34 DIFFERENTIAL AND INTEGRAL CALCULUS. 



f^ 71 " f^"" f (it \) 



So also I \og(l+ismx) dx= I log \1 + ta.nl -- xj^dx 



JQ JQ (, \ /) 



= j^ log(l 4 }=) dx - J* jlogSJ - log (1 + tan*)) dx 



7T /*4 7r 



= - Iog2 - I log (I + tan#) dx, 

 4 J 



or I log(l + tan#) dx = Iog2. 



J* 



The following makes use of both formulae, 



n-ir 



let u=i log (sin a?) dx, 



' 



r^r 

 then M= I log (cos a?) dx by (1); 



^o 



ri* 



therefore 2u = I (log sina; + log cos a?) ^a?, 



^o 



or 2M=( logf - jdx=l log (sin 2a?) ^ - Iog2. 

 J V ^ / J * 



c i 71 " r i 71 " 



But I log sin2a;^ = J I log (sin 2a?) c? (2a?) 



^ o 'o 



fir 



-tl log (sin a?) ^, 



^0 



/i 

 (log sina; + log sin (TT - a?)} fZa: by (2) 



(2 Igg sina>) db = u 5 



therefore 2u = u log 2, or u log( \] 

 and we see by the proof that also 



/* ITT 77- r TT 



I \og(cosx]dx=- log(J) ; and I log(smx]dx=7r\og(l}. 



J o * o 



The most useful application of (2) is to such as the 

 following : 



f ir rl-jr 



dx; 



