$\int_{0}^{\frac{\pi}{2}}log(\frac{4+3~sin~x}{4+3~cos~x})dx=$

Question

Accepted Answer

To solve the integral $\int_{0}^{\frac{\pi}{2}} \log\left(\frac{4+3\sin x}{4+3\cos x}\right) dx$, we can use the property of definite integrals and symmetry. Let's analyze the integral step by step:

1. **Symmetry Property**: The integral from $0$ to $\frac{\pi}{2}$ can be evaluated using the property:
   $$
   \int_{0}^{a} f(x) \, dx = \int_{0}^{a} f(a-x) \, dx
   $$
   For this problem, $a = \frac{\pi}{2}$, so we have:
   $$
   \int_{0}^{\frac{\pi}{2}} \log\left(\frac{4+3\sin x}{4+3\cos x}\right) dx = \int_{0}^{\frac{\pi}{2}} \log\left(\frac{4+3\sin\left(\frac{\pi}{2} - x\right)}{4+3\cos\left(\frac{\pi}{2} - x\right)}\right) dx
   $$

2. **Trigonometric Identities**: Using the identities $\sin\left(\frac{\pi}{2} - x\right) = \cos x$ and $\cos\left(\frac{\pi}{2} - x\right) = \sin x$, the integral becomes:
   $$
   \int_{0}^{\frac{\pi}{2}} \log\left(\frac{4+3\cos x}{4+3\sin x}\right) dx
   $$

3. **Adding the Integrals**: Now, consider the sum of the original integral and the transformed integral:
   $$
   I = \int_{0}^{\frac{\pi}{2}} \log\left(\frac{4+3\sin x}{4+3\cos x}\right) dx + \int_{0}^{\frac{\pi}{2}} \log\left(\frac{4+3\cos x}{4+3\sin x}\right) dx
   $$

Simplifying the sum:
   $$
   I = \int_{0}^{\frac{\pi}{2}} \left[\log\left(\frac{4+3\sin x}{4+3\cos x}\right) + \log\left(\frac{4+3\cos x}{4+3\sin x}\right)\right] dx
   $$

$$
   I = \int_{0}^{\frac{\pi}{2}} \log\left(\frac{4+3\sin x}{4+3\cos x} \cdot \frac{4+3\cos x}{4+3\sin x}\right) dx
   $$

$$
   I = \int_{0}^{\frac{\pi}{2}} \log(1) \, dx = \int_{0}^{\frac{\pi}{2}} 0 \, dx = 0
   $$

4. **Conclusion**: Since the sum of the two integrals is zero, and both integrals are equal, each integral must be zero. Therefore, the value of the original integral is:
   $$
   \int_{0}^{\frac{\pi}{2}} \log\left(\frac{4+3\sin x}{4+3\cos x}\right) dx = 0
   $$

Thus, the correct answer is Option 3: 0.

**Explanation for Incorrect Options**:
- **Option 1 (2)**: This option suggests a non-zero value, which contradicts the symmetry and properties of the integral.
- **Option 2 ($\frac{3}{4}$)**: This is also a non-zero value, which does not align with the result obtained through symmetry.
- **Option 4 (-2)**: This negative value is incorrect as the integral evaluates to zero based on the symmetry and properties discussed.

Therefore, the correct answer is indeed Option 3: 0.