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Need some help with computing index number (winding number). I need to compute the index number: $n(\gamma,0)$ where $\gamma(t)=\cos t + 3 i \sin t$ for $0 \leq t \leq 4\pi$

I tried to use parametrization, but the integral seem quite ugly. By the use of computer the answer is 0, but it can't be because the index counts the number of circuit on this ellipse, it should be 2 as i see it.

Maybe you have a way to compute it?

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    @Matt: You deleted the factor of $4$; I think that was included by intention.2011-11-22
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    @joriki: Unintended. Thanks, joriki.2011-11-22

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