If $\int e^u \sin 2x \, dx$ can be expressed in terms of elementary functions of $x$,then $u$ can be:

If $\int \frac{(x^2-1)}{(x+1)^2 \sqrt{x(x^2+x+1)}} dx = A \tan^{-1}\left(\sqrt{\frac{x^2+x+1}{x}}\right) + C$, where $C$ is a constant, then $A$ equals to

If $0 < a < 1$,then $\int \frac{dx}{1-2a \cos x + a^2} =$

$\int \left[ \log (1+\cos x) - x \tan \left( \frac{x}{2} \right) \right] dx =$

The value of $I = \int \frac{dx}{x^2(x^4+1)^{3/4}}$ is

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