$$ f(t):= \begin{cases} 3\cos(2t) & t<5 \\ 4\sin(3t) & 5 \leq t \leq 9 \\ 0 & t>9. \end{cases} $$
The above function I know how to solve using straight Laplace Transformation by Integration. However these integrals are not very quick to calculate by hand. I was told that there is a property involving Laplace Transformations that may make this problem quicker. Any suggestions which properties would help to find the Laplace transform in the most efficient manner?
We can rewrite your function using the Heaviside Unit Step function and then we can use the Shifting Property and Laplace Transform table. I would recommend practicing this with these examples.
Step 1: Lets plot your piece-wise function. We get:
Step 2: Lets rewrite the piece-wise function using the unit-step function as a single function.
We get:
$$f(t) = 3 \cos(2t) + (4 \sin(3t) - 3 \cos(2t)) u(t-5) + (0 - 4 \sin(3t)) u(t - 9)$$
Lets plot this function and it should be identical to the first one.
Step 3
Now, using the Laplace Table and the shifting property, we can easily write the Laplace Transform.
The final result will be:
$$\mathcal{L}\{f(t)\} = -\dfrac{3 e^{-5 s} (s \cos(10) - 2 \sin(10))}{s^2 + 4} + \dfrac{ 4 e^{-5 s} (3 \cos(15) + s \sin(15))}{s^2 + 9}$$