Posted by alsa on Wednesday, June 13, 2012 at 5:51pm.
You have not proved to me that the Integral of lnsinx w.r.t.x from 0 to pi/2 is equal to:Integral of lnsinx w.r.t.x from 0 to pi/4 plus Integral of lncosx w.r.t.x from 0 to pi/4.

Integral calculus  Damon, Wednesday, June 13, 2012 at 6:58pm
graph sin x from 0 to pi/2
on the same graph plot cos x from 0 to pi/4
Note that cos x from pi/4 to pi/2 is identical to sin x from pi/4 to pi/2 with only a reflection. 
Integral calculus  Damon, Wednesday, June 13, 2012 at 7:00pm
Thus for every value of ln sin x from pi/4 to p/2 there is an equal value of ln cos x from 0 to pi/4. Therefore the areas under will be identical.

Integral calculus  alsa, Wednesday, June 13, 2012 at 7:58pm
Damon: That is a good answer which now I really understand.I thank you very much.