Integral and Cauchy theorem

Question

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!

$$\cos t = \frac{1}{2}\left(e^{it} + \frac{1}{e^{it}}\right)$$ 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. — Cameron Williams, Apr 27 '16 at 03:00
@CameronWilliams I'm curious to see a complete answer using your index. Could you show a complete response? — , Apr 27 '16 at 03:08
@george Someone else can fill in the details, but I pretty much gave a perfect guide to follow. — Cameron Williams, Apr 27 '16 at 03:10
@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 — , Apr 27 '16 at 10:25
http://math.stackexchange.com/questions/1760520/int2-pi-0-frac1-sqrt5-cos-tdt-int2-pi-0-frac-cos2t-5 — alexjo, Apr 27 '16 at 11:29

score 1 · Answer 1 · edited Mar 09 '17 at 17:33

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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.

$z=e^{i\Theta }$

$dz=ie^{i\Theta }d\Theta$

$d\Theta =\frac{dz}{iz}$

$cos\Theta = \frac{z+z^{-1}}{2}$

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

edited Mar 09 '17 at 17:33

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answered Apr 27 '16 at 06:03

Erock Brox

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Would you be able to do the whole integral? – Apr 27 '16 at 10:44

Integral and Cauchy theorem

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