JL 



188 THEORY OF HEAT. [CHAP. III. 



taken from x = to x TT, in which i and j are integers, is 

 jj sin (* - j) x - ^-. sin (i + j) x + C. 



Since the integral must begin when x = the constant C is 

 nothing, and the numbers i and j being integers, the value of the 

 integral will become nothing when OJ = TT; it follows that each 

 of the terms, such as 



a t \ sin x sin ix da, a 2 1 sin 2x sin ix doc, a 3 (sin 5x sin ixdx t &c., 



vanishes, and that this will occur as often as the numbers i and j 

 are different. The same is not the case when the numbers i and j 



are equal, for the term - .sin (i j) x to which the integral re- 



j 



duces, becomes -^ , and its value is TT. Consequently we have 







2 I sin ix sin ix dx == TT ; 



thus we obtain, in a very brief manner, the values of a lt a z , a 3) ... 

 4 , &c., namely, 



2 f 2 f 



ttj = - /( (#) sin # dr, a 2 = - l&amp;lt; (x) sin 2 



2 f 2 r 



# 3 = - I c/&amp;gt; (a?) sin 3# &e, a, = - \$(x) sin 10 



Substituting these we have 



%7r(f&amp;gt; (x) = sin x I &amp;lt;/&amp;gt; (a?) sin # cZic + sin 2x l(f) (x) sin 2# J^? + &c. 



+ sin ix 1 (a?) sin ixdx + &c. 



222. The simplest case is that in which the given function 

 has a constant value for all values of the variable x included 



between and TT ; in this case the integral I sin ixdx is equal to 



9 



?, if the number i is odd, and equal to if the number i is even. 



