`int1/(cosx+sqrt(3)sinx)` dx equals

Options

A. `logtan((pi)/(2)+(pi)/(12))+C`
B. `logtan((x)/(2)-(pi)/(12))+C`
C. `(1)/(2)log tan ((x)/(2)+(pi)/(12))+C`
D. `(1)/(2)log tan((x)/(2)-(pi)/(12))+C`

Explanation

To solve the integral \( \int \frac{1}{\cos x + \sqrt{3} \sin x} \, dx \), we can follow these steps:

### Step 1: Rewrite the Denominator
We start by rewriting the denominator. We can factor out \( \frac{1}{2} \):

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

### Step 2: Identify Trigonometric Values
Recognizing that \( \frac{1}{2} = \cos \frac{\pi}{3} \) and \( \frac{\sqrt{3}}{2} = \sin \frac{\pi}{3} \), we can rewrite the integral as:

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

### Step 3: Use the Cosine Angle Addition Formula
Using the cosine angle addition formula, we can express the denominator:

\[ \cos \frac{\pi}{3} \cos x + \sin \frac{\pi}{3} \sin x = \cos(x - \frac{\pi}{3}) \]

Thus, the integral becomes:

\[ \int \frac{1/2}{\cos(x - \frac{\pi}{3})} \, dx = \frac{1}{2} \int \sec(x - \frac{\pi}{3}) \, dx \]

### Step 4: Integrate the Secant Function
The integral of \( \sec u \) is \( \ln | \sec u + \tan u | + C \). Therefore, we have:

\[ \frac{1}{2} \int \sec(x - \frac{\pi}{3}) \, dx = \frac{1}{2} \left( \ln | \sec(x - \frac{\pi}{3}) + \tan(x - \frac{\pi}{3}) | + C \right) \]

### Step 5: Substitute Back
Now we substitute back to get the final result:

\[ = \frac{1}{2} \ln | \sec(x - \frac{\pi}{3}) + \tan(x - \frac{\pi}{3}) | + C \]

### Final Answer
Thus, the evaluated integral is:

\[ \int \frac{1}{\cos x + \sqrt{3} \sin x} \, dx = \frac{1}{2} \ln | \sec(x - \frac{\pi}{3}) + \tan(x - \frac{\pi}{3}) | + C \]