Standard Integrals

3 Key excerpts on "Standard Integrals"

• 

  eBook - ePub

  Mathematics for Engineers and Scientists

  • Alan Jeffrey(Author)
  • 2004(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  7

  Fundamentals of integration

  The purpose of this chapter is to introduce the basic ideas underlying the definite integral, and then to prove a theorem that shows how definite integrals can be evaluated by using the idea of differentiation in reverse. The definite integral of a function f (x ) of the real variable x , evaluated between the limits x = a and x = b , and written

  ∫ a b

  f

  ( x )

  d x

  ,

  is a real number. It is shown how this number, defined as the limit of a sum, may be interpreted as the area between the graph of y = f (x ), the x -axis and the lines x = a and x = b , with areas above the x -axis being regarded as positive and those below it as negative. The first fundamental theorem of the integral calculus that is then proved provides the connection between integration and differentiation through the result

  d

  d x

  ∫ a x

  f

  ( t )

  d t = f

  ( x )

  ,

  where the upper limit of integration is the variable x , and t is a dummy variable. The concept of an antiderivative is introduced, and it is shown in the fundamental theorem of differential calculus how by using an anti-derivative it is possible to evaluate definite integrals.

  Among the simple but useful geometrical applications of definite integrals that are discussed are the determination of arc length and the area and volume of surfaces of revolution. Applications to mechanics are also introduced, and include the location of centres of mass of bodies, their moments of inertia about specified axes and the line integral

  7.1  Definite integrals and areas

  This chapter is concerned with the theory of the operation known as integration , which occupies a central position in the calculus. The connection between differentiation and integration is basic to the whole of the calculus and is contained in a result we shall prove later known as the fundamental theorem of calculus. Once again, limiting operations will play an essential part in the development of our argument. In fact we will show not only how they enable a satisfactory general theory of integration to be established, but also how they provide a tool, albeit a clumsy one, for the actual numerical integration of functions. However, aside from a number of simple but important examples, the practical details of the evaluation of integrals of specific classes of function will be deferred until Chapter 8

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  eBook - ePub

  Mathematics for Physicists

  • Brian R. Martin, Graham Shaw(Authors)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  This was discussed in detail in Section 2.1.2. We shall not repeat that discussion here, but merely note its usefulness in evaluating integrals of rational functions. For example, consider the integral of the function The denominator is x 2 + 5 x + 6 = (x + 2)(x + 3), so that by (2.17). Hence and setting x = −2 and then x = −3, gives A = 2 and B = 3, respectively. Hence the integral is Other examples are given in later sections. 4.3.5 More Standard Integrals Consider an integral of the form (4.26a) where f (x) is a given function. If we can find a substitution x = g (z), such that, where α is a constant, then equation (4.21a) gives (4.26b) where g − 1 is the inverse function of g (not). Several ‘Standard Integrals’ may be evaluated in this way. An example is (4.27a) Substituting x = a sinh z, we have so that (4.26b) gives (4.27b) where we have used the notation sinh − 1 to denote the inverse function of sinh , as an alternative to arcsinh. Other Standard Integrals of a similar type, together with the substitutions required to derive then, are given in Table 4.2. Their use is illustrated in Example 4.8 below. Table 4.2 More Standard Integrals and the substitutions used to derive them Integrand Substitution Integral x = a sin z x = a sinh z x = a cosh z x = a tan z x = a tanh z Example 4.8 Evaluate the integrals Solution On completing the square, and then setting z = x − 2, using the standard integral (4.27b), gives The integral is where we have used the logarithmic integration (4.25) and a standard integral from Table 4.2. 4.3.6 Tangent substitutions It is sometime useful to convert integrals involving sin x and cos x into integrals over rational functions by the

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  eBook - ePub

  The philosophy of mathematics

  • Auguste Comte(Author)
  • 2012(Publication Date)
  • Perlego

    (Publisher)

  singular equation which will form the most important object of the inquiry, since it alone will represent the required curve; the general integral, which thenceforth it becomes unnecessary to know, designating only the system of the tangents, or of the osculating circles of this curve. We may hence easily understand all the importance of this theory, which seems to me to be not as yet sufficiently appreciated by most geometers.

  Definite Integrals. Finally, to complete our review of the vast collection of analytical researches of which is composed the integral calculus, properly so called, there remains to be mentioned one theory, very important in all the applications of the transcendental analysis, which I have had to leave outside of the system, as not being really destined for veritable integration, and proposing, on the contrary, to supply the place of the knowledge of truly analytical integrals, which are most generally unknown. I refer to the determination of definite integrals.

  The expression, always possible, of integrals in infinite series, may at first be viewed as a happy general means of compensating for the extreme imperfection of the integral calculus. But the employment of such series, because of their complication, and of the difficulty of discovering the law of their terms, is commonly of only moderate utility in the algebraic point of view, although sometimes very essential relations have been thence deduced. It is particularly in the arithmetical point of view that this procedure acquires a great importance, as a means of calculating what are called definite integrals, that is, the values of the required functions for certain determinate values of the corresponding variables.

  An inquiry of this nature exactly corresponds, in transcendental analysis, to the numerical resolution of equations in ordinary analysis. Being generally unable to obtain the veritable integral—named by opposition the general or indefinite integral; that is, the function which, differentiated, has produced the proposed differential formula—analysts have been obliged to employ themselves in determining at least, without knowing this function, the particular numerical values which it would take on assigning certain designated values to the variables. This is evidently resolving the arithmetical question without having previously resolved the corresponding algebraic one, which most generally is the most important one. Such an analysis is, then, by its nature, as imperfect as we have seen the numerical resolution of equations to be. It presents, like this last, a vicious confusion of arithmetical and algebraic considerations, whence result analogous inconveniences both in the purely logical point of view and in the applications. We need not here repeat the considerations suggested in our third chapter. But it will be understood that, unable as we almost always are to obtain the true integrals, it is of the highest importance to have been able to obtain this solution, incomplete and necessarily insufficient as it is. Now this has been fortunately attained at the present day for all cases, the determination of the value of definite integrals having been reduced to entirely general methods, which leave nothing to desire, in a great number of cases, but less complication in the calculations, an object towards which are at present directed all the special transformations of analysts. Regarding now this sort of transcendental arithmetic

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