$$\int\limits_{\frac{\pi}{3}}^{\frac{\pi}{2}} \frac{\sqrt{1+\cos x}}{(1-\cos x)^{\frac{5}{2}}} d x=$$

If the complex number $z=x+i y$, where $i=\sqrt{-1}$, satisfies the condition $|z+1|=1$, then $z$ lies on

The general solution of the differential equation $\frac{d y}{d x}=\frac{3 e^{2 x}+3 e^{4 x}}{e^x+e^{-x}}$ is

If $\cos ^{-1} x=\alpha(0< x < 1)$ and $\sin ^{-1}\left(2 x \sqrt{1-x^2}\right)+\sec ^{-1}\left(\frac{1}{2 x^2-1}\right)=\frac{2 \pi}{3}$, then $\alpha$ is