# Stuck at integration.

I am trying to solve the following integral. The topic is Fourier Series.

$$a_k = \frac{1}{4} \int_{0}^{2} \frac{1}{2} e^{-jk \omega_o t} dt +\frac{1}{4} \int_{2}^{4} -\frac{1}{2} e^{-jk \omega_o t} dt$$ $$=\frac{1}{8} \int_{0}^{2} e^{-jk \omega_o t} dt -\frac{1}{8} \int_{2}^{4} e^{-jk \omega_o t} dt$$

Now taking $$-\frac{1}{8jk\omega_0}$$ common and applying limits we get:

$$a_k=-\frac{1}{8jk\omega_0}((e^{2jk \omega_o}-1)-(e^{-4jk \omega_o}-e^{-2jk \omega_o}))$$

What should I do after this step? I want to apply the property of sin formula which states that:

$$sin\omega_0t=\frac{1}{2j}(e^{j\omega_0t} -e^{-j\omega_0t} )$$

Hint: Let $$jk \omega_o=\alpha$$ then $$(e^{2jk \omega_o}-1)-(e^{-4jk \omega_o}-e^{-2jk \omega_o})=$$ $$(e^{2\alpha}-1)-(e^{-4\alpha}-e^{-2\alpha})=$$ $$e^{\alpha}(e^{\alpha}-e^{-\alpha})-e^{-3\alpha}(e^{-\alpha}-e^{\alpha})=$$ $$(e^{\alpha}-e^{-\alpha})e^{-\alpha}(e^{2\alpha}+e^{-2\alpha})$$