I'm trying to see if the Fourier transform of $e^{jwt}$ exists, so I am trying to evaluate this integral: $\int_{-\infty}^\infty|e^{jwt}|$ but I am not getting anywhere and $|\int_{-\infty}^\infty e^{jwt}|$ says nothing. How do I directly integrate this?
Also, just to confirm, its fourier transform doesn't exist right?
The Fourier Transform of $1$ is
$$\mathscr{F}\{1\}(\omega)=\int_{-\infty}^\infty (1)e^{j\omega t}\,dt \tag 1$$
As an improper Riemann integral or as a Lebesgue integral, the integral in $(1)$ does not exist. However, interpreted as a Distribution, the Fourier Transform of $1$ is
$$\mathscr{F}\{1\}(\omega)=2\pi \delta(\omega)$$
where $\delta$ is the Dirac Delta, which is a distribution (or generalized function) and not a function.