$\int_{\frac{-\pi}{4}}^{\frac{\pi}{3}}\left|\tan \left(x-\frac{\pi}{6}\right)\right| d x=$

$$ \int_0^\pi \frac{x \sin x}{\sin ^2 x+2 \cos ^2 x} d x= $$

$$ \mathop {\lim }\limits_{n \to \infty }\left(\frac{1}{1^2+n^2}+\frac{2}{2^2+n^2}+\frac{3}{3^2+n^2}+\ldots+\frac{n}{n^2+n^2}\right)= $$

$$ \int_0^{\frac{\pi}{2}} \log |\tan x+\cot x| d x= $$