Equations
Fourier series of x(t):
$ x(t)=\sum_{k=-\infty}^{\infty}a_ke^{jk\omega_0t} $
Signal Coefficients:
$ a_k=\frac{1}{T}\int_0^Tx(t)e^{-jk\omega_0t}dt $.
Defined Signal
$ x(t)=4sin(3t)+(1+2j)cos(2t)\! $
Solution
The fundamental period $ T\! $ is $ 2\pi\! $. Thus we use the equation $ \omega_0=\frac{2\pi}{T}\! $ to find $ \omega_0=1\! $
To find the value of $ a_0\! $ we simply plug and chug:
$ a_0=\frac{1}{2\pi}\int_0^{2\pi}[4sin(3t)+(1+2j)cos(2t)]e^{0}dt $
$ =\frac{2}{\pi}\int_0^{2\pi}sin(3t)dt+\frac{1+2j}{2\pi}\int_0^{2\pi}cos(2t)dt $
$ =\frac{-2}{3\pi}[cos(3t)]_0^{2\pi}+\frac{1+2j}{4\pi}[sin(2t)]_0^{2\pi} $
$ =\frac{-2}{3\pi}[cos(6\pi)-cos(0)]+\frac{1+2j}{4\pi}[(sin(4\pi)-sin(0)] $
$ =\frac{-2}{3\pi}[0]+\frac{1+2j}{4\pi}[0] $
$ =0\! $
The same method can be used to find each value of $ a_k\! $. To compute the rest of the values I'll use complex exponential identities, as that is much less tedious:
$ x(t)=4sin(3t)+(1+2j)cos(2t)\! $
$ =\frac{4}{2j}(e^{j3t}-e^{-j6t}+\frac{1+2j}{2}(e^{j2t}+e^{-j2t}) $
