\(\displaystyle \text{P.V}(1-e^{-\frac{2\pi}{3} i } )\int^{\infty}_{0} \frac{1}{\sqrt[3]{x}(x+1)}\, dx = 2\pi i e^{-\frac{\pi}{3} i} \)
\(\displaystyle \text{P.V}\int^{\infty}_{0} \frac{1}{\sqrt[3]{x}(x+1)}\, dx = 2\pi i \frac{e^{-\frac{\pi}{3} i}}{1-e^{-\frac{2\pi}{3} i } } \)
We can easily prove that \(\displaystyle 2\pi i \frac{e^{-\frac{\pi}{3} i}}{1-e^{-\frac{2\pi}{3} i } }=\frac{\pi }{\sin \left( \frac{\pi}{3}\right)}=\frac{2\pi }{\sqrt{3}}\)