Quick Fourier Series help?

Question

I was given a graph (shown above) and was asked to represent this as a Fourier Series. I was able to solve $a_0$ with no problem. However, when I was integrating for $a_n$ and $b_n$, I was having a little trouble.

Thanks for the help!

Answer 1

$$\begin{align*} \frac{\pi a_n}{2} &= \int_0^\pi e^{-t/2} \cos nt\,dt = \int_0^\pi e^{-t/2} \frac{e^{in t} + e^{-int}}{2}\,dt \\ &= \frac{1}{2} \int_0^\pi \exp \left( \left(-\frac{1}{2} + in \right) t \right)\,dt + \frac{1}{2} \int_0^\pi \exp \left( \left(-\frac{1}{2} - in \right) t \right)\,dt \\ \end{align*}$$ The first part can be integrated by $$\int_0^\pi \exp \left( \left(-\frac{1}{2} + in \right) t \right)\,dt = \frac{1}{-1/2 + in} \left. \exp \left( \left(-\frac{1}{2} + in \right) t \right) \right|_0^\pi = \frac{e^{-\pi/2 + i n \pi} - 1}{-1/2 + in} $$