$$ \int \frac{1}{\mathrm{e}^x+1} \mathrm{~d} x= $$

If $y=\tan ^{-1}\left(\frac{4 x}{1+5 x^2}\right)+\cot ^{-1}\left(\frac{3-2 x}{2+3 x}\right)$, then $\frac{\mathrm{d} y}{\mathrm{~d} x}$ is equal to

The differential equation $x \frac{\mathrm{~d} y}{\mathrm{~d} x}=2 y$ represents ________

$\int \mathrm{e}^x\left(\frac{x+5}{(x+6)^2}\right) \mathrm{d} x$ is