If $\int \frac{x^2-x+2}{x^2+x+2} d x=x-\log (f(x))+\frac{2}{\sqrt{7}} \operatorname{Tan}^{-1}(g(x))+c$,then $f(-1)+\sqrt{7} g(-1)=$

If $\int f(x) dx = g(x) + c$, then $\int f^{-1}(x) dx =$

If $\int \frac{2 x^{12}+5 x^9}{\left(x^5+x^3+1\right)^3} d x=\frac{1}{2} f(x)+C$, then $f(1)-f(0)=$

$\int \frac{1}{(x^2+1)^2} dx = . . . . . .$

If $\int \frac{dx}{(1+\sqrt{x}) \sqrt{x-x^2}} = \frac{A \sqrt{x}}{\sqrt{1-x}} + \frac{B}{\sqrt{1-x}} + C$, where $C$ is a real constant, then $A+B$ 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: