MATHEMATICAL NOTE*. 



Mogg+a;) 



, l+af 



THIS definite integral is evaluated in a curious manner by 

 M. Bertrand in Liouvilles Journal. The demonstration I am 

 about to give of his result, is somewhat different in form from 

 that which he made use of. 



The method employed in an ingenious paper which appeared 

 in the last volume of the Journal (in. p. 168), will apply to the 

 integral we are about to consider. 



xdx 



(l+ux) (1 + a; 2 ) (1 + x*) (l + u?) (1 + ux) (I+u*) ' 

 and therefore 



xdx u f 1 dx u f 1 dx 



ux 



d 1 f 1 xdx u f 1 dx u f 1 ds, 



du^ U ~ 1+1? J 1 + x 2 + 14- w 2 J 1 + x* 1 + ti* J 1 + 



1 7T U 



Integrate for u, from to 1, 



/(I) -/(O) = log 2 + f log 2 -/(I), 



Cambridge Mathematical Journal, No. XXI. Vol. iv. p. 143, May, 1844. 



