<s>[INST] Why is $\displaystyle\int^{1/2}_{-1/2} \cos(\tfrac{2}{3} \pi x)\, dx= 0$? Why is 
$$\int^{1/2}_{-1/2} \cos\left(\frac{2}{3} \pi x\right)\, dx= 0\qquad ?$$
Because mathematically, we have
$$\sin\left( \frac{2}{3}\pi \right) - \sin\left(-\frac{2}{3}\pi\right) = 2\sin\left(\frac{2}{3}\pi\right)$$ instead.
And what about 
$$\int^{1/2}_{-1/2} \sin\left(\frac{2}{3} \pi x\right) \,dx $$
What is this equal to ?
I know this sounds like a very dumb question. But i would like to see different approaches to tackle this problem. 
 [/INST] $$\int\limits_{-1/2}^{1/2}\cos\frac{2}{3}\pi x\,dx=\left.\frac{3}{2\pi}\sin\frac{2}{3}\pi x\right|_{-1/2}^{1/2}=\frac{3}{2\pi}\left(\sin\frac{\pi}{3}-\sin\left(-\frac{\pi}{3}\right)\right)=$$
$$=\frac{3}{\pi}\sin\frac{\pi}{3}=\frac{3\sqrt 3}{2\pi}\neq 0$$
</s>