Mr. Woodhquse on the Integration 
242 
the integral 
= 4rr7i{>+7-*" + irf.^ + ^^ + &c.}. (4) 
Which is the same series as is given in the Edinburgh Transac- 
tions, Vol. IV. p. 178, and which its ingenious author, Mr. Ivory, 
derived from a method of Lagrange, contained in the Berlin 
Acts for 1784. According to that method, dx ,J ( 1 ff) is put 
under the form Ad 9 \^ 1 + a* 2a cos. 29 jf, and its exponential 
expression substituted for cos. 2 9 . 
I have deduced the preceding series ascending by the powers 
of e' or of in order to show, that it is a particular result of the 
general method of the transformation of dx ,J ). Forpur- 
poses of computation, it will be convenient to push the transfor- 
mation farther; if, for instance, to quantities involving e", the 
integral of from x=o to x— 1, may be com- 
puted from 2 series ; or the whole integral equals 
17 ^ 7 )— { (>»*)*• e "’+ (?* + &c - ) 
— 1| 1 -l-j- (Di-iy e"*+(D n &c. } 
which expression may be derived after a manner precisely similar 
to that by which I have deduced the series ascending by the powers 
of e'. 
If the transformation of df be indefinitely continued, there 
results a form very convenient for the computation of the integral 
of dx jf (— in all values of e between o and thus. 
df = -J-.du'-'- e ' iu 
df 
U' 
i+e 7 
du' , df 
{a) 
or = du' + + 
i-\-e 1 2 JO 1 i+e 
similarly, df — . . du" + { — 
+ e" 
\ du " 1 d f 
J * U" T 1 
+ *" * 
