237 
EXAMPLES. 
(1.) 
x 
% 2% — 
ie (log xP. 22 da=(D.)" - 5 aK 
Hence, as a special case, 
: 1 
|. Gog a)». ed = (Da). (= 
0 a 
(2.) 
FANT ane 5 rin 
a (a-1) a a 
i xe dec Lot — £2 =: Lot — Ly 
Hence, as a special case, 
(3.) 
%2 LP — ay” 
ies a?) (1—a)t'de=(-1)t". At. age = 
Hence, as a special case, 
1 
fio a-ayrde=F(p, d=CI-aer.(2), 
0 
and, if y be any positive integer, we very easily derive for the value 
of the first Eulerian Integral, in this case, 
(eer engine) 
p(p+1)(p+2)...(ptg-1)" 
s (4.) Let it be proposed to reduce the integral, first discussed by 
inet, 
F (2,9)= 
1 
0 
| {(L+a) (1-a)t14(14+2)04(1-a)P3} de. 
If, in our second fuidamental theorem, we substitute m +1 for m, we ob- 
tain the theorem 
F (A,). (1+ 2)¢=F (a). (1 +2), 
a result, indeed, which cou! 1 readily have been deduced directly. 
Hence, it is easily seen that the integral proposed is equivalent to 
GIA. I, (1+ a)? dx + (1- a | (1 +ax)*!'da 
