Choose the correct answerThe value of `int_0^(pi/2) log((4+3sinx)/(4+3cosx))dx`
(A) 2 (B) `3/4` (C) 0 (D) -2

Options

A. `2`
B. `3/4`
C. `0`
D. `-2`

Explanation

To solve the integral \( I = \int_0^{\frac{\pi}{2}} \log\left(\frac{4 + 3 \sin x}{4 + 3 \cos x}\right) dx \), we will use a property of definite integrals and some logarithmic identities.

### Step-by-Step Solution:

1. **Define the Integral:**
Let \( I = \int_0^{\frac{\pi}{2}} \log\left(\frac{4 + 3 \sin x}{4 + 3 \cos x}\right) dx \).

2. **Use the Property of Integrals:**
We can use the property of integrals that states:
\[ \int_0^a f(x) dx = \int_0^a f(a - x) dx \]
Here, we will substitute \( x \) with \( \frac{\pi}{2} - x \):
\[ I = \int_0^{\frac{\pi}{2}} \log\left(\frac{4 + 3 \sin\left(\frac{\pi}{2} - x\right)}{4 + 3 \cos\left(\frac{\pi}{2} - x\right)}\right) dx \]

3. **Simplify the Sine and Cosine:**
Using the identities \( \sin\left(\frac{\pi}{2} - x\right) = \cos x \) and \( \cos\left(\frac{\pi}{2} - x\right) = \sin x \), we can rewrite the integral:
\[ I = \int_0^{\frac{\pi}{2}} \log\left(\frac{4 + 3 \cos x}{4 + 3 \sin x}\right) dx \]

4. **Add the Two Integrals:**
Now we have two expressions for \( I \):
\[ I = \int_0^{\frac{\pi}{2}} \log\left(\frac{4 + 3 \sin x}{4 + 3 \cos x}\right) dx \]
\[ I = \int_0^{\frac{\pi}{2}} \log\left(\frac{4 + 3 \cos x}{4 + 3 \sin x}\right) dx \]
Adding these two equations gives:
\[ 2I = \int_0^{\frac{\pi}{2}} \left( \log\left(\frac{4 + 3 \sin x}{4 + 3 \cos x}\right) + \log\left(\frac{4 + 3 \cos x}{4 + 3 \sin x}\right) \right) dx \]

5. **Combine the Logarithms:**
Using the property \( \log a + \log b = \log(ab) \):
\[ 2I = \int_0^{\frac{\pi}{2}} \log\left(\frac{(4 + 3 \sin x)(4 + 3 \cos x)}{(4 + 3 \cos x)(4 + 3 \sin x)}\right) dx \]
This simplifies to:
\[ 2I = \int_0^{\frac{\pi}{2}} \log(1) dx \]

6. **Evaluate the Integral:**
Since \( \log(1) = 0 \):
\[ 2I = \int_0^{\frac{\pi}{2}} 0 \, dx = 0 \]
Therefore, we find:
\[ I = 0 \]

### Conclusion:
The value of the integral \( \int_0^{\frac{\pi}{2}} \log\left(\frac{4 + 3 \sin x}{4 + 3 \cos x}\right) dx \) is \( 0 \).

### Final Answer:
The correct option is (C) \( 0 \).