NOTE ON A DEFINITE INTEGRAL. 



THE value of / log sin 6 d0, obviously the same as that of 



IT ./O 



rlog cos 6d0, was first assigned by Euler, ,and may be obtained 

 

 in the following manner. 



By Cotes's theorem, 



/ i \ / m i \ 



Z^-l^fz 2 - 1) Z 2 - 2Z COS - 7T+ 1 ... (Z*- 2Z COS 7T + 1 



' \ m J \ m J 



w- 



Let = 1, then 



^ 



m 2 m 2 



Take the logarithms of both sides, and divide by m, then 



log m + 2 (m 1) log J 

 2m 



A . 1 TT , . m 1 7r\ 1 



= log sin -+...+ log sm . 



\ * m 2 m 2J m 



Let m become infinite, and = ^- : the first side becomes equal 



to log - , for ( - j = when m = - ; and the second is trans- 

 A \ ffi / 



formed into the definite integral I log sin x ^ . dx ; therefore 



'o ^ 



I log sin x dx = log J. 

 ^o * 



Let ^ = |; 



* Cambridge Mathematical Journal, No. XII. Vol. n. p. 282, May, i84f. 



