If $\int \frac{5 \tan x}{\tan x-2} \, dx = x + a \log |\sin x - 2 \cos x| + c$ (where $c$ is a constant of integration),then the value of $a$ is

$\int \frac{1}{\sqrt{1 + \sin x}} \, dx = $

If $\int {\frac{{\sqrt {1 - {x^2}} }}{{{x^4}}}} dx\, = \,A(x)\,{(\sqrt {1 - {x^2}} )^m}\, + \,C,$ for a suitably chosen integer $m$ and a function $A(x),$ where $C$ is a constant of integration,then $(A(x))^m$ equals

Observe the following statements :
$A: \int \left(\frac{x^2-1}{x^2}\right) e^{\frac{x^2+1}{x}} d x = e^{\frac{x^2+1}{x}} + c$
$R: \int f^{\prime}(x) e^{f(x)} d x = f(x) + c$
Then which of the following is true?

$\int \sin ^{-1}\left(\sqrt{\frac{x-a}{x}}\right) d x=$

If the integral $\int \frac{\cos 8x + 1}{\cot 2x - \tan 2x} dx = A \cos 8x + k,$ where $k$ is an arbitrary constant,then $A$ is equal to