I know if we apply Fourier transform to $f(t) = \begin{cases}0,& t<0\\ e^{-\alpha t} &t>0 \end{cases} $, we get $F(\omega) = \frac{1}{\alpha+j\omega}$.
But how to get from $F(\omega)$ to $f(t)$ ?, because after writing the formula: $$f(t)=\frac{1}{2\pi} \int_{-\infty}^{\infty}\frac{1}{\alpha + j\omega}e^{j\omega t} d\omega$$ I cannot get any further. The integral seems hard to solve, as $\omega$ is in the denominator, being in the numerator it would be easy integration by parts. And also, how do I tell that the final result will have different definitions for the positive and negative parts of the real numbers.