f(t)=tsint ..........
f(0)=0
f′(t)=tcost+sint .......... f′(0)=0
f″
\mathcal{L} \left\{ f''(t) \right\} = s^2 \mathcal{L} \left\{ f(t) \right\} - s \, f(0) - f'(0)
\mathcal{L} (-t \, \sin t + 2\cos t) = s^2 \mathcal{L} (t \, \sin t) - s \, (0) - (0)
-\mathcal{L} (t \, \sin t) + 2\mathcal{L} (\cos t) = s^2 \mathcal{L} (t \, \sin t)
2\mathcal{L} (\cos t) = s^2 \mathcal{L} (t \, \sin t) + \mathcal{L} (t \, \sin t)
s^2 \mathcal{L} (t \, \sin t) + \mathcal{L} (t \, \sin t) = 2\mathcal{L} (\cos t)
(s^2 + 1) \mathcal{L} (t \, \sin t) = 2\mathcal{L} (\cos t)
(s^2 + 1) \mathcal{L} (t \, \sin t) = 2 \left( \dfrac{s}{s^2 + 1^2} \right)
(s^2 + 1) \mathcal{L} (t \, \sin t) = \dfrac{2s}{s^2 + 1}
\mathcal{L} (t \, \sin t) = \dfrac{2s}{(s^2 + 1)(s^2 + 1)}
\mathcal{L} (t \, \sin t) = \dfrac{2s}{(s^2 + 1)^2} answer