Computing $\int_{\gamma}e^zdz$, where $\gamma$ is a particular semicircle

Question

How can I compute $\int_{\gamma}e^zdz$, if $\gamma$ is the semicircular arc depicted below?

So, $\gamma=3e^{i\theta(t)}$, with $0\le\theta(t)\le\pi$, and then $$\displaystyle\int_{\gamma}e^zdz=\int_0^\pi e^{3e^{i\theta(t)}}\cdot\left|\left(3e^{i\theta(t)}\right)'\right|dt .$$

This looks awful, how can I compute the rest?

Answer 1

Hint Consider the integral of $e^z$ around the closed contour given by concatenating $\gamma$ with the line segment from $-3$ to $3$ (oriented rightward), and apply Cauchy's Integral Theorem.

Answer 2

$F(z) = e^z$ is an anti-derivative of $f(z) = e^z$. Just use the (complex version of) the fundamental theorem of calculus.