The value of definite integral `I=int_(ln((sqrt3)/(2)))^(ln((2)/(sqrt3)))ln.((2-tan^(7)x)/(2+tan^(7)x))dx` is equal to

Options

A. `ln4`
B. `ln2`
C. 0
D. `ln((1)/(2))`

Answer

To solve the definite integral

\[ I = \int_{\ln(\frac{\sqrt{3}}{2})}^{\ln(\frac{2}{\sqrt{3}})} \ln\left(\frac{2 - \tan^7 x}{2 + \tan^7 x}\right) dx, \]

we can follow these steps:

### Step 1: Change of Variables
We can use the property of logarithms and the symmetry of the integral. Notice that the limits of integration are symmetric about \( \ln(1) = 0 \). We can perform a substitution \( x = -u \), which gives \( dx = -du \). The limits change accordingly:

- When \( x = \ln(\frac{\sqrt{3}}{2}) \), \( u = -\ln(\frac{\sqrt{3}}{2}) \).
- When \( x = \ln(\frac{2}{\sqrt{3}}) \), \( u = -\ln(\frac{2}{\sqrt{3}}) \).

Thus, the integral becomes:

\[ I = \int_{-\ln(\frac{2}{\sqrt{3}})}^{-\ln(\frac{\sqrt{3}}{2})} \ln\left(\frac{2 - \tan^7(-u)}{2 + \tan^7(-u)}\right)(-du). \]

### Step 2: Simplifying the Function
Using the property of the tangent function, we know that \( \tan(-u) = -\tan(u) \). Therefore, we have:

\[ \tan^7(-u) = -\tan^7(u). \]

Substituting this into the integral gives:

\[ I = \int_{-\ln(\frac{2}{\sqrt{3}})}^{-\ln(\frac{\sqrt{3}}{2})} \ln\left(\frac{2 + \tan^7 u}{2 - \tan^7 u}\right) du. \]

### Step 3: Combining the Integrals
Now we have two expressions for \( I \):

1. The original integral:
\[ I = \int_{\ln(\frac{\sqrt{3}}{2})}^{\ln(\frac{2}{\sqrt{3}})} \ln\left(\frac{2 - \tan^7 x}{2 + \tan^7 x}\right) dx. \]

2. The transformed integral:
\[ I = \int_{-\ln(\frac{2}{\sqrt{3}})}^{-\ln(\frac{\sqrt{3}}{2})} \ln\left(\frac{2 + \tan^7 u}{2 - \tan^7 u}\right) du. \]

### Step 4: Adding the Two Integrals
Adding both expressions for \( I \):

\[ 2I = \int_{\ln(\frac{\sqrt{3}}{2})}^{\ln(\frac{2}{\sqrt{3}})} \left[ \ln\left(\frac{2 - \tan^7 x}{2 + \tan^7 x}\right) + \ln\left(\frac{2 + \tan^7 x}{2 - \tan^7 x}\right) \right] dx. \]

Using the property of logarithms \( \ln(a) + \ln(b) = \ln(ab) \):

\[ 2I = \int_{\ln(\frac{\sqrt{3}}{2})}^{\ln(\frac{2}{\sqrt{3}})} \ln(1) dx = 0. \]

### Step 5: Conclusion
Thus, we find that:

\[ I = 0. \]

### Final Answer
The value of the definite integral is

\[ \boxed{0}. \]