$\int \frac{2 \sin x - 3 \cos x}{4 \cos x - 3 \sin x} dx = $

$k \in N, \int \frac{1-k \cos ^2 x}{\sin ^k x \cdot \cos ^2 x} d x=$

$\int \frac{dx}{x^2 \sqrt{4+x^2}}$ is equal to

If $\int \frac{5 \tan (x)}{\tan (x)-2} d x = x + a \log |\sin (x) - 2 \cos (x)| + k$,then $a$ is equal to

If $\int \operatorname{cosec}^5 x \, dx = \alpha \cot x \operatorname{cosec} x \left(\operatorname{cosec}^2 x + \frac{3}{2}\right) + \beta \log_e \left|\tan \frac{x}{2}\right| + C$,where $\alpha, \beta \in R$ and $C$ is the constant of integration,then the value of $8(\alpha + \beta)$ is equal to:

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