I Evaluate two complex integrals along a parabolic contour

[elias001]

TL;DR Summary

How to evaluate the following two complex integrals along a parabolic contour?

I need to evaluate the following two complex integrals along a parabolic contour, and I am not sure I know how to deal with the $\log\bar{z}$ in both cases. Since for the log term, I do not know how to find $i\arg\bar{z}$. Also, if I used either the line integral approach vs parameterize the contour as $(t, 1-t^2)$. Would I get the same answer?

$\int_{i}^{1}\ e^{(i\log\bar{z})} dz$

$\int_{i}^{1} \log\bar{z}dz$ both on the contour $y=1-x^2$

Thank you in advance

 

[anuttarasammyak]

Say
z=r e^{i\theta}
The first integral would be
\int_0^1 dt e^{i \log r(t)}e^{\theta(t)}
where
r(t)=\sqrt{t^4-t^2+1}
\theta(t)=arctan\frac{1-t^2}{t}
Numerical evaluation seems applicable. The second integral would be
\int_0^1 dt (\log r (t)- i\theta(t))

 

[elias001]

@anuttarasammyak I just saw your reply. For some reasons, I did not get the notifications that you made a reply. Thank you so much for your solutions. I really appreciate it.