Contents P II ART Foreword v Preface vii 7. Integrals 287 7.1 Introduction 288 7.2 Integration as an Inverse Process of Differentiation 288 7.3 Methods of Integration 300 7.4 Integrals of some Particular Functions 307 7.5 Integration by Partial Fractions 316 7.6 Integration by Parts 323 7.7 Definite Integral 331 7.8 Fundamental Theorem of Calculus 334 7.9 Evaluation of Definite Integrals by Substitution 338 7.10 Some Properties of Definite Integrals 341 8. Application of Integrals 359 8.1 Introduction 359 8.2 Area under Simple Curves 359 8.3 Area between Two Curves 366 9. Differential Equations 379 9.1 Introduction 379 9.2 Basic Concepts 379 9.3 General and Particular Solutions of a 383 Differential Equation 9.4 Formation of a Differential Equation whose 385 General Solution is given 9.5 Methods of Solving First order, First Degree 391 Differential Equations 10. Vector Algebra 424 10.1 Introduction 424 10.2 Some Basic Concepts 424 10.3 Types of Vectors 427 10.4 Addition of Vectors 429 xiv 10.5 Multiplication of a Vector by a Scalar 432 10.6 Product of Two Vectors 441 11. Three Dimensional Geometry 463 11.1 Introduction 463 11.2 Direction Cosines and Direction Ratios of a Line 463 11.3 Equation of a Line in Space 468 11.4 Angle between Two Lines 471 11.5 Shortest Distance between Two Lines 473 11.6 Plane 479 11.7 Coplanarity of Two Lines 487 11.8 Angle between Two Planes 488 11.9 Distance of a Point from a Plane 490 11.10 Angle between a Line and a Plane 492 12. Linear Programming 504 12.1 Introduction 504 12.2 Linear Programming Problem and its Mathematical Formulation 505 12.3 Different Types of Linear Programming Problems 514 13. Probability 531 13.1 Introduction 531 13.2 Conditional Probability 531 13.3 Multiplication Theorem on Probability 540 13.4 Independent Events 542 13.5 Bayes' Theorem 548 13.6 Random Variables and its Probability Distributions 557 13.7 Bernoulli Trials and Binomial Distribution 572 Answers 588 7 Chapter INTEGRALS Just as a mountaineer climbs a mountain – because it is there, so (cid:0) a good mathematics student studies new material because it is there. — JAMES B. BRISTOL (cid:0) 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 G .W. Leibnitz primitive) of the function. Further, the formula that gives (1646-1716) 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. 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: d We know that (sin x) =cos x ... (1) dx d x3 ( ) =x2 ... (2) dx 3 d and (ex)=ex ... (3) dx We observe that in (1), the function cos x is the derived function of sin x. We say x3 that sin x is an anti derivative (or an integral) of cos x. Similarly, in (2) and (3), and 3 ex are the anti derivatives (or integrals) of x2 and ex, 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 : d d x3 d (sin x+C) cosx, ( +C) x2and (ex +C) ex dx (cid:0) dx 3 ✁ 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. d More generally, if there is a function F such that F(x)= f (x), x I (interval), dx ✄ ☎ then for any arbitrary real number C, (also called constant of integration) d F(x)+C =f(x), x I dx✆ ✝ ☎ 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 – h defined by f(x) = g(x) – h(x), x I (cid:0) ✄ df Then =f = g – h giving f (x) = g (x) – h (x) x I dx ✂ ✂ ✂ ✂ ✂ ✂ (cid:0) ✄ or f (x) = 0, x I by hypothesis, i.e., the rate of ch✂ange of f w✁ith✄ 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(x)dx=F(x)+C. ☎ dy Notation Given that f (x), we write y = f (x)dx. dx ✆ ✝ For the sake of convenience, we mention below the following symbols/terms/phrases with their meanings as given in the Table (7.1). Table 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) d ✁ xn(cid:0)1✂ xn ; xn dx x n✠1 C, n –1 dx☎n 1✆✄ ✡ n 1 ☛ ✌ ✞ ✝ ✟ ☞ ☛ Particularly, we note that d x 1 ; dx x C dx✍ ✎✏ ✄ ✝ ✑ d (ii) sin x cosx ; cosxdx sin x C dx✒ ✓✔ ✄ ✝ ✑ d (iii) –cosx sin x ; sin xdx –cosx C dx✒ ✓✔ ✄ ✝ ✑ d (iv) tan x sec2x ; sec2 xdx tan x C dx✒ ✓✔ ✄ ✝ ✑ d (v) –cot x cosec2x ; cosec2 xdx –cot x C dx✒ ✓✔ ✄ ✝ ✑ d (vi) secx secxtan x ; secxtanxdx secx C dx✕ ✖✏ ✄ ✝ ✑ d (vii) –cosecx cosecxcot x ; cosecxcotxdx –cosecx C dx✗ ✘✏ ✄ ✝ ✑ d 1 dx sin–1 x sin–1 x C (viii) ; dx✙ ✚✛ 1– x2 ✢ 1– x2 ✛ ✜ d 1 dx –cos–1 x –cos–1 x C (ix) ; dx✣ ✤✛ 1– x2 ✢ 1– x2 ✛ ✜ d 1 dx tan–1 x tan–1 x C (x) ; dx✥ ✦✧1 x2 ✩1 x2 ✧ ★ ★ ★ d 1 dx –cot–1 x –cot–1 x C (xi) ; dx✪ ✫✬1 x2 ✮1 x2 ✬ ✭ ✭ ✭ INTEGRALS 291 d 1 dx sec–1 x sec–1 x C (xii) ; dx(cid:0) ✁✂ x x2 –1 ☎x x2 –1✂ ✄ d 1 dx –cosec–1 x –cosec–1x C (xiii) ; dx✆ ✝✂ x x2 –1 ☎x x2 –1✂ ✄ d (xiv) (ex) ex ; exdx ex C dx ✞ ✟ ✠ ✡ d 1 1 (xv) log|x| ; dx log|x| C dx☛ ☞✌ x x ✌ ✍ ✎ d ax ax (xvi) ✏ ✑ ax ; axdx C dx✒log a✓✟ ✘ ✖loga ✗ ✔ ✕ 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 f(x)dx x2 C. For different values of C, we get different ✟ ✠ ✡ integrals. But these integrals are very similar geometrically. Thus, y = x2 + C, where C is arbitrary constant, represents a family of integrals. By assigning different values to C, we get different 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 = x2, a parabola with its vertex on the origin. The curve y = x2 + 1 for C = 1 is obtained by shifting the parabola y = x2 one unit along y-axis in positive direction. For C = – 1, y = x2 – 1 is obtained by shifting the parabola y = x2 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 = x2, y = x2 + 1, y = x2 + 2, y = x2 – 1, y = x2 – 2 at P , P, P , P , P etc., 0 1 2 –1 –2 dy respectively, then at these points equals 2a. This indicates that the tangents to the dx curves at these points are parallel. Thus, 2xdx x2 C F (x)(say), implies that ✟ ✠ ✟ C ✡ 292 MATHEMATICS Fig 7.1 the tangents to all the curves y = F (x), C R, at the points of intersection of the C ✄ curves by the line x = a, (a R), are parallel. ✄ Further, the following equation (statement) f(x)dx F(x) C y(say), (cid:0) ✁ (cid:0) ✂ represents a family of curves. The different values of C will correspond to different members of this family and these members can be obtained by shifting any one of the curves parallel to itself. This is the geometrical interpretation of indefinite integral. 7.2.2 Some properties of indefinite integral In this sub section, we shall derive some properties of indefinite integrals. (I) The process of differentiation and integration are inverses of each other in the sense of the following results : d f(x)dx =f(x) dx ☎ and f (x)dx =f(x) + C, where C is any arbitrary constant. ✆ ✂ INTEGRALS 293 Proof Let F be any anti derivative of f, i.e., d F(x) =f(x) dx Then f(x)dx =F(x) + C (cid:0) d d Therefore f(x)dx = F(x)+C dx dx✂ ✄ ✁ d = F(x)= f(x) dx Similarly, we note that d f (x) = f(x) ☎ dx and hence f (x)dx =f(x) + C ✆ (cid:0) where C is arbitrary constant called constant of integration. (II) Two indefinite integrals with the same derivative lead to the same family of curves and so they are equivalent. Proof Let f and g be two functions such that d d f(x)dx = g(x)dx dx dx ✁ ✁ d or f(x)dx– g(x)dx = 0 dx✝ ✞ ✟✡ ✡ ✠ Hence f(x)dx– g(x)dx= C, where C is any real number (Why?) (cid:0) (cid:0) or f(x)dx = g(x)dx C ☛ (cid:0) (cid:0) So the families of curves f(x)dx C ,C R ☞✏ ✍ 1 1✎ ✌ and g(x)dx C ,C R are identical. ✑✏ ✍ 2 2✎ ✒ Hence, in this sense, f(x)dxand g(x)dx are equivalent. (cid:0) (cid:0)