Complex Analysis Residue Problem

[tylerc1991]

Homework Statement

The question asks me to find the integral from 0 to infinity of 1/(x^3 + 1), where I have to use the specific contours that they specify. Now I know that I need to use residues (in fact just one here) and the singular point is (1+sqrt(3)*i)/2. Once I can factor the (x^3 + 1) part then I can take the problem from there. Can someone check me to see if I factored this right?

The Attempt at a Solution

(x^3 + 1) has zeros at x = -1, (1+sqrt(3)*i)/2, and (1-sqrt(3)*i)/2, so when I factor this it becomes (x + 1)(x - (1+sqrt(3)*i)/2)(x - (1-sqrt(3)*i)/2). Then I can continue with the residue and solve the problem, but I need to have factored this correctly. Thank you anyone for your help!

 

[ideasrule]

Yes, you factored correctly. You could check by multiplying all the factors back together.