My pdf is defined as follows:
$$f_X(x) = \frac{1}{\tau} e^{-x/\tau}$$
At first I started finding the characteristic function like so:
$$\hat{f}_X(\xi) = \mathbb{E}[e^{i\xi X}] = \frac{1}{\tau}\int_{\mathbb{R}} e^{i\xi x}e^{-x/\tau}dx$$
I then wrote $e^{i\xi x}$ as $\cos\xi x + i \sin \xi x$, so that I have:
$$\frac{1}{\tau} \int_{\mathbb{R}}\cos (\xi x) e^{-x/\tau} dx + \frac{i}{\tau} \int_{\mathbb{R}}\sin(\xi x)e^{-x/\tau}dx$$
Which in hindsight is not the best of the ideas (even though I can find the integrals, I will have problems evaluating say $\cos(\xi x)$ at infinity. So how shall I actually solve this?
On request i checked if OP got the right CF. Almost. Just the wrong sign.
$$ \begin{align} \hat{f}_X(\xi) & = \mathbb{E}[e^{i\xi X}] \\ & \\ & = \frac{1}{\tau}\int_{0}^{\infty} e^{i\xi x}e^{-x/\tau}\:\:\text{dx}\\ & \\ & = \frac{1}{\tau}\int_{0}^{\infty} e^{-(1/\tau - i\xi) x}\:\:\text{dx}\\ & \\ & = -\frac{1}{\tau}\frac{1}{1/\tau - i\xi} \left[e^{-(1/\tau - i\xi) x}\:\right]_0^{\infty}\\ & \\ & = \frac{1}{\tau}\frac{1}{1/\tau - i\xi} \\ & \\ & = \frac{1}{1 - i\tau\xi} \\ \end{align} $$