obtained by shifting the parabola y = x 2 one unit along y-axis in positive direction. For C = â€“ 1, y = x 2 â€“ 1 is obtained by shifting the parabola y = x 2 one unit along y-axis in the negative direction. Thus, for each positive value of C, each parabola of the family has its vertex on the positive side of the y-axis and for negative values of C, each has its vertex along the negative side of the y-axis. Some of these have been shown in the Fig 7.1. Let us consider the intersection of all these parabolas by a line x = a. In the Fig 7.1, we have taken a > 0. The same is true when a < 0. If the line x = a intersects the parabolas y = x 2 , y = x 2 + 1, y = x 2 + 2, y = x 2 â€“ 1, y = x 2 â€“ 2 at P 0 , P 1 , P 2 , P â€“1 , P â€“2 etc., respectively, then dy dx at these points equals 2a. This indicates that the tangents to the curves at these points are parallel. Thus, 2 C 2 C F ( ) x dx x x = + = ? (say), implies thatRead More

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