1882.] certain Definite Integrals. 261 



, dx(l-x)x n ~ l , 



Jo + 



jo n(n + l)(n + 2) 



&o. = &c, we have, if 0(6>) =A + A^ A 3 3 + . . . , 



[ 1 0(l-^)^-i=^Q+— ^— + 7 1 '*' ±*— + &c. . (228). 

 Jo n n(n+l) n(n + l)(n + 2) 



This formula is most useful when n is very large, or when (n) is 

 very small. Suppose ^=100, then the tenth term is less than 



A 9 



60,000,000,000,000 



and after this the terms continue to diminish, though not so rapidly. 



When n is very small, we shall come to an integer (r) which doea 

 not differ sensibly from n+r, and therefore we are able to write the 

 integral : — 



P0(l- g )(faai»-l = ^o+ A l , b-llh + . . . 



Jo K ' n n(n + l) n(n + l)(n + 2) 



1.2. 3 

 n(n + l) . . . (n + r) 



— — t — i — \ C Aj ' ~f~ A >'+i A »*+2 • 



a o_l A i i 1.2.A 3 1.2. 3. A 



+ ^ -f- ^ + • _ " ' t3* + . . . 



n n(n + l) n(n + \)(n + 2) n(n + l)(n + 2)(n+S) 



- | -r i' 1 V 8 'r' r . ^ (°)- A o- A i • • • - Am) • (229). 

 w(w + l) . . . (n-i-r) 



In the same way we are able to find : — 



— x)dxx n ~ 1 (log e xy (230) 



(since IWaog^)^^-^); 



also j\;(l— a)cto i_1 0(log e a) (231). 



P. 



Jo 



dxx n ~ 1 (j)(l—x,log e x) (232). 



We must, however, observe that the functions involving log x are 

 supposed convergent, whatever the value of log e x, when expanded. 



