How can I calculate the Fourier transform of $$ f(x)= \begin{cases} e^{ax}, & -\infty \lt x \le 0 \\ e^{-ax}, & 0 \lt x \lt \infty \\ \end{cases} $$ where $a\gt 0$ is a constant?
I know that the Fourier transform is defined as $$ \mathcal{F} f (\omega)=\int_{-\infty}^{\infty} f(x)e^{-i\omega x} \,\mathrm{d}x$$ but i just don't know how to tackle this.