`int(1)/(5+2 cos x)dx`

Answer

To evaluate the integral \( I = \int \frac{1}{5 + 2 \cos x} \, dx \), we can follow these steps:

### Step 1: Rewrite the Cosine Function
We start by rewriting the cosine function using the identity:
\[ \cos x = \frac{1 - \tan^2\left(\frac{x}{2}\right)}{1 + \tan^2\left(\frac{x}{2}\right)} \]
Substituting this into the integral gives:
\[ I = \int \frac{1}{5 + 2 \left(\frac{1 - \tan^2\left(\frac{x}{2}\right)}{1 + \tan^2\left(\frac{x}{2}\right)}\right)} \, dx \]

### Step 2: Simplify the Denominator
We can simplify the denominator:
\[ 5 + 2 \cos x = 5 + 2 \left(\frac{1 - \tan^2\left(\frac{x}{2}\right)}{1 + \tan^2\left(\frac{x}{2}\right)}\right) = \frac{5(1 + \tan^2\left(\frac{x}{2}\right)) + 2(1 - \tan^2\left(\frac{x}{2}\right))}{1 + \tan^2\left(\frac{x}{2}\right)} \]
This simplifies to:
\[ \frac{(5 + 2) + (5 - 2) \tan^2\left(\frac{x}{2}\right)}{1 + \tan^2\left(\frac{x}{2}\right)} = \frac{7 + 3 \tan^2\left(\frac{x}{2}\right)}{1 + \tan^2\left(\frac{x}{2}\right)} \]

### Step 3: Substitute and Change Variables
Now, substituting this back into the integral gives:
\[ I = \int \frac{1 + \tan^2\left(\frac{x}{2}\right)}{7 + 3 \tan^2\left(\frac{x}{2}\right)} \, dx \]
Let \( t = \tan\left(\frac{x}{2}\right) \). Then, we have:
\[ dx = \frac{2}{1 + t^2} \, dt \]
Substituting this into the integral:
\[ I = \int \frac{1 + t^2}{7 + 3t^2} \cdot \frac{2}{1 + t^2} \, dt = 2 \int \frac{1}{7 + 3t^2} \, dt \]

### Step 4: Evaluate the Integral
The integral \( \int \frac{1}{7 + 3t^2} \, dt \) can be solved using the formula:
\[ \int \frac{1}{a^2 + x^2} \, dx = \frac{1}{a} \tan^{-1}\left(\frac{x}{a}\right) + C \]
In our case, \( a^2 = 7 \) and \( x^2 = 3t^2 \), so \( a = \sqrt{7} \). Thus:
\[ I = 2 \cdot \frac{1}{\sqrt{7}} \cdot \frac{1}{\sqrt{3}} \tan^{-1}\left(\frac{t}{\sqrt{\frac{7}{3}}}\right) + C \]

### Step 5: Back Substitute for \( t \)
Since \( t = \tan\left(\frac{x}{2}\right) \), we substitute back:
\[ I = \frac{2}{\sqrt{21}} \tan^{-1}\left(\frac{\tan\left(\frac{x}{2}\right)}{\sqrt{\frac{7}{3}}}\right) + C \]

### Final Answer
Thus, the final answer for the integral is:
\[ I = \frac{2}{\sqrt{21}} \tan^{-1}\left(\frac{\tan\left(\frac{x}{2}\right)}{\sqrt{\frac{7}{3}}}\right) + C \]