EVALUATION OF CERTAIN DEFINITE 

 INTEGRALS*. 



WHEN the value of a definite integral is known, we may, if 

 it involve an arbitrary parameter, integrate it (under certain 

 conditions) with respect to this quantity. The result thus ob- 

 tained involves an arbitrary constant of integration; in order 

 to eliminate it, we may ascribe two different values to the 

 quantity for which the integration has been effected, and then, 

 of the two corresponding equations thus got, subtract one from 

 the other. In other words, we integrate between limits for the 

 arbitrary parameter, and thus get a new definite integral, in- 

 volving two arbitrary quantities, namely, the two limiting 

 values ascribed to the single one involved in the original in- 

 tegral. We may integrate again, with respect to either of these, 

 and so on. But this method of proceeding, though it will lead 

 to a variety of particular results, is not well fitted to show the 

 nature of the class of definite integrals to which they all belong, 

 and which may be obtained by repeated integrations for an 

 arbitrary parameter. 



If we integrate n times successively, we shall introduce n 

 constants. These may be eliminated at once, in the manner 

 I am about to point out. The result thus got, includes for 

 every original definite integral, all that can be deduced from 

 it by n integrations for an arbitrary parameter. 



The following theorem will serve to illustrate the general 

 method. 



If Fx is a rational and integral function of circular func- 

 tions of x (sines and cosines), then we may express in finite 



r+ M F x 

 terms the value of I - dx, n being a positive integer, and 



J-oo X 



such that -[ is not infinite. 

 a n Jo 



* Cambridge Mathematical Journal, No. XVI. Vol. in. p. 185, Nov. i8j 



