The value of `int_(0)^(4pi)log_(e)|3sinx+3sqrt(3) cos x|dx` then the value of I is equal to

Options

A. `pi log_(e)3`
B. `2 pi log_(e)3`
C. `4pilog_(e)3`
D. `8pilog_(e)3`

Explanation

To solve the integral \( I = \int_{0}^{4\pi} \log_{e} |3 \sin x + 3 \sqrt{3} \cos x| \, dx \), we can follow these steps:

### Step 1: Simplify the Integral
First, we can factor out the constant from the logarithm:
\[ I = \int_{0}^{4\pi} \log_{e} (3 |\sin x + \sqrt{3} \cos x|) \, dx \]
This can be simplified to:
\[ I = \int_{0}^{4\pi} \log_{e} 3 \, dx + \int_{0}^{4\pi} \log_{e} |\sin x + \sqrt{3} \cos x| \, dx \]
The first integral is straightforward:
\[ \int_{0}^{4\pi} \log_{e} 3 \, dx = \log_{e} 3 \cdot (4\pi) = 4\pi \log_{e} 3 \]

### Step 2: Evaluate the Second Integral
Now we need to evaluate:
\[ J = \int_{0}^{4\pi} \log_{e} |\sin x + \sqrt{3} \cos x| \, dx \]
Notice that \( \sin x + \sqrt{3} \cos x \) can be expressed in a different form. We can write:
\[ \sin x + \sqrt{3} \cos x = 2 \left( \frac{1}{2} \sin x + \frac{\sqrt{3}}{2} \cos x \right) \]
This can be rewritten using the angle addition formula:
\[ \sin x + \sqrt{3} \cos x = 2 \sin \left( x + \frac{\pi}{3} \right) \]
Thus, we have:
\[ |\sin x + \sqrt{3} \cos x| = 2 |\sin \left( x + \frac{\pi}{3} \right)| \]
Now substituting this back into the integral:
\[ J = \int_{0}^{4\pi} \log_{e} (2 |\sin \left( x + \frac{\pi}{3} \right)|) \, dx \]
This can be split into:
\[ J = \int_{0}^{4\pi} \log_{e} 2 \, dx + \int_{0}^{4\pi} \log_{e} |\sin \left( x + \frac{\pi}{3} \right)| \, dx \]
The first part evaluates to:
\[ \int_{0}^{4\pi} \log_{e} 2 \, dx = \log_{e} 2 \cdot (4\pi) = 4\pi \log_{e} 2 \]

### Step 3: Evaluate the Integral of the Logarithm of Sine
The integral of \( \log_{e} |\sin x| \) over a full period (from \( 0 \) to \( 2\pi \)) is known:
\[ \int_{0}^{2\pi} \log_{e} |\sin x| \, dx = -2\pi \log_{e} 2 \]
Thus, over \( 0 \) to \( 4\pi \):
\[ \int_{0}^{4\pi} \log_{e} |\sin \left( x + \frac{\pi}{3} \right)| \, dx = 2 \int_{0}^{2\pi} \log_{e} |\sin x| \, dx = 2(-2\pi \log_{e} 2) = -4\pi \log_{e} 2 \]

### Step 4: Combine Everything
Now, substituting back into \( J \):
\[ J = 4\pi \log_{e} 2 - 4\pi \log_{e} 2 = 0 \]

### Step 5: Final Calculation of \( I \)
Now we can find \( I \):
\[ I = 4\pi \log_{e} 3 + J = 4\pi \log_{e} 3 + 0 = 4\pi \log_{e} 3 \]

### Final Answer
Thus, the value of \( I \) is:
\[ \boxed{4\pi \log_{e} 3} \]