So the question is:
Determine the fourier series representations for the following 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?