Other then as a fantastic tool to evaluate some difficult real integrals, complex integrals have many purposes.
Firstly, contour integrals are used in Laurent Series, generalizing real power series.
The argument principle can tell us the difference between the poles and roots of a function in the closed contour $C$:
$$\oint_{C} {f'(z) \over f(z)}\, dz=2\pi i (\text{Number of Roots}-\text{Number of Poles})$$
and this has been used to prove many important theorems, especially relating to the zeros of the Riemann zeta function.
Noting that the residue of $\pi \cot (\pi z)f(z)$ is $f(z)$ at all the integers. Using a square contour offset by the integers by $\frac{1}{2}$, we note the contour disappears as it gets large, and thus
$$\sum_{n=-\infty}^\infty f(n) = -\pi \sum \operatorname{Res}\, \cot (\pi z)f(z)$$
where the residues are at poles of $f$.
While I have only mentioned a few, basic uses, many, many others exist.
I have a hard time avoiding blatant self-promotion here...
I don't know Lang. Ahlfors is of course a classic. I have a lot of issues with Conway. (My complaints are with the first volume, which it turns out he wrote as a student! The second volume is full of great stuff.) Conway was the standard text here for years - I hated it so much I started using my own notes instead, which eventually became Complex Made Simple (oops. Well, there are things in there that are not in any other elementary text that I know of.)
Two examples that spring to mind regarding Conway:
He spends almost a page using the power series for $\log(1+z)$ to show that $\lim_{z\to0}\frac{\log(1+z)}{z}=1,$ evidently not recalling the definition of the derivative.
There's a chapter or at least a section on the Perron solution to the Dirichlet problem. There's an exercise, like the first or second exercise in the chapter, which a few decades ago I was unable to do. I sent him a letter explaining why it was harder than he seemed to think.
In the next edition the words "This exercise is hard" were added. A year or so later I realized the exercise was not just hard, it was impossible. Asks us to prove something false.
Seems very unimpressive - I complain I don't know how to do the exercise and he doesn't even bother to make sure it's correct.
Best Answer
Note
$$\int_0^{2\pi} e^{\cos x}\cos(nx - \sin x)\, dx = \operatorname{Re} \int_0^{2\pi} e^{\cos x + i(nx - \sin x)}\, dx = \operatorname{Re} \int_0^{2\pi} e^{e^{-ix}} e^{inx}\, dx.$$
Using the parametrization $z = e^{-ix}$, $0 \le x \le 2\pi$ for the unit circle $|z| = 1$, we have
$$\int_0^{2\pi} e^{e^{-ix}}e^{inx}\, dx = \frac{1}{i}\int_{|z| = 1} e^{z} \frac{dz}{z^{n+1}}.$$
Now show that this contour integral is $\frac{2\pi}{n!}$.