I need help to write nicely the transition from Riemann's integral to the curvilinear integral.
Finally, I received a good result, only the problem is probably with the record. And here is my teacher's commentary

It looks as if you were performing a substitution in the starting integral
Riemann and as a result of this substitution he received a curvilinear integral. But
it does not work like that

https://i.imgur.com/C5PowQB.png
i decide to put url bcs img its too big.

For good help i will pay a fair amount of fg.

Dead line = 12h

I'm not sure what's he's looking for. Perhaps he simply wants you to start with the definition of a complex curvilinear integral instead of directly making a substitution, in which case it'd start like this



But I didn't do much here... do you want help for the actual computation? Also, I think there's a tiny error in your red rectangle, it should read (z^2+1/z^2-2), and not (z^2+1/z^2-2)^2 (even though you corrected it later and the calculation looks correct I guess)