The value of $\int_{0}^{\pi / 2} \log (\operatorname{cosec} x) d x$ is

If $f : R \rightarrow R$ is a continuous function satisfying $\int \limits_0^{\pi / 2} f(\sin 2x) \cdot \sin x \, dx + \alpha \int \limits_0^{\pi / 4} f(\cos 2x) \cdot \cos x \, dx = 0$,then $\alpha$ is equal to

$\int_{-1}^{1} x \tan^{-1} x \, dx$ equals

$\int_0^{\frac{\pi}{4}} \frac{\cos ^2 x}{\cos ^2 x+4 \sin ^2 x} d x=$

For $U_n = \int\limits_0^1 x^n (2 - x)^n \, dx$ and $V_n = \int\limits_0^1 x^n (1 - x)^n \, dx$,where $n \in N$,which of the following statements is true?

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