220 
Mr. Woodhouse on the Integration 
puttin gfx — f(a + x — a), the integral of dx .fx is calculated 
from the series Jdx .fa . (x — a) [dx . vfa . [x — a[ -f- &c. 
But, although the integrals of many expressions can thus be 
exhibited, yet such series are useless for the purpose of arith- 
metical computation, except their terms continually decrease, 
and except the limits of the ratio of the decrease of the terms 
can be determined; and the invention of series adapted to 
arithmetical computation, has not been the least of the difficulties 
encountered by modern analysts. 
Although the differential expressions that admit no finite inte- 
gration have not been reduced into classes, yet there are some, 
from their simplicity, and frequent occurrence in analytical in- 
vestigation, more conspicuously known and attentively consi- 
dered : such are the expressions » anc ^ compu- 
tation of their integrals, in other words, is the determination of 
the logarithms of numbers, and the lengths of circular arcs. 
The necessity of calculating the integrals of expressions such 
as must soon have obtruded itself on the attention 
i+* V(i— * ) 
of the early analysts : for several expressions, as ~t^== , 
dx -r, &c. apparently dissimilar, are easily reduced 
•; and besides, the difficulty of inte- 
I + X 
to the forms 
dx 
dx 
i -f x y v' : 
grating a variety of forms, is soon reduced to that of the integra- 
tion of^-, : such, for instance, are the forms ^77 
1 -J- X r I — X X\ \[\ 
■x 2 ) 
dx 
•r 3 v'(i— X 2 -) 
the forms 
, and all that are comprehended under 
dx 
x * dx x*dx 
V(i -xy vc* — - **)* 
+ ‘V(i -**)’ 
&c. and all that are comprehended 
under 
x ’ lm . dx 
V 
