2 j 6 Mr. Woodhouse on the Integration 
to a series of terms, as Z'-f- Z"-f Z'" &c. 4 - fdz (a-{-b . cos. 3)14- 
-j; and, after this reduction, he must have found the 
Si 
dz 
(a+b cos. z)z 
integral of dz (#+& cos. z) f, which, as he then could only do by 
resolving it into a series, was a problem not more easy, than the 
finding of the integral of dz (a-{-b cos. z)-r from its immediate 
resolution into a series; consequently, the reduction of 
dz (a-f & cos. 3 ) 4 -, into Z'-J-Z"-f &e. would have been useless and 
unprofitable labour. Had a certain and easy method of computing 
(tf+ b.cos.z)l been known to D’alembert, he would not have 
asserted the reduction of f(a-\-b . cos. z)-r into Z'-fiZ " -f &c„ 
J (a-\-b cos.)f dz , to be a method “ plus curieuse que commode.” 
The integral of ^ jpq furnishes an easy instance for illustration. 
Suppose it were necessary to compute it, a value of x being given 
less than 1 ; resolve it into 
v'(t-rr){A**+B**+o}+/E. 7 ^i 
then, from such expression, may the integral be easily found, 
since we have tables that exhibit the value of for all values 
of x between o and 1 ; but, if the zeal and ability of former com- 
putists had not enabled us, in all cases, to assign the value of 
it would be, practically, more easy and convenient, for a 
single instance, to compute an expression as immediately 
from 
fx*dx[ 1— di ^ 2 -j-D 9 i d 3 i *a; 6 -f&;c. j, 
or 
5 
D 1 Z .X' 
D 1 
c 
&c. 
than from ( 1 — a® ) | Aa ,3 + O j -j- 
■**) * 
