Laplace transform

Definition

\[ F(s) = \int_0^{\infty} f(t)e^{-st}\,dt = \mathcal{L}\{f(t)\} \]

Find the Laplace transforms

1. \( 3e^{-4t} \)
2. \( 2t^2 \)
3. \( 4\cos(5t) \)
4. \( \sin(\pi t) \)
6. \( \sin(t)\cos(t) \)

Find the inverse Laplace transforms

1. \[ F(s) = \frac{2s^{2} - 4}{(s - 2)(s + 1)(s - 3)} \]
2. \[ F(s) = \frac{3s + 1}{(s - 1)(s^{2} + 1)} \]