Integrate: `int (dx)/(2+cosx)`

Answer

To solve the integral \( \int \frac{dx}{2 + \cos x} \), we can follow these steps:

### Step 1: Use the half-angle identity for cosine
We know that:
\[ \cos x = \frac{1 - \tan^2\left(\frac{x}{2}\right)}{1 + \tan^2\left(\frac{x}{2}\right)} \]
Let \( t = \tan\left(\frac{x}{2}\right) \). Then, we can express \( \cos x \) in terms of \( t \):
\[ 2 + \cos x = 2 + \frac{1 - t^2}{1 + t^2} = \frac{(2 + 1 + t^2) + (1 - t^2)}{1 + t^2} = \frac{3 + t^2}{1 + t^2} \]

### Step 2: Rewrite the integral
Now substituting this into the integral, we have:
\[ \int \frac{dx}{2 + \cos x} = \int \frac{dx}{\frac{3 + t^2}{1 + t^2}} = \int \frac{(1 + t^2) \, dx}{3 + t^2} \]

### Step 3: Change of variable
Next, we need to express \( dx \) in terms of \( dt \). The derivative of \( t = \tan\left(\frac{x}{2}\right) \) gives us:
\[ dt = \frac{1}{2} \sec^2\left(\frac{x}{2}\right) dx \quad \Rightarrow \quad dx = 2 \frac{dt}{\sec^2\left(\frac{x}{2}\right)} = 2 \frac{dt}{1 + t^2} \]

### Step 4: Substitute \( dx \) into the integral
Substituting \( dx \) into the integral:
\[ \int \frac{(1 + t^2)(2 \frac{dt}{1 + t^2})}{3 + t^2} = 2 \int \frac{dt}{3 + t^2} \]

### Step 5: Solve the integral
Now we can solve the integral:
\[ 2 \int \frac{dt}{3 + t^2} = 2 \cdot \frac{1}{\sqrt{3}} \tan^{-1}\left(\frac{t}{\sqrt{3}}\right) + C \]

### Step 6: Substitute back for \( t \)
Recall that \( t = \tan\left(\frac{x}{2}\right) \):
\[ = \frac{2}{\sqrt{3}} \tan^{-1}\left(\frac{\tan\left(\frac{x}{2}\right)}{\sqrt{3}}\right) + C \]

### Final Answer
Thus, the integral evaluates to:
\[ \int \frac{dx}{2 + \cos x} = \frac{2}{\sqrt{3}} \tan^{-1}\left(\frac{\tan\left(\frac{x}{2}\right)}{\sqrt{3}}\right) + C \]