Integrating with indented contour

[jimmycricket]

Homework Statement

Evaluate the following integral by integrating the corresponding complex function.

$\int_{-\infty}^\infty \frac{dx}{x(x^2+x+1)}$

Homework Equations

Cauchy's Residue Theorem for simple pole at a:$Res(f;a)=\displaystyle\lim_{z\rightarrow a} (z-a)f(z)$

The Attempt at a Solution

I have used the definite real integral widget on wolfram which states that the integral does not converge. Will I be able to show this is the case by integrating around the semi circular contour indented at 0?

 

[jackmell]

Homework Statement

Evaluate the following integral by integrating the corresponding complex function.

$\int_{-\infty}^\infty \frac{dx}{x(x^2+x+1)}$

Homework Equations

Cauchy's Residue Theorem for simple pole at a:$Res(f;a)=\displaystyle\lim_{z\rightarrow a} (z-a)f(z)$

The Attempt at a Solution

I have used the definite real integral widget on wolfram which states that the integral does not converge. Will I be able to show this is the case by integrating around the semi circular contour indented at 0?

Did you look at it first? I mean plot it say from -10 to 10? Looks to me it has the right shape to converge in the Cauchy Principal Value sense. That is the only way it can converge since it has a pole on the path of integration. Alpha is telling you it diverges in the Riemann sense. Did you try:

Integrate[1/(x*(x^2 + x + 1)), 
  {x, -Infinity, Infinity}, PrincipalValue -> True]

However, if you're not familiar with Principal-valued integrals, you may want to look that up.

Edit: made a mistake with the function. It's +1 and I corrected it above but still everything I said in regards to the function with -1 applies to this function as well.