Integrate $\int_C \tan z \,dz $, where $C$ is the parabola arc $y=x^2$ that connects the points $z=0$ and $z=1+i$.

Question

I have already seen a similar question on the site, but this did not help with the solution, I think that answer is not correct. There came to mind

$\int \tan(1+i)x\, dx $.

Help with a further decision

Answer 1

Writing your complex variable $z=x+iy$, the arc $C$ will be parametrized by the complex function

$\gamma(t) = xt + iyt^2$

for $0\le t\le 1$. Now use the definition of the

$\int _{\gamma }f(z)\,dz=\int _{a}^{b}f{\big (}\gamma (t){\big )}\gamma '(t)\,dt $

and you should be done.

Edit: Renamed the parametrization function to $\gamma$ to avoid conflicting notation.