How do you use the fourier transform to evaluate the integral?

Question

I need to use the Fourier Transform to confirm that this result is true:

$$\int_0^\infty2te^{-at}*\cos(t)\,dt=2(a^2-1)/((a^2-1)^2+4a^2)$$

How can I do so? I'm using fourier transform identities such as $x(t)*\cos(w_0t)$ where $x(t)=2*t*e^{-at}$, but the end result contains terms of $\omega$. What am I supposed to do with omega and how do I get rid of it to make it look like that?

Answer 1

Hint. By using a Laplace Transform table one has

$$ \mathscr{L}(\cos t) = \int_0^\infty e^{-at}\cdot\cos(t)\,dt=\dfrac{a}{a^2 + 1} \tag 1$$

differentiating both sides of $(1)$ with respect to $a$ yields

$$\mathscr{L}(t \cos t) = \int_0^\infty t\:e^{-at}\cdot\cos(t)\,dt=\dfrac{a^2-1}{(a^2 + 1)^2} \tag 2$$

which gives the announced result since $(a^2+1)^2=(a^2-1)^2+4a^2$.