what is the fourier transfrom coefficient of $e^{-j\omega t}$?

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Note:$$j=i=\sqrt{-1}$$

The Fourier Transform series coefficient formula: $$X(jw)=\int^{\infty}_{-\infty}x(t)e^{-j\omega t}dt$$

We choose $$x(t)=e^{j\omega_0 t}$$

$$X(jw)=\int^{\infty}_{-\infty}e^{j{w_0-w}t}=\frac{e^{j(w_0-w)\infty}-e^{j(w_0-w)-\infty}}{j(\omega_0-\omega)t}=\frac{\infty-0}{j(\omega_0-\omega)t}$$

I know the right answer is $2\pi\delta(\omega-\omega_0)$. But how do you see it?