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So the question is:
Determine the fourier series representations for the following signal:
Signal

Here the formula for the fourier series
$$C_k=\frac{1}{T}\int_T \! x(t)e^\frac{-j2\pi kt}{T} \, \mathrm{d} t.$$

The period of that signal is 2. If your bound is from $\frac{-1}{2}$ to $\frac{3}{2}$ then your $x(t)$ is just two dirac deltas $\delta(t)-2\delta(t-1)$, therefore we have the equation:

$$C_k=\frac{1}{2}\int_\frac{-1}{2}^\frac{3}{2} \! (\delta(t)-2\delta(t-1))e^\frac{-j2\pi kt}{2} \, \mathrm{d} t.$$

And then I'm stuck. The answer is:
$$\frac{1}{2}-e^{-j\pi k} \mbox{ or } \frac{1-2e^{-j\pi k}}{2}$$

Which I can kind of understand but really, I don't know how they went from the integration step to the answer. Any ideas?

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The Dirac Delta Function $\delta(x)$ is defined as

$$ \int_{-\infty}^\infty f(x) \delta(x - a)\,dx = f(a). $$

This definition can be extended to finite intervals in several ways, and the above holds if $a$ is in the domain of integration, wich is your case. Hence

$$ C_k = \frac{1}{2}\int_{-\frac{1}{2}}^\frac{3}{2}\big(\delta(t) - 2 \delta(t-1)\big)e^{-j\pi k t}\,dt = \frac{1}{2}\Big(e^{-j\pi k t}\Big|_{t=0} - 2 e^{-j\pi k t}\big|_{t=1}\Big) $$ $$ = \frac{1}{2}\big(1 - 2 e^{-j\pi t}\big) $$

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