and .
Now let then
For the last integral, see here http://www.mathhelpforum.com/math-he...lp-needed.html
and .
Now let then
For the last integral, see here http://www.mathhelpforum.com/math-he...lp-needed.html
I'll take it from here: in fact consider whereas by putting this leads to whereat by adding these (provided that for each is ) we get and the integral equals which holds for all
Alas, this statement does not hold for all besides, by differentiating your result, we don't get the integrand. (Actually, there's no primitive in terms of elementary functions.)
I=integrate{ln(xtanx) dx}[lim pi/6 to pi/3]
I=integrate{ln((pi/2-x)cotx) dx}[lim pi/6 to pi/3]
so,adding,2I=integrate{ln(x*(pi/2-x)) dx)[lim pi/6 to pi/3]
2I=integrate {lnx}[lim pi/6 to pi/3] + integrate {ln(pi/2-x)}[lim pi/6 to pi/3]
the rest is easy.