DIFFERENTIAL AND INTEGRAL CALCULUS. 35 



r 7T r ^7T 



/ F(cosx}dx = {F (cos x) + F (- cos a?) } dx, 



J o o 



and of (1) I .F(sina;) cfojj* I jP(cos;e) cfo. 



* * a 



Thus 



ir r&ir rifir rlir 



_ 2_ 7- / . -_jr_ / 2 i/i ._' 2 \_7_ 1_ / 2 _ 



r** 1 2 r** 1 a r^ [* 



I I I I 



JO J o ^0 



r ^TT 



therefore I a slrfxdx = JTT. 



^ 



Integrating by parts 



T^TT 



I (log sin a?) dx 



= (x log sinz)^ - [ " x cotxdx = - f -- cfo, 



the former vanishing at both limits ; therefore 



Jo tana: 



~ U tana?/ 3 J sm*aj ' " 2 J sin 2 oj 



therefore I -T-JT~ dx = 7r log 2. 



/o sm ^ 



r^r 

 The integral I log (sin a?) dx may be evaluated thus : 



Jo 



= \ I log (sin 2 #) dx = % I log (cos 2 #) ^o; 



o J o 



fi* 



= i I log^l sin 2 ic) (7ic 



^ 



ri 71 " 

 ~ " ' J }sm x + ^ sm ^ + ^ sm ^ + -1 



1 1.3 1 1.3.5 



T - '^ 1 ' 3 , 



But 



