What is ∫ydx equal to? 

where c is the constant of integration.

Calculation:

Given,

The function is \(y = \left(1 - \cos(x)\right)^{-1} \), where\(x \neq 2n\pi \) and n is an integer.

The given function can be written as:

\( y = \frac{1}{1 - \cos(x)} \)

Using the identity \(1 - \cos(x) = 2\sin^2\left(\frac{x}{2}\right) \), we rewrite the function as:

\( y = \frac{1}{2\sin^2\left(\frac{x}{2}\right)} \)

The integral we need to evaluate is:

\( \int \frac{dx}{2\sin^2\left(\frac{x}{2}\right)} \)

\( \frac{1}{2} \int \csc^2\left(\frac{x}{2}\right) dx \)

The integral of \(\csc^2(x) \) is known to be \( -\cot(x) \), so we get:

\( \frac{1}{2} \left( -\cot\left(\frac{x}{2}\right) \right) + C \)

\( -\frac{1}{2} \cot\left(\frac{x}{2}\right) + C \)

The final result is \( -\cot\left(\frac{x}{2}\right) + C \)

Hence, the correct answer is Option 2.