SECT. I.] 



PROPERTIES OF DEFINITE INTEGRALS. 



347 



the two terms add together and the value of the integral is J TT. 



Hence the definite integral 1 - sin a cos qx is a function of x 



vrJo q 



equal to 1 if the variable x has any value included between 1 and 

 1 ; and the same function is nothing for every other value of x 

 not included between the limits 1 and 1. 



358. We might deduce also from the transformation of series 

 into integrals the properties of the two expressions 2 



2 r dq cos qx , 2 f qdq sin qx t 

 vJt 1 + &amp;lt;f FC W 1 + 2 2 



the first (Art. 350) is equivalent to e~ x when x is positive, and to 

 e x when x is negative. The second is equivalent to e~ x if x is positive, 

 and to e x if x is negative, so that the two integrals have the 

 same value, when x is positive, and have values of contrary sign 

 when x is negative. One is represented by the line eeee (fig. 19), 

 the other by the line eeee (fig. 20). 



Fig. 19. Fig. 20. 



The equation 

 1 . TTX __ sin a sin x sin 2a sin 2# sin 3 a sin 3x 



&amp;gt; olLL ^ o v &quot;T&quot; 2 V 2 &quot; 2 O52 2 ~1 O^Cij 



which we have arrived at (Art. 226), gives immediately the integral 



2 f dqsinqTTsmqx ,., 3 . . , . .. 



- I ^ 2 ? which expression is equivalent to sin x, if x 



is included between and TT, and its value is whenever x ex 

 ceeds 7T. 



1 At the limiting values of x the value of this integral is | ; Eiemann, 15. 



2 Cf. Eiemann, 16. 



3 The substitutions required in the equation are for , dq for -, q for -. 



We then have sin x equal to a series equivalent to the above integral for values of x 

 between and TT, the original equation being true for values of x between and a. 



[A.F.] 



