The value of `int (cos 8x-cos 7x)/(1+2 cos 5x)dx`, is

Options

A. `(sin 2x)/(2)+(cos 3x)/(3)+C`
B. `sin x - cos x + C`
C. `(sin 2x)/(2)-(cos 3x)/(3)+C`
D. None of these

Answer

To solve the integral \( I = \int \frac{\cos 8x - \cos 7x}{1 + 2 \cos 5x} \, dx \), we will follow these steps:

### Step 1: Use the Cosine Difference Formula
We can use the formula for the difference of cosines:
\[ \cos c - \cos d = -2 \sin\left(\frac{c + d}{2}\right) \sin\left(\frac{c - d}{2}\right) \]
Here, let \( c = 8x \) and \( d = 7x \). Thus, we have:
\[ \cos 8x - \cos 7x = -2 \sin\left(\frac{8x + 7x}{2}\right) \sin\left(\frac{8x - 7x}{2}\right) = -2 \sin\left(\frac{15x}{2}\right) \sin\left(\frac{x}{2}\right) \]

### Step 2: Rewrite the Integral
Substituting this into the integral gives:
\[ I = \int \frac{-2 \sin\left(\frac{15x}{2}\right) \sin\left(\frac{x}{2}\right)}{1 + 2 \cos 5x} \, dx \]

### Step 3: Simplify the Denominator
We can rewrite \( \cos 5x \) using the identity \( \cos 2a = 1 - 2 \sin^2 a \):
\[ \cos 5x = \cos\left(2 \cdot \frac{5x}{2}\right) = 1 - 2 \sin^2\left(\frac{5x}{2}\right) \]
Thus, the denominator becomes:
\[ 1 + 2 \cos 5x = 1 + 2(1 - 2 \sin^2\left(\frac{5x}{2}\right)) = 3 - 4 \sin^2\left(\frac{5x}{2}\right) \]

### Step 4: Substitute Back into the Integral
Now, substituting this back into the integral:
\[ I = \int \frac{-2 \sin\left(\frac{15x}{2}\right) \sin\left(\frac{x}{2}\right)}{3 - 4 \sin^2\left(\frac{5x}{2}\right)} \, dx \]

### Step 5: Use the Identity for Sine
Using the identity \( \sin 3a = 3 \sin a - 4 \sin^3 a \), we can express \( -4 \sin^2\left(\frac{5x}{2}\right) \) in terms of \( \sin 3\left(\frac{5x}{2}\right) \):
\[ -4 \sin^2\left(\frac{5x}{2}\right) = 3 \sin\left(\frac{15x}{2}\right) - \sin\left(\frac{15x}{2}\right) \]

### Step 6: Cancel Terms
Now, substituting this back into our integral:
\[ I = \int \frac{-2 \sin\left(\frac{15x}{2}\right) \sin\left(\frac{x}{2}\right)}{3 \sin\left(\frac{15x}{2}\right)} \, dx \]
This simplifies to:
\[ I = \int \frac{-2 \sin\left(\frac{x}{2}\right)}{3} \, dx \]

### Step 7: Integrate
Now we can integrate:
\[ I = -\frac{2}{3} \int \sin\left(\frac{x}{2}\right) \, dx \]
The integral of \( \sin\left(kx\right) \) is \( -\frac{1}{k} \cos\left(kx\right) + C \). Thus:
\[ I = -\frac{2}{3} \left(-2 \cos\left(\frac{x}{2}\right)\right) + C = \frac{4}{3} \cos\left(\frac{x}{2}\right) + C \]

### Final Solution
The value of the integral is:
\[ I = \frac{4}{3} \cos\left(\frac{x}{2}\right) + C \]