If you have the symbolic toolbox use ifourier() and multiply your function by the Heaviside function to give you an integration for [0 \infty)
syms t v
ifourier(exp(-v)*heaviside(v),v)
Keep in mind the convention used here for the Fourier and inverse pair.
The Fourier transform is
X(\omega) = \int_{-\infty}^{\infty} x(t) e^{-i \omega t} dt
and the inverse is
x(t) = \dfrac{1}{2\pi} \int_{-\infty}^{\infty} X(\omega) e^{i\omega t} d\omega
