144 EVALUATION OF CERTAIN DEFINITE INTEGRALS. 



This theorem applies to several remarkable definite integrals, 

 some of which occur in the theory of probabilities; there are 

 others which do not seem to have been noticed. 



DEM. Fx, as every function of x, may be considered the 

 sum of two functions, one of which remains unchanged when 

 x changes its sign, and the other changes its sign with that of 

 x; its value aux signes pres remaining unaltered. Hence, 

 whether n is odd or even, we may write 



fx_fa fa 

 x n ~x n + x n ' 



where _ 



It is obvious that 



I 

 J 



and that if n is odd, fa, which is of course a rational and in- 

 tegral function of circular functions (sines and cosines) of x, 

 must be developable in a series of sines exclusively, and if n 

 is even in a series of cosines exclusively. 

 Thus, we may assume 



Now, as J is not infinite when x = 0, the lowest power of 



x which can enter into fx must be not < n, call it m, and 

 develope in powers of x every sine or cosine which appears on 

 the second side of the last written equation. We must have 

 ?<Aa m ~* = 0, S-4a m ~ 4 = 0, &c. ... J m 1 equations if m is even, 

 and (m 1) if it is odd. 



Let us now consider the definite integral 



Integrating it repeatedly for r, we get 



^ LX sm rx , _j r 



e - cto = tan l - , 



x a' 



