Here, f(t) = $\frac{sint}{t}$

We first take,

L[sint] = $\frac{1}{s^2 + 1}$

then,

$L \bigg[\frac{sint}{t} \bigg] = \int\limits_{s}^{\infty} \frac{1}{s^2 + 1} ds$

$\therefore L \bigg[\frac{sint}{t} \bigg] = \big[tan^ {-1} (s) \big]_{s}^{\infty}$

$\therefore L \bigg[\frac{sint}{t} \bigg] = tan^{-1} (\infty) - tan^{-1}(s)$

$\therefore L \bigg[\frac{sint}{t} \bigg] = \frac{\pi}{2} - tan^{-1}(s)$

$\therefore L \bigg[\frac{sint}{t} \bigg] = cot^{-1}(s)$

Now, $L\bigg[\frac{d}{dt} \bigg(\frac{sint}{t} \bigg) \bigg] = sf(s) - f(0)$

$\therefore f(0) = cot^{-1} (0) = \frac{\pi}{2}$

$L\bigg[\frac{d}{dt} \bigg(\frac{sint}{t} \bigg) \bigg] = s cot^{-1} (s) - \frac{\pi}{2}$