To find the Laplace transform of $f(t) = e^{2t} \sin(3t)$, we can use the Laplace transform shifting theorem. The shifting theorem states that if:

$\mathcal{L}\{e^{at} f(t)\}(s) = F(s-a)$

We will start by finding the Laplace transform of $\sin(3t)$ and then apply the shifting rule.

Step 1: Find the Laplace transform of $\sin(3t)$

The Laplace transform of $\sin(bt)$ is:

$\mathcal{L}\{\sin(bt)\}(s) = \frac{b}{s^2 + b^2}$

For $\sin(3t)$, we substitute $b = 3$:

$\mathcal{L}\{\sin(3t)\}(s) = \frac{3}{s^2 + 9}$

Step 2: Apply the shifting theorem

Now, using the shifting theorem for $e^{2t} \sin(3t)$, we replace $s$ with $s - 2$ in the Laplace transform of $\sin(3t)$:

$\mathcal{L}\{e^{2t} \sin(3t)\}(s) = \frac{3}{(s - 2)^2 + 9}$

Thus, the Laplace transform of $f(t) = e^{2t} \sin(3t)$ is:

$\mathcal{L}\{f(t)\}(s) = \frac{3}{(s - 2)^2 + 9}$

Would you like more details or have any questions?

Here are 5 related questions to explore further:

1. How do you apply the Laplace transform to exponential functions?
2. What is the inverse Laplace transform of $\frac{3}{(s - 2)^2 + 9}$?
3. How does the shifting theorem work for different kinds of functions?
4. Can you find the Laplace transform of $e^{3t} \cos(4t)$?
5. How does the convolution theorem relate to the Laplace transform?

Tip: Always check for initial conditions when applying the Laplace transform to differential equations!