Why Multiply by Exponential Terms in Fourier Series Calculations?

[lottotlyl]

Homework Statement

how did my prof get the last term after the third equal sign

Relevant Equations

fourier series coeffecient equation

i tired using complex identity equation for sin(pi*k/3) but it doesn't work out

 

[Gaussian97]

Applying the fundamental theorem of calculus which says that
$$\int_a^b f(t)dt=F(t)\Big|_a^b=F(b)-F(a)$$ where $F(t)$ is a function satisfying $$\frac{dF(t)}{dt}=f(t).$$
Then you also need to use elemental algebra and complex analysis.

 

[lottotlyl]

I got this, but I don't know the rest of the steps

 

[Gaussian97]

Well, notice that in the final answer you have the term $$e^{\frac{-j\pi k}{3}}$$ so it would be a good idea to multiply by $$1=e^{\frac{-j\pi k}{3}}e^{\frac{j\pi k}{3}}.$$