Evaluate the following integrals:
`int frac{dx}{1+cosx}`

Answer

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

### Step 1: Use the Trigonometric Identity
We start with the identity:
\[ 1 + \cos x = 2 \cos^2 \left(\frac{x}{2}\right) \]
Thus, we can rewrite the integral as:
\[ I = \int \frac{dx}{2 \cos^2 \left(\frac{x}{2}\right)} \]

### Step 2: Simplify the Integral
This simplifies to:
\[ I = \frac{1}{2} \int \sec^2 \left(\frac{x}{2}\right) dx \]

### Step 3: Integrate
The integral of \( \sec^2 u \) is \( \tan u \), where \( u = \frac{x}{2} \). Therefore, we have:
\[ I = \frac{1}{2} \cdot 2 \tan \left(\frac{x}{2}\right) + C \]
Here, the factor of \( 2 \) comes from the derivative of \( \frac{x}{2} \).

### Step 4: Final Result
Thus, the final result simplifies to:
\[ I = \tan \left(\frac{x}{2}\right) + C \]

### Summary of the Steps:
1. Use the identity \( 1 + \cos x = 2 \cos^2 \left(\frac{x}{2}\right) \).
2. Rewrite the integral as \( \frac{1}{2} \int \sec^2 \left(\frac{x}{2}\right) dx \).
3. Integrate to get \( \tan \left(\frac{x}{2}\right) + C \).
4. Conclude with the final result.