Consider
$$\int^{\frac{1}{2}}\_{-\frac{1}{2} } e^{i2\pi f} \,df = \int^{\frac{1}{2} }\_{-\frac{1}{2} } \cos(2 \pi f)\, df$$

Why do we only look at the real part? What about the imaginary part $i\sin(2\pi f)$?

What is the reasoning behind this?