If $\int \frac{a \cos x+3 \sin x}{5 \cos x+2 \sin x} d x=\frac{26}{29} x-\frac{k}{29} \log |5 \cos x+2 \sin x|+c$,then $|a+k|=$

$\int \frac{1}{(x^2+1)^2} dx = . . . . . .$

The value of $I = \int_{0}^{\frac{\pi}{4}} \tan^{n+1} x \, dx + \frac{1}{2} \int_{0}^{\frac{\pi}{2}} \tan^{n-1} \left( \frac{x}{2} \right) \, dx$ is

If $\int \frac{1}{\cos 4x \cos 2x} dx = \frac{1}{2\sqrt{2}} \log \left(\frac{1+f(x)}{1-f(x)}\right) - \frac{1}{2} \log g(x) + C$,then find the value of $g\left(\frac{\pi}{6}\right) - \sqrt{2} f\left(\frac{\pi}{6}\right)$.

$\int \frac{\sec x}{3(\sec x+\tan x)+2} d x=$

If $\int \sin ^{-1}\left(\sqrt{\frac{x}{1+x}}\right) d x=A(x) \tan ^{-1}(\sqrt{x})+B(x)+C$ where $C$ is a constant of integration,then the ordered pair $(A(x), B(x))$ can be