$\int \frac{\sin x+8 \cos x}{4 \sin x+6 \cos x} d x$ is equal to

If $\int \frac{1 - (\cot x)^{2021}}{\tan x + (\cot x)^{2022}} dx = \frac{1}{A} \log |(\sin x)^{2023} + (\cos x)^{2023}| + c$,then $A = . . . . . .$

$\begin{aligned} & \int \frac{x \, dx}{\sqrt[15]{\left(1+x^2\right)^{12}\left(2+x^2\right)^{18}}}=\alpha\left(\frac{1+x^2}{2+x^2}\right)^{1 / n}+C \Rightarrow \\ & \frac{n}{\alpha}= \end{aligned}$

$\int \frac{dx}{\sin(x-a) \cos(x-b)} = $

If the graph of the anti-derivative $g(x)$ of $f(x) = \log(\log x) + (\log x)^{-2}$ passes through $(e, 2023 - e)$ and the term independent of $x$ in $g(x)$ is $k$,then the sum of all the digits of $k$ is

Let $I(x) = \int \frac{dx}{(x-11)^{\frac{11}{13}}(x+15)^{\frac{15}{13}}}$. If $I(37) - I(24) = \frac{1}{4} \left( \frac{1}{b^{\frac{1}{13}}} - \frac{1}{c^{\frac{1}{13}}} \right)$,where $b, c \in \mathbb{N}$,then $3(b+c)$ is equal to