Find the Laplace transforms
d) \( f(t) = \sin(\pi t) \)
\[
F(s) = \mathcal{L}\{ \sin(\pi t) \}
= \frac{\pi}{s^2 + \pi^2}
\]
\[
\boxed{\frac{\pi}{s^2 + \pi^2}}
\]
e) \( f(t) = \begin{cases} 3, & 0 < t < 2 \\ -1, & 2 \le t < 4 \\ 0, & t \ge 4 \end{cases} \)
Use the Laplace transform definition:
\[
F(s) = \int_{0}^{\infty} f(t)e^{-st}\,dt
\]
\[
F(s) = \int_{0}^{2} 3e^{-st}\,dt + \int_{2}^{4} (-1)e^{-st}\,dt + 0
\]
\[
F(s) = \frac{3 - 4e^{-2s} + e^{-4s}}{s}
\]
\[
\boxed{\frac{3 - 4e^{-2s} + e^{-4s}}{s}}
\]
f) \( f(t) = \sin(t)\cos(t) \)
\[
f(t) = \tfrac{1}{2}\sin(2t)
\]
\[
F(s) = \tfrac{1}{2}\mathcal{L}\{\sin(2t)\}
= \tfrac{1}{2}\frac{2}{s^2 + 4}
= \frac{1}{s^2 + 4}
\]
\[
\boxed{\frac{1}{s^2 + 4}}
\]