Let

be the unit circle with the positive sense.

Evaluate

.

My attempt :

Ok so first I notice that

only when

(at least in the circle, if I'm not wrong).

So there is one singularity in the domain of integration.

My idea was to determine the order of the pole (I'm all confused with poles and singularities) of

at

.

I found the Laurent series of

to be worth

thus the zero of

is of order 3, hence the order of the pole of

is of order

.

This implies that the integral is worth

.

Am I right?

(sorry for all these questions, tomorrow I'm going to ask help at university since I've a lecture. I might be asking here at night though

)