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EVALUATION OF DEFINITE INTEGRALS.
When we are able to effect the integration of any function, the deterniination of its value between certain limits of the independent variable offers in general no difficulty, as we have merely to subtract its value at one limit from its value at another. There are however many functions, the Definite Integrals of which we are able to find, although the indefinite integral cannot be expressed in finite terms. The evaluation of these integrals has become one of the most important branches of the Integral Calculus, in consequence of the numerous applications which are made of them both in pure mathematics and in physics : it is to functions of this kind that the examples in the following paper refer.
The methods for evaluating those definite integrals whose general values cannot be found are very various, but they can generally be classed under the following heads.
(1) Expansion of the function into series, integration of each term separately, and summation of the result.
(2) Differentiation and integration with respect to some quantity not affected by the original sign of integration.
(3) Integration by parts of a known definite integral, so as to obtain a relation between it and an unknown one.
(4) Multiplication of several definite integrals together, so as to obtain a multiple integral, and, by a change of the variables in this, converting it into another multiple integral, coinciding with the first at the limits, and admitting of integration. By this means a relation is found between the definite integrals multiplied together, which frequently enables us to discover their values.
(5) Conversion of the function by means of impossible quantities into a form admitting of integration.
These different methods will be best understood by their application to the following examples.
We shall begin with the function known as the Second Eulerian Integral, because, though its exact value cannot be found generally, its properties have been much studied, and to it a number of other integrals are reduced.
The definite integral Ldx e-*0*-, when n is a whole number, is easily seen by the method of reduction in Ex. (13), Chap. 11. of the Integ. Calc. to be
(n − 1)... 3.2.1. When, however, n is a fraction, its value can be found only in certain cases, but it possesses many remarkable properties which render it of the greatest importance in the Theory of Definite Integrals. It was first studied by Euler, who seems at an early period to have seen its importance, and has devoted several memoirs to the investigation of its properties ; on this account Legendre has named it after him, at once for the purposes of characterizing the function and honouring that great mathematician. To distinguish it from another integral with which also Euler had much occupied himself, and of which we shall afterwards treat, it is usually called the “ Second Eulerian Integral,” and Legendre has affixed to it the characteristic symbol r, applied to the index, so that he writes
* dx 6-***-1 = (n), which notation we shall adopt. Throughout the following investigations n is supposed to be greater than 0.
In the first place we remark that by a change of the independent variable this integral may be put under other forms which are sometimes more convenient in practice than that which we have used. Thus if we put e-s= y, the corresponding limits are
x = 0, y = 1; x = 0, y = 0,
and the integral takes the form (a)
I (n) = ['dv (log) This is the shape under which the integral has been usually treated both by Euler and Lagrange, but it is scarcely so convenient as the preceding.
Again, if we put x = x, the limits remain the same as before, and we have
T(n) = + (*dze
This last form is the most convenient for determining the value of the integral in one remarkable case when it can be found in finite terms. If n=1
6* dx e-% x-1 = (1) = 2 6* dx e-. Let
k = f*dxe": then as the value of the definite integral is independent of the variable, we have also
k = f*dy e=y", and therefore multiplying these together,
kʻ = 6* dx e-**. Lody e-y = * l *dy dz e-(y2+z2); since y and x are independent. Now assume
z = r cos 0, y = r sin 0, then
con + yé = poi, dydx = r dr do. To determine the limits we observe that y and never become negative, and therefore 0 must vary from o to , while r varies from 0 to “, so that we have
k = 6*$drdor e->2 = 1; whence (c)
k = 1 and r (1) = 7%. We shall now demonstrate the more important properties of the function r (n) referring the reader who wishes, for a more detailed ex position of them to Legendre, Exercices de Calcul Intégral, Tom. I. and 11.
ist. Let n be even : Then there are in - 1 pairs of factors of the form r(s) (1 – ), and a middle factor which is r (1) = 7). Therefore by the preceding theorem the product is equal to
sin (5) sin (*sin (**) ... sin (= 2)
* This demonstration of a Theorem discovered by Euler is given by Mr Greatheed in the Camb. Math. Journal, Vol. 1. p. 17.