$ \displaystyle \int ^{\infty}_{-\infty} 12 \cos ( 2 π ) \dfrac{\sin ( 4 π t )}{ 4 π t} d t =?$

we know, $ {\displaystyle \int _{-\infty }^{\infty }{\frac {\sin(\pi x)}{\pi x}}\,dx=\operatorname {rect} (0)=1} $

by using, $ 2\sin(A) \cos (B) = \sin(A+B) + \sin(A-B) $ we get,

$$ \displaystyle \int ^{\infty}_{-\infty} 6 \bigg[\dfrac{\sin ( 2 π t )}{ 4 π t} + \dfrac{\sin ( 6 π t )}{ 4 π t}\bigg] d t \\

6 \bigg[\dfrac{1}{ 4} + \dfrac{1}{ 4}\bigg] = 3 $$