`int (cosx-cos2x)/(1-cosx) dx`

Explanation

To solve the integral \( \int \frac{\cos x - \cos 2x}{1 - \cos x} \, dx \), we will follow these steps:

### Step 1: Use the double angle formula
We know that \( \cos 2x = 2 \cos^2 x - 1 \). We can substitute this into our integral:

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

### Step 2: Simplify the expression
Now, simplify the numerator:

\[ \cos x - (2 \cos^2 x - 1) = \cos x - 2 \cos^2 x + 1 \]

Thus, the integral becomes:

\[ \int \frac{\cos x - 2 \cos^2 x + 1}{1 - \cos x} \, dx \]

### Step 3: Factor out terms
We can factor out \(-1\) from the numerator:

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

### Step 4: Rewrite the numerator
We can rewrite the numerator \(2 \cos^2 x - \cos x - 1\) as:

\[ = 2 \cos^2 x - \cos x - 1 = 2 \cos x (\cos x - 1) + (\cos x - 1) \]

So we can factor it as:

\[ = (2 \cos x + 1)(\cos x - 1) \]

### Step 5: Substitute back into the integral
Now substituting this back into the integral gives:

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

### Step 6: Cancel terms
The \( \cos x - 1 \) in the numerator and \( 1 - \cos x \) in the denominator can be simplified. Remember \( 1 - \cos x = -(\cos x - 1) \):

\[ = \int -(2 \cos x + 1) \, dx \]

### Step 7: Integrate term by term
Now we can integrate:

\[ = -\int (2 \cos x + 1) \, dx = -\left(2 \sin x + x\right) + C \]

### Final Result
Thus, the final result of the integral is:

\[ = -2 \sin x - x + C \]

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