If $\int \frac{1-(\cot x)^{2019}}{\tan x+(\cot x)^{2020}} dx = \frac{1}{n} \ln |(f(x))^n + (g(x))^n| + c$, then the value of $n[(f(x))^4 + (g(x))^4]_{x=\frac{\pi}{3}}$ is:

If $\int \frac{dx}{x^{7/2}(x^4+1)^{3/8}} = m \left(\frac{x^4+1}{x^4}\right)^n + c$, where $c$ is a constant of integration, then the value of $n$ is...

The integral $\int\left(\left(\frac{x}{2}\right)^x+\left(\frac{2}{x}\right)^x\right) \log _2 x \, dx$ is equal to

$\int \frac{x+\sin x}{1+\cos x} d x$ is equal to

$\int \frac{d x}{\sqrt{\left(5+2 x+x^2\right)^3}}$ is equal to

Let $\alpha \in (0, \pi /2)$ be fixed. If the integral $\int \frac{\tan x + \tan \alpha}{\tan x - \tan \alpha} dx = A(x) \cos 2\alpha + B(x) \sin 2\alpha + C$,where $C$ is a constant of integration,then the functions $A(x)$ and $B(x)$ are respectively: