Fourier Transformation – Convolution quick question

Okay the question is to find the fourier transform of:


=F[itex]^{\infty}_{\infty}[/itex] [itex]\int[/itex]rect([itex]\frac{x’}{5}[/itex])([itex]\delta[/itex](x+3-x’)-[itex]\delta[/itex](x-3-x’)) dx’ [1]

– where F represents a fourier transform.
My Issue
Okay I am fine doing this using the convolution theorem, that the fourier transform of a convultion is given by the product of the two individual fourier transforms, but I am having trouble doing it explicitly

So from [1] integrating over each delta function, I deduce that the first term collapses everywhere except x’=x+3, and the second everywhere except x’=x-3, . So I get:

= (5sinc[itex]\frac{5k}{2}[/itex]exp[itex]^{\frac{3ik}{5}}[/itex]exp[itex]^{\frac{-3ik}{5}}[/itex])
using the properties that F(rect([itex]\frac{x}{1}[/itex]))=asinc([itex]\frac{ka}{2}[/itex]) and that F(f(x+a))=F(f(x))exp[itex]^{ika}[/itex]

Which does not agree with the convultion theorem were I get :


Thanks alot in advance for any assistance !

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