Examples

1. $x(t)=e^{-at}u(t) \hspace{1cm} a>0$

(1)
\begin{aligned} X(jw) & =\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt=\int_{-\infty}^{\infty}e^{-at}u(t)e^{-j\omega t}dt \\ \nonumber & =\int_{0}^{\infty}e^{-at}e^{-j\omega t}dt=\int_{0}^{\infty}e^{-(a+j\omega) t}dt \\ \nonumber & =\bigg[\frac{e^{-(a+j\omega) t}}{-(a+j\omega)}\bigg]_{0}^{\infty}=\frac{0-1}{-(a+j\omega)} \\ \nonumber & =\frac{1}{a+j\omega}=\frac{a-j\omega}{a^2+\omega^2} \nonumber \end{aligned}
(2)
\begin{aligned} |X(jw)| & =\bigg|\frac{1}{a+j\omega}\bigg| \nonumber & =\frac{1}{|a+j\omega |}=\frac{1}{\sqrt{a^2+\omega^2}} \\ \nonumber \end{aligned}

$\angle X(jw)=0-tan^{-1}\big(\frac{w}{a}\big)$

2. $x(t)=e^{-a|t|} \hspace{1cm} a>0$

(3)
\begin{cases} x(t) = e^{-at} & \text{if}\ t>0 \\ \nonumber e^{at} & \text{if}\ t<0\\ \nonumber \end{cases}
(4)
\begin{aligned} X(jw) & =\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt=\int_{-\infty}^{0}e^{at}e^{-j\omega t}dt+\int_{0}^{\infty}e^{-at}e^{-j\omega t}dt \\ \nonumber & =\int_{-\infty}^{0}e^{(a-j\omega) t}dt+\int_{0}^{\infty}e^{-(a+j\omega t)}dt \\ \nonumber & =\bigg[\frac{e^{(a-j\omega) t}}{(a-j\omega)}\bigg]_{-\infty}^{0}+\bigg[\frac{e^{-(a+j\omega) t}}{-(a+j\omega)}\bigg]_{0}^{\infty} \\ \nonumber & =\frac{1-0}{a-j\omega}+\frac{0-1}{-(a+j\omega)}=\frac{1}{a-j\omega}+\frac{1}{a+j\omega} \\ \nonumber & =\frac{2a}{a^2+\omega^2} \\ \nonumber \end{aligned}

3. $x(t)=\delta(t)$

$X(jw) =\int_{-\infty}^{\infty}\delta(t)e^{-j\omega t}dt=e^{-j\omega t}\big|_{t=0} =1$

Let, $e^{-j\omega t}=g(t)$.
Recall, $\int_{-\infty}^{\infty}\delta(t)g(t)dt=g(0)=g(t)\big|_{t=0}$.
Therefore,

$X(j\omega)=1 \hspace{0.2cm} \text{i.e} \hspace{0.2cm} \delta(t) \longleftrightarrow 1$