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I find integration with branch cuts difficult to grasp.

For example, I can understand that [tex]\sqrt{x}[/tex] is mutivalued and has 2 branches if we take a branch cut from 0 to +infty. But given it to be integrated from -infty to +infty what is the the meaning of taking a branch of [tex]\sqrt{x}[/tex] ?

Could u pls give me some pointers to look or guide me with this example?

I hope to find answers to integrals like [tex]\sqrt{(x-1){x-2}}[/tex]