THE SINE SERIES 



381 



Then 



and between and - 



1 



but between - and TC 



I 



2c 



TC 



2c 



TC 



2c, 



y = W(TC a;) - (TC a?) 

 Then y = A x sin a; + A 2 sin 2a; -f . . . A,, sin nx + ... 



and 



sin 2 nx dx = -TcA_ 

 2 



I y sin nx dx = A n I sii 

 Jo Jo 



But in taking the integral I ysinnx dx it must be remem- 



Jo 



bered that between and y = x, but between and TC 



2 TC 2 



2c. . 



__ / .. __ Ail 



' TC V 



Therefore I w sin nx dx 



I ?/ si 

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2c f 2c f' . 



= V x sin M<r dx H -- I (TC #) si 



^J 



sin 



2c r f- p' f* ~i 



2 x sin nx dx + TT I, sin nx dx \ x sin nx dx \ 



K IJo J J 



2c /T a? cos w.2; sin nx~& r cos n<r~l* r x cos 



-ill * J: + *L- JrL n 





sm 



2c f I 



TC cos mt 

 n 



cos TJTC 1 nTc\1 

 n -- ^ Sm TJ) 



Ten 



sin 



flTC 



TC. 4c . WTC 

 But -A n = 5 sm - 

 2 ' Ten 2 2 



8c 



Sln 



vtir Rf 1 



when n = 1 sin - = 1 and A, = ^ -r 



2 TC 2 I 2 



when n = 2 sin = and A 2 = 



