I'm a physics student and learning complex analysis (to do definite integrals). There is a contour integral problem but I cannot solve it. Fortunately, I found the same problem on this stack exchange but I cannot understand the logic. There is the problem in this link. Evaluate by contour integration $\int_0^1\frac{dx}{(x^2-x^3)^{1/3}}$ and I want to ask about the first answer.
What I cannot understand is how to get a Laurent expansion of given integrand. (to get a residue at infinity) He/She said what I should do but I cannot see how x^(2/3) vanishes and e^(2pi*i/3) appears.
And another point confusing me is that I've never seen doing contour integral using residue at infinity. I googled for a bit and found that there is a property that total residue sum (including at infinity) is zero. (which is not written on my textbook.) My question is, can we just find the residues at 0 and 1 instead of finding residue at infinity?
I added a another confusing point in reply.