Complex Integral of $(\exp(z)-\exp(-z))/z^n$

How should I evaluate the following integral: $$\int_\gamma \frac{e^z-e^{-z}}{z^n}dz,$$where $\gamma$ is the unit circle and $n\in\Bbb N$.

My work on the integral has been to parametrize the unit circle, so I get the integral to $$\int_0^{2\pi}\frac{e^{e^{i\theta}}-e^{-e^{i\theta}}}{e^{in\theta}}ie^{i\theta}\,d\theta=i\int_0^{2\pi}\frac{e^{e^{i\theta}}-e^{-e^{i\theta}}}{e^{i(n-1)\theta}}\,d\theta.$$I got to this point, and I wasn't sure how to finish evaluating the integral.

Will someone provide only a hint?

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Hint: the integrand has a pole at $z=0$. What is the Laurent series?

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Laurent series have not been introduced in the course. Officially, we're about two chapters into Stein and Shakarchi's book, but this is not an exercise from the book, just from the professor. –  Clayton Feb 8 '13 at 4:42
@Clayton, if you are two chapters into that book, you probably know Cauchy's theorem already. Now write $e^z = 1 + z + \cdots + z^n/n! + g(z)$, where $g(z)$ is the tail. (Note that $g(z)/z^n$ is holomorphic). Do the same thing for $e^{-z}$, what can you conclude? –  user27126 Feb 8 '13 at 4:59
@Sanchez: Ah, now that makes sense! Let me work on it. –  Clayton Feb 8 '13 at 5:03

Hint: Use Cauchy's integral formula.

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