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

Options

A. `log|secx+tanx|-tan((x)/(2))+c`
B. `log|secx+tanx|+tan((x)/(2))+c`
C. `log|secx+tanx|-cot((x)/(2))+c`
D. `log|secx+tanx|+cot((x)/(2))+c`

Answer

To solve the integral

\[ I = \int \frac{dx}{\cos x (1 - \cos x)} \]

we can break it down into simpler parts. Here’s a step-by-step solution:

### Step 1: Rewrite the Integral
We start with the integral:

\[ I = \int \frac{dx}{\cos x (1 - \cos x)} \]

To make integration easier, we can separate the terms in the denominator.

### Step 2: Separate the Denominator
We can rewrite the integrand as follows:

\[ I = \int \left( \frac{1}{\cos x} + \frac{1}{1 - \cos x} \right) dx \]

This is done by expressing \( \frac{1}{\cos x (1 - \cos x)} \) as:

\[ \frac{1}{\cos x} \cdot \frac{1}{1 - \cos x} \]

### Step 3: Split the Integral
Now we can split the integral into two parts:

\[ I = \int \sec x \, dx + \int \frac{1}{1 - \cos x} \, dx \]

### Step 4: Solve the First Integral
The first integral \( \int \sec x \, dx \) can be solved using the standard formula:

\[ \int \sec x \, dx = \ln | \sec x + \tan x | + C_1 \]

### Step 5: Solve the Second Integral
For the second integral \( \int \frac{1}{1 - \cos x} \, dx \), we can use the identity \( 1 - \cos x = 2 \sin^2 \left( \frac{x}{2} \right) \):

\[ \int \frac{1}{1 - \cos x} \, dx = \int \frac{1}{2 \sin^2 \left( \frac{x}{2} \right)} \, dx \]

This can be rewritten as:

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

The integral of \( \csc^2 \) is:

\[ \int \csc^2 z \, dz = -\cot z + C \]

So, substituting back, we have:

\[ \int \csc^2 \left( \frac{x}{2} \right) \, dx = -2 \cot \left( \frac{x}{2} \right) + C_2 \]

Thus,

\[ \int \frac{1}{1 - \cos x} \, dx = -\cot \left( \frac{x}{2} \right) + C_2 \]

### Step 6: Combine the Results
Now we can combine the results of both integrals:

\[ I = \ln | \sec x + \tan x | - \cot \left( \frac{x}{2} \right) + C \]

where \( C = C_1 + C_2 \) is the constant of integration.

### Final Answer
Thus, the final result for the integral is:

\[ I = \ln | \sec x + \tan x | - \cot \left( \frac{x}{2} \right) + C \]

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