Fourier transform of cos(x)

Question

I've been stuck at an exercise that wants us to find the Fourier transform for f(x) = cos(x).

The solution being this

Using this formula

I don't know how you go from an integral of sines and cosines to a Dirac delta function, please help.

Answer 1

Use the fact that $\cos(x)=\frac{1}{2}(e^{ix}+e^{-ix})$, and that the Dirac delta function is the Fourier transform of unity:

$$\delta(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{ikx}dk$$

From here hopefully things will be pretty straightforward.