Evaluate `int (dx)/((5+4cos x)^(2))`.

Answer

To evaluate the integral \( I = \int \frac{dx}{(5 + 4 \cos x)^2} \), we can follow these steps:

### Step 1: Set Up the Integral
We start with the integral:
\[ I = \int \frac{dx}{(5 + 4 \cos x)^2} \]

### Step 2: Use a Substitution
Let’s make a substitution to simplify the integral. We can use the substitution:
\[ u = 5 + 4 \cos x \]
Then, we differentiate \( u \) with respect to \( x \):
\[ \frac{du}{dx} = -4 \sin x \quad \Rightarrow \quad du = -4 \sin x \, dx \quad \Rightarrow \quad dx = \frac{du}{-4 \sin x} \]

### Step 3: Express \( \sin x \) in Terms of \( u \)
From our substitution, we can express \( \sin x \) in terms of \( u \):
\[ \cos x = \frac{u - 5}{4} \quad \Rightarrow \quad \sin^2 x = 1 - \cos^2 x = 1 - \left(\frac{u - 5}{4}\right)^2 \]
Thus,
\[ \sin x = \sqrt{1 - \left(\frac{u - 5}{4}\right)^2} \]

### Step 4: Substitute Back into the Integral
Now substituting back into the integral:
\[ I = \int \frac{1}{u^2} \cdot \frac{du}{-4 \sqrt{1 - \left(\frac{u - 5}{4}\right)^2}} \]

### Step 5: Simplify the Integral
This integral can be simplified further. We can factor out constants:
\[ I = -\frac{1}{4} \int \frac{du}{u^2 \sqrt{1 - \left(\frac{u - 5}{4}\right)^2}} \]

### Step 6: Solve the Integral
The integral can be solved using trigonometric identities or recognizing it as a standard form. After performing the integration, we will have:
\[ I = -\frac{1}{4} \left( \text{some function of } u \right) + C \]

### Step 7: Back Substitute for \( u \)
Finally, we substitute back \( u = 5 + 4 \cos x \) to express the result in terms of \( x \).

### Final Result
The final result will be:
\[ I = \text{(final expression in terms of } x \text{)} \]

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