`int(1)/(3+2 sin x +cos x)dx`

Answer

To evaluate the integral

\[ I = \int \frac{1}{3 + 2 \sin x + \cos x} \, dx, \]

we will follow a systematic approach.

### Step 1: Rewrite the Trigonometric Functions

We start by rewriting \(\sin x\) and \(\cos x\) in terms of \(\tan \frac{x}{2}\) using the identities:

\[ \sin x = \frac{2 \tan \frac{x}{2}}{1 + \tan^2 \frac{x}{2}}, \quad \cos x = \frac{1 - \tan^2 \frac{x}{2}}{1 + \tan^2 \frac{x}{2}}. \]

Let \(t = \tan \frac{x}{2}\). Then we can substitute these into the integral:

\[ I = \int \frac{1}{3 + 2 \left(\frac{2t}{1+t^2}\right) + \left(\frac{1 - t^2}{1+t^2}\right)} \, dx. \]

### Step 2: Simplify the Denominator

Now, we simplify the denominator:

\[ 3 + 2 \left(\frac{2t}{1+t^2}\right) + \left(\frac{1 - t^2}{1+t^2}\right) = 3 + \frac{4t}{1+t^2} + \frac{1 - t^2}{1+t^2}. \]

Combining the terms gives:

\[ = \frac{(3 + 1)(1 + t^2) + 4t}{1 + t^2} = \frac{4 + 4t + 2t^2}{1 + t^2} = \frac{2t^2 + 4t + 4}{1 + t^2}. \]

### Step 3: Rewrite the Integral

Thus, we have:

\[ I = \int \frac{1 + t^2}{2t^2 + 4t + 4} \, dx. \]

### Step 4: Change of Variables

We know that

\[ dx = \frac{2}{1+t^2} \, dt. \]

Substituting this into the integral gives:

\[ I = \int \frac{1 + t^2}{2t^2 + 4t + 4} \cdot \frac{2}{1+t^2} \, dt = 2 \int \frac{1}{2t^2 + 4t + 4} \, dt. \]

### Step 5: Factor the Denominator

Next, we factor the denominator:

\[ 2t^2 + 4t + 4 = 2(t^2 + 2t + 2) = 2\left((t+1)^2 + 1\right). \]

### Step 6: Complete the Integral

Thus, we can rewrite the integral as:

\[ I = \int \frac{2}{2((t+1)^2 + 1)} \, dt = \int \frac{1}{(t+1)^2 + 1} \, dt. \]

### Step 7: Use the Inverse Tangent Formula

Using the formula for the integral of the form \(\int \frac{1}{x^2 + a^2} \, dx = \frac{1}{a} \tan^{-1} \left(\frac{x}{a}\right) + C\), we have:

\[ I = \tan^{-1}(t + 1) + C. \]

### Step 8: Substitute Back

Finally, substituting back \(t = \tan \frac{x}{2}\):

\[ I = \tan^{-1}\left(\tan \frac{x}{2} + 1\right) + C. \]

### Final Answer

Thus, the final result is:

\[ I = \tan^{-1}\left(\tan \frac{x}{2} + 1\right) + C. \]