Let x(t)= $ cos(t) $
Then
$ x(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}X(\omega)e^{j\omega t}d\omega $
$ x(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}cos(t)e^{j\omega t}d\omega $
$ x(t)=\frac{1}{2\pi}cos (t)\int_{-\infty}^{\infty}e^{j\omega t}d\omega $
$ x(t)=\frac{1}{2\pi}cos (t){\left.\frac{e^{j\omega t}}{jt}\right]_{-\infty}^{\infty}} $
$ x(t)=\frac{1}{2j\pi t}cos (t){\left.e^{j\omega t}\right]_{-\infty}^{\infty}} $
