ON EXPONENTIAL FUNCTIONS. 1o. If X = ƒ (a), then the function Xda, if we make a = u win become f (u) du dX 2o. Differentiating Xe*, we have e2 dæ (X+) so that every exponential function in which the factor of e* dæ is composed of two parts, one of which is the first differential co-efficient of the other, will be easily integrated. For example In like manner, if we make 1+x=z, we shall find In every other case, however, we must have recourse to the method of integration by parts. u = fa dr. 2 and considering a" in .he first instance as constant. n log. a log. a Treating a 2-1 dx, &c. in the same manner, we shall finally have It is manifest that the same method is applicable to Xa* dx, where X is any entire algebraical function of x. But if the exponent n be negative, it is manifest that the exponent of a must go on increasing; and therefore, in the integration by parts we must consider a' as constant in the first instance, in this manner, if a* dx Integrating in the same manner, we shall finally have We cannot, however, proceed with our calculation beyond this point, because we should obtain a result = a We can, however, approximate to it in the following manner If n is fractional, one or other of the above methods will enable us to reduce the exponent of x until its value lies between 0 and 1, or 1, and we shall then be enabled to approximate to the required integral by series. On Logarithmic Functions. Let it be required to integrate Xda log." x where X is any algebraic function of x. If n is a positive whole number we may integrate by the method of parts, regarding log." x as constant in the first instance. We shall then have X fx de log." x = log." x ƒ x dx - nf (log."-fx dx) and since fx dx is supposed to be known by the principles already establish ed, we perceive that the integration of the proposed function is reduced to that of one whose form is the same, and in which the exponent of the logarithm is reduced by unity. The same process is applicable to this new function, and thus the integration will be completed step by step. Adding the successive results obtained in this manner, we find, ƒ xTM då log." x = xm+1 Șlog."x_nlog."-1x_n(n—1)log."—2x xm {m+1 (m + 1)2 + (m+1)3 But if n be integral and negative, we perceive that, as in the case of exponential functions, in performing the integration by parts of fX log." x dã, we must in the first instance suppose X constant. dx we shall divide X log." x dx into the two factors X x log.", hence X a formula which manifestly attains the object in view. In order, however, to understand the difficulties which occur, let us apply repeating the calculation for this last term, and performing the successive openations in the same manner, we shall find upon adding the different results to We cannot, however, proceed with our calculation beyond this point, because our result would become = α In this manner we reduce the proposed quantity to the function already treated of in the chapter on Exponential Functions, which can be integrated by approximation only. When n is a fraction either positive or negative, one or other of the above methods will enable us to reduce the integral of X dæ log." x, to that of a function of the same form, in which the value of n lies between 1 and We must then approximate to the value of the required integral by series. 1. On Circular Functions. These may always be reduced to algebraic functions by assuming sin. é or cos. = z, but with a few exceptions we shall obtain the integrals of these quantities by the method of parts. Proceeding by the method of parts, and supposing sin.m-1 constant in the first instance. Similarly, if we integrate for the cosine in the same manner as we have done for the sine, we shall have These integrals will.. by successive reduction become de cos." 0, de sin." / or de sin. cos." e, de cos. e sin." 8, according as m or n are odd or even. Now, suppose m, or n, to be negative, making ʼn negative in (1) m-1 sin.m θ m-l de sin.m-2 COS. 0 + S cosin."0 |
