I need help to integrate this, i tried changing variable and it didn't work, i tried integration by parts, and it failed too.
$\frac{1}{2π}∫e^{jx\omega}\frac{(1/6)}{(1/6-j\omega)}d\omega$
I need to integrate from $-\infty$ to $\infty$
Can someone give me a hint where to start.
Thank you
The Fourier transform has the following property:
$$f(-\omega)=2\pi\mathcal{F}(G(t))$$
where $G(\omega)=\mathcal{F}(f(t))$
Let $f(t)=\text{sgn}(t)$ then $$\mathcal{F}(f(t))=\frac 1{2\pi}\int_{-\infty}^{\infty} \text{sgn}(t)e^{-j\omega t}d\omega=\frac 1{2\pi}\left[\int_{0}^{\infty}e^{-j\omega t}d\omega-\int_{-\infty}^{0}e^{-j\omega t}d\omega\right]$$ $$=\frac 1\pi \left[\frac{e^{-j\omega t}}{-j\omega}\right]_0^\infty=\frac j{\pi\omega}$$
In addition, if $\mathcal F(h)=G(\omega-\omega_0)$ then $h(t)=e^{j\omega_0 t}f(t)$
Combining these three hints gives you another way of solving the integral.