ON EXPONENTIAL FUNCTIONS. 1o. If X = ƒ (a*), then the function Xdz, if we make aa = u wil become 2o. Differentiating Xe, we have e* dx (X+ so that every exponential function in which the factor of e* dx is composed of two parts, one of which is the first differential co-efficient of the other, will be easily integrated. For examplo 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. Treating a 2-1 dx, &c. in the same manner, we shall finally have N x2-1 n(n−1)r-2 1.2.3..n u =a* { log.a ... It is manifest that the same method is applicable to Xa* dr, where X is any entire algebraical function of x. But if the exponent n be negative, it is manifest that the exponent of z 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 dx Integrating in the same manner, we shall finally have 24-1 |