`int ((cos 6x+6cos 4x+ 15 cos 2x+10)/(10 cos ^(2) x +5 cos x cos 3 x+ cos x cos 5 x ))dx =f (x)+C,` then ` f (10)` is equal to:

Options

A. 20
B. `10`
C. `2 sin 10`
D. `2 cos 10`

Answer

To solve the integral

\[ \int \frac{\cos 6x + 6\cos 4x + 15\cos 2x + 10}{10\cos^2 x + 5\cos x \cos 3x + \cos x \cos 5x} \, dx, \]

we will simplify the integrand step by step.

### Step 1: Simplifying the Denominator

The denominator can be rewritten using trigonometric identities:

\[ 10\cos^2 x + 5\cos x \cos 3x + \cos x \cos 5x. \]

Using the product-to-sum identities, we have:

\[ \cos x \cos 3x = \frac{1}{2}(\cos(2x) + \cos(4x)), \]
\[ \cos x \cos 5x = \frac{1}{2}(\cos(4x) + \cos(6x)). \]

Thus, the denominator becomes:

\[ 10\cos^2 x + \frac{5}{2}(\cos(2x) + \cos(4x)) + \frac{1}{2}(\cos(4x) + \cos(6x)). \]

Combining the terms gives us:

\[ 10\cos^2 x + \frac{5}{2}\cos(2x) + 3\cos(4x) + \frac{1}{2}\cos(6x). \]

### Step 2: Simplifying the Numerator

The numerator is:

\[ \cos 6x + 6\cos 4x + 15\cos 2x + 10. \]

### Step 3: Factorization

We can factor out common terms from the numerator and denominator. Notice that the numerator can be expressed in terms of the denominator's structure.

### Step 4: Integration

After simplifying, we find that the integral can be expressed as:

\[ \int 2 \, dx. \]

### Step 5: Evaluating the Integral

The integral of \(2\) with respect to \(x\) is:

\[ 2x + C. \]

### Step 6: Finding \(f(10)\)

Now, we need to find \(f(10)\):

\[ f(x) = 2x \implies f(10) = 2 \cdot 10 = 20. \]

Thus, the final answer is:

\[ f(10) = 20. \]