Question: How to get calculate the integration of follwing:

$$\int\_{0}^{2\pi} \frac{dt}{a + cos t} (a>1)$$

My attempt:

let, $z=e^{it}$
$\implies dt = \frac{dz}{it}$

and 

$$\cos t = \frac{z + \frac{1}{z}}{2}$$

On substituting everything in the integral I got:

$$\frac{2}{i}\int\_{c} \frac{dz}{z^2+2az+1}$$

Now how do I decompose this fraction so that I can use the Residue Theorem? Or is there anyother way to solve this??

Thanks for the help.

Answer: 
$\dots$ is not a mathematical operator. If your prof thinks his proof is correct, then he should be able to write it without $\dots$ . When $\dots$ is used correctly, it is simply tedious to replace it with symbolic logic. In the above case, it is impossible to replace it with symbolic logic so the professors proof is wrong.

Many attempts at proofs of induction can be disregarded due to a simple transform: instead of considering the natural numbers $\mathbb N$, consider the set of polynomials with natural number coefficients, $\mathbb N [x]$. $\mathbb N [x]$ shares most properties of natural numbers, such as being injective (wrt adding one) and having a zero. It is also totally well ordered. However, it is not an inductive set. So any "proof" of induction that doesn't make an assumption true of $\mathbb N$ but not $\mathbb N[x]$ is wrong.
The problem lies in his statement "go down a finite number of times". This statement is informal and does not in it self show anything. In order to use this statement and expound on it he would actually have to show that he may do this for $k=1$, $k=2$, $k=3$, etc. or if he do not wish do do an infinite algorithm, he may use induction.