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

Answer

To solve the integral \( \int \frac{1 - \cos x}{\cos x (1 + \cos x)} \, dx \), we can break it down into simpler parts. Here’s a step-by-step solution:

### Step 1: Split the Integral
We can split the integral into two parts:
\[ \int \frac{1 - \cos x}{\cos x (1 + \cos x)} \, dx = \int \frac{1}{\cos x (1 + \cos x)} \, dx - \int \frac{\cos x}{\cos x (1 + \cos x)} \, dx \]

### Step 2: Simplify the Second Integral
The second integral simplifies as follows:
\[ \int \frac{\cos x}{\cos x (1 + \cos x)} \, dx = \int \frac{1}{1 + \cos x} \, dx \]
Thus, we rewrite the integral:
\[ \int \frac{1 - \cos x}{\cos x (1 + \cos x)} \, dx = \int \frac{1}{\cos x (1 + \cos x)} \, dx - \int \frac{1}{1 + \cos x} \, dx \]

### Step 3: Change of Variables for the First Integral
For the first integral \( \int \frac{1}{\cos x (1 + \cos x)} \, dx \), we can use the identity \( 1 + \cos x = 2 \cos^2\left(\frac{x}{2}\right) \):
\[ \int \frac{1}{\cos x (1 + \cos x)} \, dx = \int \frac{1}{\cos x \cdot 2 \cos^2\left(\frac{x}{2}\right)} \, dx \]
This can be rewritten as:
\[ \frac{1}{2} \int \frac{1}{\cos x} \cdot \frac{1}{\cos^2\left(\frac{x}{2}\right)} \, dx \]

### Step 4: Solve Each Integral
1. The first integral becomes:
\[ \frac{1}{2} \int \sec x \cdot \sec^2\left(\frac{x}{2}\right) \, dx \]
To integrate \( \sec x \), we can multiply and divide by \( \sec x + \tan x \):
\[ \int \sec x \, dx = \ln | \sec x + \tan x | + C \]

2. The second integral \( \int \frac{1}{1 + \cos x} \, dx \) can be simplified using the same identity:
\[ \int \frac{1}{1 + \cos x} \, dx = \int \frac{1}{2 \cos^2\left(\frac{x}{2}\right)} \, dx = \frac{1}{2} \int \sec^2\left(\frac{x}{2}\right) \, dx \]
This integral evaluates to:
\[ \frac{1}{2} \cdot 2 \tan\left(\frac{x}{2}\right) + C = \tan\left(\frac{x}{2}\right) + C \]

### Step 5: Combine the Results
Combining the results from both integrals, we have:
\[ \int \frac{1 - \cos x}{\cos x (1 + \cos x)} \, dx = \frac{1}{2} \ln | \sec x + \tan x | - \tan\left(\frac{x}{2}\right) + C \]

### Final Answer
Thus, the final answer is:
\[ \int \frac{1 - \cos x}{\cos x (1 + \cos x)} \, dx = \frac{1}{2} \ln | \sec x + \tan x | - \tan\left(\frac{x}{2}\right) + C \]