Problem. Evaluate the integral, using appropriately chosen contour as shown in the figure:
$$ \int_{-1}^{1} \mathop{\mathrm{d}x} \frac{(1-x)^{-2/3}(1+x)^{-1/3}}{4+x^2} $$
Remember that when complexifying keep the argument between $-\pi$ and $\pi$
I'm learning complex integration and I am facing difficulties in solving this problem after trying several times.
I know that the function is analytic on this contour so the closed integral on the green contour would be zero. In my calculation the residue of f(x) at x= 1 and x= -1 is coming out to be zero. I used a "dog bone" contur to integrate it.
How to calculate it ?
Someone please help me out.