I'm trying to solve the following integral equation using Fourier transforms: $$u(t)+ \int_{-\infty}^{t} e^{\tau-t} u(\tau)\,d\tau=e^{-2|\tau|}$$
I tried to transform both sides of the equation using $\mathfrak{F}\left[\int_{\infty}^{t} f(\tau)\,d\tau\right]=\frac{F(\omega)}{i\omega}+\pi F(0)\delta(\omega)$ which I found in a table of Fourier transforms but this did not seem to help me.
Is using Fourier transform a good approach for this equation? If it is, how do I proceed?