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

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

$$ \int_0^\pi x \cdot \sin ^5 x \cdot \cos ^6 x d x= $$

$$ \int_{\frac{1}{2}}^{\frac{1}{\sqrt{2}}} \frac{1}{\left(x+\sqrt{1-x^2}\right)\left(1-x^2\right)} d x= $$