Evaluating ∫(−∞,∞) x/(x³+1) with a singularity on the real line

[NT123]

Homework Statement

Need to evaluate int(-inf,inf)(x/(x^3+1)).

Homework Equations

The Attempt at a Solution

I don't believe finding the residues will be a problem. However, the integral is over the whole real line, and there is a singularity at -1 on the real line, so I'm not sure how to draw an appropriate contour to integrate around. Am I supposed to integrate over a keyhole contour avoiding the singularity at -1 from the right and then from the left? Any help / advice will be much appreciated.

 

[gabbagabbahey]

You might want to check if the integral actually exists/converges before worrying about contours.

 

[NT123]

You might want to check if the integral actually exists/converges before worrying about contours.

My book says it is equal to pi/sqrt(3) :)

 

[gabbagabbahey]

My book says it is equal to pi/sqrt(3) :)

Hmm...

$$\lim_{\epsilon\to 0} \left[\int_{-\infty}^{-1-\epsilon} \frac{x}{x^3+1}dx+\int^{\infty}_{-1+\epsilon} \frac{x}{x^3+1}dx\right]=\frac{\pi}{\sqrt{3}}$$

But, I don't think you can say that is the same thing as $\int_{-\infty}^{\infty} \frac{x}{x^3+1}dx$. Mathematica seems to agree with me, that the integral doesn't converge. I don't think $x=-1$ is a removable singularity.

 

[NT123]

Yeah... MATLAB seems to say the same thing.