[Math] How to find the Laplace transform of $t\cos{t}$

Question

I need to find the Laplace transform of $f(t) = t \cos{t}$. I tried using the Taylor series expansion for $\cos{t}$ but I got stuck since the resulting expression is again a series which I could not simplify.

I would like to know if there is an easier method to find this transform without using Taylor series. I don't want the answer, just need to learn how to find it.

Thanks.

Answer 1

We have:

$$\mathcal{L}(\cos(t)) = \dfrac{s}{s^2+1}$$

$$\mathcal{L}(t \cos(t)) = -\dfrac{d}{ds} \left(\dfrac{s}{s^2+1}\right) = \dfrac{s^2-1}{(s^2+1)^2}$$

Are you familiar with the rule I am using?

If

$$\mathcal{L}(f(t)) = F(s)$$

Then

$$\mathcal{L}(t^nf(t)) = (-1)^n\dfrac{d^n}{ds^n}F(s)$$