Let $a_n=\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{1-\cos 2nx}{1-\cos 2x}dx,$ then $a_1,a_2,a_3,......$ are in

If $x$ satisfies the equation $\left({\int }_0^1\frac{dt}{t^2+2t\cos \alpha +1}\right)x^2-\left(\underset{-3}{\overset{3}{\int }}\frac{t^2\sin 2t}{t^2+1}\right)x-2=0, \left(0<\alpha <\mathrm{\pi }\right)$, then the value of $x$ is

The value of $f\left(k\right)={\int }_0^{\frac{\mathrm{\pi }}{2}}\log \left({\sin }^2\theta +k^2{\cos }^2\theta \right)d\theta$ is equal to