How can I compute the Fourier transform a a product of two functions?

Question

I think I understood the basic rule but cannot apply it in practice with more difficult examples than the ones consisting in basic functions. Convention for Fourier transform I use is the following: \begin{equation} \hat{f}(k)=\int_{\mathbb{R}}e^{ikx}f(x)dx \end{equation} If I should find what is $\hat{u}(k)$ of the function $u(x)=sign(x)e^{-\left|x\right|}$, what can I do once written the integral: \begin{equation} \hat{u}(k)=\int_{\mathbb{R}}e^{ikx}sign(x)e^{-\left|x\right|}dx \text{ ?} \end{equation}