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

Explanation

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

### Step 1: Rewrite the integral
Let \( I = \int \frac{dx}{1 + \cos x} \).

### Step 2: Use a trigonometric identity
We can use the identity \( 1 + \cos x = 2 \cos^2 \left( \frac{x}{2} \right) \). Therefore, we can rewrite the integral as:
\[ I = \int \frac{dx}{2 \cos^2 \left( \frac{x}{2} \right)} = \frac{1}{2} \int \sec^2 \left( \frac{x}{2} \right) dx \]

### Step 3: Integrate using the known integral
The integral of \( \sec^2 u \) is \( \tan u + C \). Thus, we have:
\[ I = \frac{1}{2} \cdot \tan \left( \frac{x}{2} \right) + C \]

### Step 4: Final result
So, the final result for the integral is:
\[ I = \tan \left( \frac{x}{2} \right) + C \]