To find the value of a constant when the Laplace transform of a function is given

Question

This question is regarding my previous post

https://math.stackexchange.com/questions/1616836/find-the-value-of-a-constant-when-the-laplace-transform-of-a-function-is-given

where the hint was given by Moo to find the laplace transform of $\frac{-sin~t}{t}$..

My sol:

$\mathcal{L}(-sin ~t) = \frac{-1}{s^2 +1}$ and thus

$\mathcal{L}(\frac{-sin ~t}{t}) = \int_{s}^{\infty}\frac{-1}{s^2 +1}$ which implies that

$\mathcal{L}(\frac{-sin ~t}{t}) =\frac{\pi}{2} + tan^{-1}s$

which gives $k = \frac{\pi}{2}$.

Is my answer correct? If not then help me. Thanks for the time and help.