EQUATION AND ITS TRANSFORMATIONS. 
79 5 
27. In a similar manner we may obtain the value of the integral 
[ a x sin bx 
dx 
i 0 (a 2 +x 2 ) 
which is finite for all values of n greater than unity. For, differentiating (15) with 
respect to b, we have 
% sin bx , , a/ 7r x51 f” 0 . , UU , 
— —dx=j;-—-a 2 " +3 /d .x~" 4 e r dx 
J 0 (cd+x^y 2 r(7Q J n 
. . . . (16) 
=i nS'{ r < ,i - |)o " 2 " +8 { 1+ ^ 
( A4&c. 
71 — 4 
(2ti—4)(2ti — 5) 2! 
+ r(-w+»(*&)«{ 1+ — + ^+&c. 
271-2 
(2ti—2)(2ti —1) 2! 
J 
where, as before, if the first series terminates, the finite portion of it represents the 
value of the integral. 
If n is a positive integer, and the terms of the series are written in the reverse 
order, we find 
f" sd sin bx , _ it b n 4 f (n-l)(n—2) ( 1 \ n(n-l)(n-2)(n-3) [ 1 \ 3 0 1 6 
\ 0 (a* + x*y 2»(w-l)! 2 \ab)'^ 2.4 \ab) J 5 
which is a known result (Schlomilch, ‘Anal. Stud.,’ loc. cit., p. 97). 
It follows at once by combining (15) and (16) that, for all values of n such that the 
integrals are not infinite, viz., if n— or >1, 
x sin bx -j l b 
Ovt* Q 
0 (a? -J- x~) n+l 
,0 ° cos bx 
0 (a?+x 2 )’ 1 
dx. 
It would be strange if this equation were new, but I have not met with it anywhere: 
it is readily proved in the case of n an integer, for 
— 2a 
[°° x sin bx 
1 0 (a* + z*y ( 
and 
f* cos bx 
1 o | 0 ( 
J 0 flv "r X" 
whence 
1 
f 00 x sin bx 
1 
I c (a 2 +A) 2< 
/< 
dad 
dx—^-e ah 
la 
dx—\b 
cos bx 
a" + or 
„dx, 
which, differentiated n times with regard to a 3 , gives the relation in question. 
5 k 2 
