Example of Computation of Fourier transform of a CT SIGNAL

A practice problem on CT Fourier transform


$ x(t) = u(t)\frac{d}{dt}cos(t-2\pi) $

$ X(\omega) = j\omega\int\limits_{-\infty}^{\infty} cos(t-2\pi)u(t)e^{-j\omega t}dt $

    $ =  j\omega\int\limits_{0}^{\infty} cos(t-2\pi)e^{-j\omega t}dt $
    $ \tau  = t - 2\pi $
    $ =  j\omega\int\limits_{0}^{\infty} cos(\tau)e^{-j\omega(\tau -2\pi)}dt $
    $ =  j\omega\int\limits_{0}^{\infty} cos(\tau)e^{-j\omega \tau}e^{-j\omega 2\pi}dt $
    $ =  j\omega e^{-j\omega 2\pi} \int\limits_{0}^{\infty} cos(\tau)e^{-j\omega \tau}dt $
    $ =  j\omega e^{-j\omega 2\pi} \int\limits_{0}^{\infty}frac{1}{2}(e^{j\tau} e^{-j\omega \tau}dt $

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