The value of the integral $\int\limits_{\frac{\pi}{6}}^{\frac{\pi}{3}}\left(\frac{4-\operatorname{cosec}^2 x}{\cos ^4 x}\right) d x$ is :

If $\alpha=1$ and $\beta=1+i \sqrt{2}$, where $i=\sqrt{-1}$ are two roots of the equation

$x^3+a x^2+b x+c=0, a, b, \in \mathbb{R}$, then $\int_{-1}^1\left(x^3+a x^2+b x+c\right) d x$ is equal to:

The integral $\int\limits_0^1 \cot ^{-1}\left(1+x+x^2\right) d x$ is equal to :

Let $f$ be a real polynomial of degree $n$ such that $f(x)=f^{\prime}(x) f^{\prime \prime}(x)$, for all $x \in \mathbb{R}$. If $f(0)=0$, then $36\left(f^{\prime}(2)+f^{\prime \prime}(2)+\int_0^2 f(x) d x\right)$ is equal to: