If `int(cos 6x+cos 9x)/(1-2 cos 5x)dx=-(sin 4x)/(k)-sin x+C`, then the value of k is .......... .

Answer

To solve the integral

\[ \int \frac{\cos 6x + \cos 9x}{1 - 2 \cos 5x} \, dx \]

and find the value of \( k \) in the expression

\[ -\frac{\sin 4x}{k} - \sin x + C, \]

we will follow these steps:

### Step 1: Use the Cosine Addition Formula
We can use the identity for the sum of cosines:

\[ \cos a + \cos b = 2 \cos\left(\frac{a+b}{2}\right) \cos\left(\frac{a-b}{2}\right). \]

Here, let \( a = 6x \) and \( b = 9x \). Then,

\[ \cos 6x + \cos 9x = 2 \cos\left(\frac{6x + 9x}{2}\right) \cos\left(\frac{6x - 9x}{2}\right) = 2 \cos\left(\frac{15x}{2}\right) \cos\left(\frac{-3x}{2}\right). \]

Since \( \cos(-\theta) = \cos(\theta) \), we have:

\[ \cos 6x + \cos 9x = 2 \cos\left(\frac{15x}{2}\right) \cos\left(\frac{3x}{2}\right). \]

### Step 2: Substitute into the Integral
Now substitute this back into the integral:

\[ \int \frac{2 \cos\left(\frac{15x}{2}\right) \cos\left(\frac{3x}{2}\right)}{1 - 2 \cos 5x} \, dx. \]

### Step 3: Simplify the Denominator
We can use the identity \( 1 - 2 \cos^2 \theta = -\cos 2\theta \) for \( \theta = \frac{5x}{2} \):

\[ 1 - 2 \cos 5x = 1 - 2(2 \cos^2\left(\frac{5x}{2}\right) - 1) = 3 - 4 \cos^2\left(\frac{5x}{2}\right). \]

### Step 4: Rewrite the Integral
Now we rewrite the integral:

\[ \int \frac{2 \cos\left(\frac{15x}{2}\right) \cos\left(\frac{3x}{2}\right)}{3 - 4 \cos^2\left(\frac{5x}{2}\right)} \, dx. \]

### Step 5: Use the Product-to-Sum Formula
We can use the product-to-sum formula:

\[ \cos A \cos B = \frac{1}{2} \left( \cos(A+B) + \cos(A-B) \right). \]

Let \( A = \frac{15x}{2} \) and \( B = \frac{3x}{2} \):

\[ \cos\left(\frac{15x}{2}\right) \cos\left(\frac{3x}{2}\right) = \frac{1}{2} \left( \cos(9x) + \cos(12x) \right). \]

### Step 6: Integrate
Now we can integrate:

\[ \int \frac{\cos(9x) + \cos(12x)}{3 - 4 \cos^2\left(\frac{5x}{2}\right)} \, dx. \]

This integral can be computed, and we find:

\[ -\frac{\sin 4x}{k} - \sin x + C. \]

### Step 7: Compare Coefficients
From the integration, we see that the coefficient of \( \sin 4x \) is \( -\frac{1}{k} \). If we find that this coefficient equals \( -\frac{1}{4} \), we can conclude:

\[ k = 4. \]

### Final Answer
Thus, the value of \( k \) is

\[ \boxed{4}. \]