`int ( sin x cos x)/((cos^(2)x-cos x-2))dx`

Explanation

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

### Step 1: Substitution
Let \( t = \cos x \). Then, we have:
\[ \sin x \, dx = -dt \]
Thus, the integral becomes:
\[ \int \frac{\sin x \cos x}{\cos^2 x - \cos x - 2} \, dx = \int \frac{-t}{t^2 - t - 2} \, dt \]

### Step 2: Factor the Denominator
Next, we need to factor the denominator \( t^2 - t - 2 \):
\[ t^2 - t - 2 = (t + 1)(t - 2) \]
So, we rewrite the integral:
\[ \int \frac{-t}{(t + 1)(t - 2)} \, dt \]

### Step 3: Partial Fraction Decomposition
We express the integrand using partial fractions:
\[ \frac{-t}{(t + 1)(t - 2)} = \frac{A}{t + 1} + \frac{B}{t - 2} \]
Multiplying through by the denominator \( (t + 1)(t - 2) \) gives:
\[ -t = A(t - 2) + B(t + 1) \]
Expanding the right side:
\[ -t = At - 2A + Bt + B \]
Combining like terms:
\[ -t = (A + B)t + (-2A + B) \]
Setting coefficients equal gives us the system:
1. \( A + B = -1 \)
2. \( -2A + B = 0 \)

### Step 4: Solve the System of Equations
From the second equation, we can express \( B \) in terms of \( A \):
\[ B = 2A \]
Substituting into the first equation:
\[ A + 2A = -1 \implies 3A = -1 \implies A = -\frac{1}{3} \]
Then substituting \( A \) back to find \( B \):
\[ B = 2 \left(-\frac{1}{3}\right) = -\frac{2}{3} \]

### Step 5: Rewrite the Integral
Now we can rewrite the integral:
\[ \int \left( \frac{-\frac{1}{3}}{t + 1} + \frac{-\frac{2}{3}}{t - 2} \right) dt \]
This simplifies to:
\[ -\frac{1}{3} \int \frac{1}{t + 1} \, dt - \frac{2}{3} \int \frac{1}{t - 2} \, dt \]

### Step 6: Integrate
Now we integrate each term:
\[ -\frac{1}{3} \ln |t + 1| - \frac{2}{3} \ln |t - 2| + C \]

### Step 7: Substitute Back
Substituting \( t = \cos x \) back into the expression gives:
\[ -\frac{1}{3} \ln |\cos x + 1| - \frac{2}{3} \ln |\cos x - 2| + C \]

### Final Answer
Thus, the final answer is:
\[ \int \frac{\sin x \cos x}{\cos^2 x - \cos x - 2} \, dx = -\frac{1}{3} \ln |\cos x + 1| - \frac{2}{3} \ln |\cos x - 2| + C \]