The value of $\cot \left(\operatorname{cosec}^{-1} \frac{5}{3}+\tan ^{-1} \frac{2}{3}\right)$ is

$\int\left(1+x-\frac{1}{x}\right) e^{x+\frac{1}{x}} d x$ equal to

The function $f(x)=\frac{\log _e(\pi+x)}{\log _e(e+x)}$ is

Let $y=y(x)$ be the solution of the differential equation $\sin x \frac{\mathrm{~d} y}{\mathrm{~d} x}+y \cos x=4 x, x \in(0, \pi)$. If $y\left(\frac{\pi}{2}\right)=0$, then $y\left(\frac{\pi}{6}\right)$ is equal to