If $\bar{a}$ and $\bar{b}$ are two unit vectors such that $5 \bar{a}+4 \bar{b}$ and $\bar{a}-2 \bar{b}$ are perpendicular to each other, then the between $\bar{a}$ and $\bar{b}$ is

The value of $\int \frac{\cos ^3 x}{\sin ^2 x+\sin x} \mathrm{~d} x$ is

Let $\mathrm{P} \equiv(-5,0), \mathrm{Q} \equiv(0,0)$ and $\mathrm{R} \equiv(2,2 \sqrt{3})$ be three points. Then the equation of the bisector of the angle $P Q R$ is

$$\cos \left[\sin ^{-1}\left(\frac{3}{5}\right)+\cos ^{-1}\left(\frac{12}{13}\right)\right]=$$