Evaluate the integral $\int \frac{x^2 + \cos^2 x}{1 + x^2} \operatorname{cosec}^2 x \, dx$,where $c$ is the constant of integration.

$\int \frac{\sin x \cdot \sec ^2 x-\tan x \cdot \sin x+\cos x}{(1-\cos 2 x)} d x=$

$\int \frac{6x+5}{\sqrt{6+x-2x^2}} dx =$

If $n$ is a positive integer greater than $1$ and $I_{n}=\int \frac{\sin n x}{\sin x} d x$, then $I_{n+1}-I_{n-1}=$

For $\alpha, \beta, \gamma, \delta \in \mathbb{N}$,if $\int \left( \left( \frac{x}{e} \right)^{2x} + \left( \frac{e}{x} \right)^{2x} \right) \log_{e} x \, dx = \frac{1}{\alpha} \left( \frac{x}{e} \right)^{\beta x} - \frac{1}{\gamma} \left( \frac{e}{x} \right)^{\delta x} + C$,where $e = \sum_{n=0}^{\infty} \frac{1}{n!}$ and $C$ is the constant of integration,then $\alpha + 2\beta + 3\gamma - 4\delta$ is equal to:

Let $g(x)$ be an antiderivative for $f(x)$. Then $\ln(1 + (g(x))^2)$ is an antiderivative for