`int ((1+cosx)/(1-cosx)) dx`

Explanation

To solve the integral \( \int \frac{1 + \cos x}{1 - \cos x} \, dx \), we can follow these steps:

### Step 1: Rewrite the integral
We start with the integral:
\[ \int \frac{1 + \cos x}{1 - \cos x} \, dx \]

### Step 2: Use trigonometric identities
We can use the identities for \(1 + \cos x\) and \(1 - \cos x\):
\[ 1 + \cos x = 2 \cos^2\left(\frac{x}{2}\right) \]
\[ 1 - \cos x = 2 \sin^2\left(\frac{x}{2}\right) \]
Substituting these into the integral gives:
\[ \int \frac{2 \cos^2\left(\frac{x}{2}\right)}{2 \sin^2\left(\frac{x}{2}\right)} \, dx \]
This simplifies to:
\[ \int \frac{\cos^2\left(\frac{x}{2}\right)}{\sin^2\left(\frac{x}{2}\right)} \, dx = \int \cot^2\left(\frac{x}{2}\right) \, dx \]

### Step 3: Substitute
Let \( t = \frac{x}{2} \), then \( dx = 2 \, dt \). The integral becomes:
\[ \int \cot^2(t) \cdot 2 \, dt = 2 \int \cot^2(t) \, dt \]

### Step 4: Use the identity for cotangent
Using the identity \( \cot^2(t) = \csc^2(t) - 1 \), we can rewrite the integral:
\[ 2 \int \cot^2(t) \, dt = 2 \int (\csc^2(t) - 1) \, dt \]
This separates into two integrals:
\[ 2 \left( \int \csc^2(t) \, dt - \int 1 \, dt \right) \]

### Step 5: Integrate
The integral of \( \csc^2(t) \) is \( -\cot(t) \) and the integral of \( 1 \) is \( t \):
\[ 2 \left( -\cot(t) - t \right) + C = -2\cot(t) - 2t + C \]

### Step 6: Substitute back
Now substitute back \( t = \frac{x}{2} \):
\[ -2\cot\left(\frac{x}{2}\right) - 2\left(\frac{x}{2}\right) + C = -2\cot\left(\frac{x}{2}\right) - x + C \]

### Final Answer
Thus, the final result of the integral is:
\[ \int \frac{1 + \cos x}{1 - \cos x} \, dx = -2\cot\left(\frac{x}{2}\right) - x + C \]