Sin/cos integrals multiplying results (fourier transform).

Okay, I am trying to determine the fourier transform of cos (2[itex]\pi[/itex]x)=f(x)



Where F(k)=[itex]^{\infty}_{\infty}[/itex][itex]\int[/itex]f(x)exp[itex]^{-ikx}[/itex] dx,



So I use eulers relation to express the exponential term in terms of cos and sin, and then I want to use sin/cos multiplication integral results, such as:



[itex]^{\infty}_{-\infty}[/itex][itex]\int[/itex] cos(nx)cos(mx) dx =[itex]\pi[/itex] if m=n≠0

= 2[itex]\pi[/itex] if m=n=0

=0 if m≠n



- But these are only defined for limits [itex]\pm[/itex][itex]\pi[/itex].

So my question is , what are these results for [itex]\pm[/itex][itex]\infty[/itex].



Is there a obvious natural extension?



Many thanks for any assistance !