Faster way of solving Laplace problems like this

Question

I have been practicing some Laplace problems for my exam. I keep coming across problems like the following:
$$L[\frac{d}{dt}e^{-at}cos^{2}(\omega t)u(t)]$$

I have done the differentiation bit here:

The final derivative looks messy and I don't think I'll have enough time to do the Laplace transform for a problem like this fully if it's in my exam. Is there a quicker way of doing problems like these? I can work out the Laplace transform but all I am asking for is if there is a quicker way of solving the problem.

Answer 1

Here's where all of those Laplace transform properties come in handy.

$$\mathcal{L}\left(\frac{d}{dt}f(t)\right) = sF(s) - f(0)$$

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

$$\mathcal{L}(a\cdot f(t)+b\cdot g(t)) = a\cdot F(s) + b\cdot G(s)$$

$$\mathcal{L}(\cos at) = \frac{s}{s^2+a^2}$$

Using the fact that $\cos^2 \omega t = \frac{1}{2} + \frac{1}{2}\cos 2\omega t$, we have that

$$\frac{1}{2}\mathcal{L}\left(\frac{d}{dt}e^{-at}(1+\cos 2\omega t)\right) = \frac{1}{2}s\left(\frac{1}{s+a} + \frac{(s+a)}{(s+a)^2+4\omega^2}\right) - 1$$