Compute $\int_0^{2\pi} e^{e^{ix}} dx$.
For the definition of the contour integral we have $\int_\gamma f(z) dz = \int_a^b f(\gamma(x)) \ \gamma'(x) dx$ where $\gamma : [a,b] \to \mathbb C$ is a curve.
We note $\gamma(x) = ix$ and $\gamma'(x) = i$ with $0 \le x \le 2\pi$ and $f(z) = e^{e^z}$. Then $$ \int_\gamma f(z) dz = \int_0^{2\pi} e^{e^{ix}} i \ dx $$ from which we have $\int_0^{2\pi} e^{e^{ix}} dx = \frac 1i \int_\gamma f(z) dz = \frac 1i \int_\gamma e^{e^z} dz$.
$$ \begin{align*} \int_0^{2\pi} e^{e^{ix}} dx &= \frac 1i \int_\gamma e^{e^z} dz\\ &= \frac{2\pi}{i2\pi} \int_\gamma \frac{e^w}{w}dw \ \ \ \ \text{(Cauchy's integral formula)} \\ &= 2 \pi e^0 = 2 \pi \end{align*} $$ where $w = e^z$ and $dw = e^z dz = w dz$.
Is this complex u-substitution with $w$ okay here? If it is okay, why can I use it here, and if not why not and what should I do instead?