Inverse Laplace transform of $\tan^{−1}\left(\frac{1}{s}\right)$ I'm studying Laplace transformations, but I don't understand where $-\frac{1}{t}$ comes from. And what is the relationship between the corollary and the example?


Think of $\tan^{-1} \left( \frac 1 s\right)$ as the antiderivative of $\frac{-1}{s^2+1}$.
Then in your example, $n=1$. This introduces a factor of $1/t$ on the left hand side, and a negative sign on the right hand side.
