To evaluate $\int \frac{\sec^2 x}{(1 + \tan x)(2 + \tan x)} \, dx$,the most suitable substitution is

$\int \frac{d x}{(x+100) \sqrt{x+99}}=f(x)+c \Rightarrow f(x)$

If $f(x)=\int \frac{x^2 \, dx}{(1+x^2)(1+\sqrt{1+x^2})}$ and $f(0)=0$,then $f(1)$ is

If $\int \frac{\sin x}{\sin (x-\alpha)} dx = Ax + B \log |\sin (x-\alpha)| + c$,then the values of $A$ and $B$ are respectively (where $c$ is a constant of integration).

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

$\int \frac{dx}{\sqrt{e^x-1}} = 2 \tan^{-1}(f(x)) + c$,where $x > 0$ and $c$ is a constant of integration,then $f(x)$ is