This integral has stumped me for quite a bit.

$$\int_0^{\infty} \frac{\sqrt{x}}{x^3+1} \mathrm{d}x$$

I have identified poles at $x=e^{-\frac{i\pi}{3}}, e^{\frac{i\pi}{3}}, -1$.

Edit: I have changed my approach to this question thanks to Hans Lundmark's comment.

Since $\sqrt{x}$ is a multivalued function, I'm using a keyhole contour with a branch cut at $[0,\infty)$.

However, I have issues computing the residue at $x=-1$.

By L'Hopital's Rule: $$\text{Res}_{x=-1}=\lim_{x \to -1}\frac{\sqrt{x}}{3x^2}$$

But $\sqrt{-1}=\pm i$ since $-1=e^{\pm i\pi}$.

Which value should I be taking in cases like these?


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