The problem is:

Find the fourier transform of:

$\displaystyle f(t)=\int{cos(t)*e^{-t^2}}$

My question is: can i solve this problem using convolution, and how? I have some problems understanding when to apply convolution, can I choose

$\displaystyle f(t)=g(t)*h(t)$ with $\displaystyle g(t)=cos(t)$ and $\displaystyle h(t)=e^{-t^2}$, and then say that $\displaystyle F(f) = F(g*h) = \sqrt{2*Pi}*F(g)*F(h)$ ?

Or is this completely wrong?