The value of `int_(0)^(pi//2) log ((4+3 sin x)/(4+3 cos x)) dx` is

Options

A. 2
B. `3//4`
C. 0
D. None of these

Answer

To solve the integral \( I = \int_{0}^{\frac{\pi}{2}} \log\left(\frac{4 + 3 \sin x}{4 + 3 \cos x}\right) \, dx \), we can follow these steps:

### Step 1: Define the Integral
Let
\[ I = \int_{0}^{\frac{\pi}{2}} \log\left(\frac{4 + 3 \sin x}{4 + 3 \cos x}\right) \, dx \]

### Step 2: Use the Property of Logarithms
Using the property of logarithms, we can rewrite the integral:
\[ I = \int_{0}^{\frac{\pi}{2}} \left( \log(4 + 3 \sin x) - \log(4 + 3 \cos x) \right) \, dx \]

### Step 3: Define a New Integral
Now, we will define a new integral \( J \):
\[ J = \int_{0}^{\frac{\pi}{2}} \log\left(\frac{4 + 3 \cos x}{4 + 3 \sin x}\right) \, dx \]

### Step 4: Relate the Two Integrals
Notice that:
\[ J = \int_{0}^{\frac{\pi}{2}} \left( \log(4 + 3 \cos x) - \log(4 + 3 \sin x) \right) \, dx \]

### Step 5: Add the Two Integrals
Now, we can add \( I \) and \( J \):
\[ I + J = \int_{0}^{\frac{\pi}{2}} \left( \log(4 + 3 \sin x) - \log(4 + 3 \cos x) + \log(4 + 3 \cos x) - \log(4 + 3 \sin x) \right) \, dx \]

### Step 6: Simplify the Expression
The expression simplifies to:
\[ I + J = \int_{0}^{\frac{\pi}{2}} 0 \, dx = 0 \]
Thus, we have:
\[ I + J = 0 \]

### Step 7: Conclude the Value of the Integral
Since \( J \) is just \( -I \), we can conclude that:
\[ I = -I \implies 2I = 0 \implies I = 0 \]

### Final Answer
The value of the integral is:
\[ \boxed{0} \]