Evaluate: `int1/(sinx+sqrt(3)cosx)\ dx`

Answer

To evaluate the integral \( \int \frac{1}{\sin x + \sqrt{3} \cos x} \, dx \), we will follow a systematic approach.

### Step-by-Step Solution:

1. **Rewrite the Integral**:
We start with the integral:
\[ I = \int \frac{1}{\sin x + \sqrt{3} \cos x} \, dx \]

2. **Multiply and Divide by 2**:
To simplify the expression, we multiply and divide by 2:
\[ I = \int \frac{2}{2(\sin x + \sqrt{3} \cos x)} \, dx = \int \frac{2}{\sin x + \sqrt{3} \cos x} \cdot \frac{1}{2} \, dx \]

3. **Use the Angle Sum Identity**:
We recognize that \( \sin x + \sqrt{3} \cos x \) can be expressed in terms of a single sine function using the angle sum identity:
\[ \sin x + \sqrt{3} \cos x = 2 \left( \sin x \cos \frac{\pi}{3} + \cos x \sin \frac{\pi}{3} \right) = 2 \sin \left( x + \frac{\pi}{3} \right) \]
Therefore, we can rewrite the integral as:
\[ I = \int \frac{1}{2 \sin \left( x + \frac{\pi}{3} \right)} \, dx \]

4. **Integrate**:
The integral of \( \frac{1}{\sin u} \) is \( \ln | \tan \frac{u}{2} | + C \). Thus, we have:
\[ I = \frac{1}{2} \int \frac{1}{\sin \left( x + \frac{\pi}{3} \right)} \, dx = \frac{1}{2} \ln \left| \tan \left( \frac{x + \frac{\pi}{3}}{2} \right) \right| + C \]

5. **Final Result**:
Simplifying gives us:
\[ I = \frac{1}{2} \ln \left| \tan \left( \frac{x}{2} + \frac{\pi}{6} \right) \right| + C \]

### Final Answer:
\[ \int \frac{1}{\sin x + \sqrt{3} \cos x} \, dx = \frac{1}{2} \ln \left| \tan \left( \frac{x}{2} + \frac{\pi}{6} \right) \right| + C \]