Integrability of $e^{iax}$ on $\mathbb{R}$

Question

I have the following function $f_\epsilon(x)=e^{iax-\epsilon|x|}$

I need to show that $f_\epsilon$ is in $L^1(\mathbb{R})$ and that $e^{iax}$ defines a tempered distribution on $\mathbb{R}$.

In both cases I need to study the integrablity of $e^{iax}$, which as I far as I know isn't integrable on $\mathbb{R}$ ? Am I missing something ?

Thanks.