If $\int \frac{\cos 4x + 1}{\cot x - \tan x} dx = k \cos 4x + c$,then $k$ is

If $\int \frac{x-\sin x}{1+\cos x} dx = x \tan \left(\frac{x}{2}\right) + p \log \left|\sec \left(\frac{x}{2}\right)\right| + C$,then $p$ is equal to

For $\alpha, \beta, \gamma, \delta \in \mathbb{N}$,if $\int \left( \left( \frac{x}{e} \right)^{2x} + \left( \frac{e}{x} \right)^{2x} \right) \log_{e} x \, dx = \frac{1}{\alpha} \left( \frac{x}{e} \right)^{\beta x} - \frac{1}{\gamma} \left( \frac{e}{x} \right)^{\delta x} + C$,where $e = \sum_{n=0}^{\infty} \frac{1}{n!}$ and $C$ is the constant of integration,then $\alpha + 2\beta + 3\gamma - 4\delta$ is equal to:

If $\int \frac{a \cos x-2 \sin x}{b \sin x+5 \cos x} d x=\frac{7}{41} x+\frac{22}{41} \log |b \sin x+5 \cos x|+C, (a>0, b>0)$, then $\int \frac{d x}{b+a \cos x}=$

If $\int \frac{d x}{\cos ^4 x+\sin ^4 x}=\frac{1}{\sqrt{2}} \tan ^{-1}[g(x)]+C$,then $g(x)$ equals

If $\int \frac{\cos x \, dx}{\sin ^{3} x \left(1+\sin ^{6} x\right)^{2 / 3}} = f(x) \left(1+\sin ^{6} x\right)^{1 / \lambda} + c$ where $c$ is a constant of integration,then $\lambda f\left(\frac{\pi}{3}\right)$ is equal to