386 PRACTICAL MATHEMATICS 



(2) If we multiply throughout by cos nx and integrate each 

 term with respect to x between the limits and 2n, 



Then I y cos nx dx 

 Jo 



f 2 * f 2 ' -. p' 



= B | cos nx dx + T&A cosxcosnxdx + . . . BJ cos 2 nx dx+ . . . 

 Jo Jo Jo 



+ AJ sinxcosnxdx + . . . A n \ sinnxcosnxdx+ ... 

 Jo Jo 



and all the integrals on the right-hand side vanish except 



f 2 * 



B m l cos 2 nx dx, which becomes 7zB n . 



Jo 



Hence 7iB n = I y cos nx dx 



Jo 



1 f 2 ' 



or B n = I y cos nx dx 



(3) If we multiply throughout by sin nx and integrate each 

 term with respect to x between the limits and 2TC, 



ft, 



Then I y sin nx dx 

 Jo 



f2r f2 f2 



= B I sinnircte + Bji cosa?sinwo;dir + . . . B w l cosnxsinnxdx+. . 

 Jo Jo Jo 



+ A.A sinxsinnxdx + . . . A n l sin 2 w# <c + . . . 

 Jo Jo 



and all the integrals on the right-hand side vanish except 



A n I sin 2 nx dx, which becomes 7rA n . 

 Jo 



r- 

 y sin nx dx 

 



I f 2 * 

 or A n = I y sin nx dx 



TlJo 



Working with the same series in a similar manner between the 

 limits n and TC, it can be shown that 



1 f ff 



^ = ~~ I y cos 



71 J TT 

 1 f* 



and A M = I y sin 



TC J TT 



