If $I_n = \int \frac{\sin nx}{\cos x} dx$,then $I_n =$

If $\int {\frac{{\csc^2 x}}{{{{\left( {\csc x + \cot x} \right)}^{\frac{9}{2}}}}}\,dx} = {\left( {\csc x - \cot x} \right)^{\frac{7}{2}}}\left( {\frac{1}{\alpha } + \frac{{{{\left( {\csc x - \cot x} \right)}^2}}}{{11}}} \right) + C$ (where $C$ is the constant of integration and $\alpha \in N$),then $\alpha$ is:

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:

If $\int \frac{3e^x - 5e^{-x}}{4e^x + 5e^{-x}} dx = px + q \cdot \log |4e^x + 5e^{-x}| + C$,then

$\int \frac{d x}{4 \sin x+3 \cos x}=$

The value of $\int_{\frac{1}{3}}^1 (x - x^3)^{\frac{1}{3}} dx$ is