The value of the definite integral `I=int_(-1)^(1)ln((2-sin^(3)x)/(2+sin^(3)x))dx` is equal to

Options

A. ln 4
B. ln 2
C. 0
D. `ln((1)/(2))`

Answer

To solve the definite integral \( I = \int_{-1}^{1} \ln\left(\frac{2 - \sin^3 x}{2 + \sin^3 x}\right) dx \), we can use the property of odd functions.

### Step 1: Identify the function
Let \( f(x) = \ln\left(\frac{2 - \sin^3 x}{2 + \sin^3 x}\right) \).

### Step 2: Check if \( f(x) \) is an odd function
To determine if \( f(x) \) is odd, we need to compute \( f(-x) \):
\[ f(-x) = \ln\left(\frac{2 - \sin^3(-x)}{2 + \sin^3(-x)}\right) \]
Using the property of sine, \( \sin(-x) = -\sin(x) \):
\[ f(-x) = \ln\left(\frac{2 - (-\sin x)^3}{2 + (-\sin x)^3}\right) = \ln\left(\frac{2 + \sin^3 x}{2 - \sin^3 x}\right) \]

### Step 3: Relate \( f(-x) \) to \( f(x) \)
We can rewrite \( f(-x) \):
\[ f(-x) = \ln\left(\frac{2 + \sin^3 x}{2 - \sin^3 x}\right) = -\ln\left(\frac{2 - \sin^3 x}{2 + \sin^3 x}\right) = -f(x) \]
This shows that \( f(x) \) is an odd function since \( f(-x) = -f(x) \).

### Step 4: Use the property of definite integrals
Since \( f(x) \) is an odd function and we are integrating over a symmetric interval from \(-1\) to \(1\):
\[ I = \int_{-1}^{1} f(x) \, dx = 0 \]

### Final Answer
Thus, the value of the definite integral is:
\[ I = 0 \]