What is the Contour Integral of Log(z) on a Specific Contour?

[hadroneater]

Homework Statement

Find the contour integral of Log(z). The contour is defined as: x^2 + 4y^2 = 4, x>= 0, y>=0

Homework Equations

The Attempt at a Solution

parametrize the contour as z(t) = 2cos(t) + isin(t)
0 <= t <= pi/2
The contour integral = ∫Log(z(t))z'(t)dt
I am having trouble finding Log(z(t)).
Log(z(t)) = ln|z(t)| + iArg(z(t))

Would ln|z(t)| = ln|sqrt(1 + 3cos(t)^2)| and Arg(z(t)) = t?

 

[jackmell]

What's wrong with antiderivatives? I mean why not use the Fundamental Theorem of Calculus for this? You know the starting point of integration and the ending point so poke-a-poke right?