# Integral and Cauchy theorem

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. Apr 27 '16 at 3:00
• @CameronWilliams I'm curious to see a complete answer using your index. Could you show a complete response?
– user316765
Apr 27 '16 at 3:08
• @george Someone else can fill in the details, but I pretty much gave a perfect guide to follow. Apr 27 '16 at 3:10
• @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
– user316765
Apr 27 '16 at 10:25
• math.stackexchange.com/questions/1760520/… Apr 27 '16 at 11:29

## 1 Answer

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.

$z=e^{i\Theta&space;}$

$dz=ie^{i\Theta&space;}d\Theta$

$d\Theta&space;=\frac{dz}{iz}$

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

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

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