Evaluate: $\int \frac{2 \cos x+1}{(2+\cos x)^2} d x - \frac{\sin x}{2+\cos x}$

$\int {\frac{{\sec x \cdot \csc x}}{{2\cot x - \sec x \cdot \csc x}}} dx$ is equal to (where $c$ is the constant of integration).

$\text{If } \int \frac{1}{\operatorname{cosec} x+\cos x} d x = \frac{1}{2 \sqrt{3}} \log |f(x)| - \int \frac{\cos x-\sin x}{2+\sin 2 x} d x + c, \text{ then at } x = \frac{\pi}{3}, |f(x)| = $

For any integer $n \geq 2$,let $I_n = \int \tan^n x \, dx$. If $I_n = \frac{1}{a} \tan^{n-1} x - b I_{n-2}$ for $n \geq 2$,then the ordered pair $(a, b)$ is equal to

If $\int {\frac{{dx}}{{{x^3}{{\left( {1 + {x^6}} \right)}^{2/3}}}} = xf\left( x \right){{\left( {1 + {x^6}} \right)}^{\frac{1}{3}}} + C} $ where $C$ is a constant of integration,then the function $f(x)$ is equal to

If $\int \frac{5 \tan x}{\tan x-2} dx=ax+b \log |\sin x-2 \cos x|+c$,then $a+b=$