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Question : Compute the integral of $$ \int^{2 \pi}_0 \frac{1}{3+2\cos t}dt. $$ Indication: take the path $\gamma: [0,2 \pi] \to \mathbb{C}$, $\gamma(t)=e^{it}$ and the integral of $$ \int_{\gamma} \frac{1}{z^2+3z+1} dz. $$

I am stucked on this problem since a good while. Is there someone who could solve this problem for me?

Thanks!

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  • $\begingroup$ $$\cos t = \frac{1}{2}\left(e^{it} + \frac{1}{e^{it}}\right)$$ Make the substitution they suggested and massage it a bit to get the second integral, then break the integrand into two pieces with partial fractions, and finally integrate. $\endgroup$ – Cameron Williams Apr 27 '16 at 3:00
  • $\begingroup$ @CameronWilliams I'm curious to see a complete answer using your index. Could you show a complete response? $\endgroup$ – user316765 Apr 27 '16 at 3:08
  • $\begingroup$ @george Someone else can fill in the details, but I pretty much gave a perfect guide to follow. $\endgroup$ – Cameron Williams Apr 27 '16 at 3:10
  • $\begingroup$ @CameronWilliams The integral I have to do is an real integral; so $\cos t$ is the real cosinus and not the complex cosinus as you consider in your previous sending $\endgroup$ – user316765 Apr 27 '16 at 10:25
  • $\begingroup$ math.stackexchange.com/questions/1760520/… $\endgroup$ – alexjo Apr 27 '16 at 11:29
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Note that this is a real integral. The basic idea here is to convert the real integral into a complex one and make it a contour integral.

Since the bounds are 0 to 2pi we can use the unit circle as the contour.

The the following substitutions.

Now make these subs and factor the integrand. Now look for poles inside the unit circle and use residue theorem on them.

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  • $\begingroup$ Would you be able to do the whole integral? $\endgroup$ – user316765 Apr 27 '16 at 10:44

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