If $$\mathrm{f}^{\prime}(x)=\tan ^{-1}(\sec x+\tan x),-\frac{\pi}{2} < x < \frac{\pi}{2}$$ and $$f(0)=0$$, then $$\mathrm{f}(1)$$ is

If $$\int \frac{\cos \theta}{5+7 \sin \theta-2 \cos ^2 \theta} d \theta=A \log _e|f(\theta)|+c$$ (where $$c$$ is a constant of integration), then $$\frac{f(\theta)}{A}$$ can be

If $$\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}$$ are unit vectors and $$\theta$$ is angle between $$\overline{\mathrm{a}}$$ and $$\bar{c}$$ and $$\bar{a}+2 \bar{b}+2 \bar{c}=\overline{0}$$, then $$|\bar{a} \times \bar{c}|=$$

The integral $$\int_\limits{\frac{\pi}{6}}^{\frac{\pi}{3}} \sec ^{\frac{2}{3}} x \operatorname{cosec}^{\frac{4}{3}} x d x$$ is equal to