If `f(x)` is the antidervative of `(1+2 tanx(tanx+secx))^((1)/(2)) and f((pi)/(6))=log2`, then the value of `f(0)` is

Answer

To solve the problem, we need to find the value of \( f(0) \) given that \( f(x) \) is the antiderivative of the function \( (1 + 2 \tan x (\tan x + \sec x))^{\frac{1}{2}} \) and that \( f\left(\frac{\pi}{6}\right) = \log 2 \).

### Step-by-Step Solution:

1. **Identify the Function to Integrate**:
We need to integrate the function:
\[ (1 + 2 \tan x (\tan x + \sec x))^{\frac{1}{2}}. \]

2. **Simplify the Expression Inside the Square Root**:
We can rewrite \( 1 + 2 \tan x (\tan x + \sec x) \):
\[ 1 + 2 \tan x \tan x + 2 \tan x \sec x = 1 + 2 \tan^2 x + 2 \tan x \sec x. \]
We know that \( 1 + \tan^2 x = \sec^2 x \), so:
\[ 1 + 2 \tan^2 x + 2 \tan x \sec x = \sec^2 x + 2 \tan x \sec x. \]

3. **Factor the Expression**:
The expression can be factored as:
\[ \sec^2 x + 2 \tan x \sec x = (\sec x + \tan x)^2. \]

4. **Integrate**:
Therefore, we can rewrite the integral:
\[ \int (1 + 2 \tan x (\tan x + \sec x))^{\frac{1}{2}} \, dx = \int \sqrt{(\sec x + \tan x)^2} \, dx = \int (\sec x + \tan x) \, dx. \]
The integral of \( \sec x + \tan x \) is:
\[ \int (\sec x + \tan x) \, dx = \log |\sec x + \tan x| + C. \]
Thus, we have:
\[ f(x) = \log |\sec x + \tan x| + C. \]

5. **Use the Given Condition**:
We know that \( f\left(\frac{\pi}{6}\right) = \log 2 \). Let's calculate \( f\left(\frac{\pi}{6}\right) \):
\[ \sec\left(\frac{\pi}{6}\right) = \frac{2}{\sqrt{3}}, \quad \tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}}. \]
Therefore,
\[ f\left(\frac{\pi}{6}\right) = \log\left(\frac{2}{\sqrt{3}} + \frac{1}{\sqrt{3}}\right) + C = \log\left(\frac{2 + 1}{\sqrt{3}}\right) + C = \log\left(\frac{3}{\sqrt{3}}\right) + C = \log\sqrt{3} + C. \]
Setting this equal to \( \log 2 \):
\[ \log\sqrt{3} + C = \log 2. \]
Thus,
\[ C = \log 2 - \log\sqrt{3} = \log\left(\frac{2}{\sqrt{3}}\right). \]

6. **Final Function**:
Now we can express \( f(x) \):
\[ f(x) = \log |\sec x + \tan x| + \log\left(\frac{2}{\sqrt{3}}\right) = \log\left(\frac{2}{\sqrt{3}} (\sec x + \tan x)\right). \]

7. **Calculate \( f(0) \)**:
Now we need to find \( f(0) \):
\[ \sec(0) = 1, \quad \tan(0) = 0. \]
Thus,
\[ f(0) = \log\left(\frac{2}{\sqrt{3}} (1 + 0)\right) = \log\left(\frac{2}{\sqrt{3}}\right). \]

### Conclusion:
The value of \( f(0) \) is:
\[ f(0) = \log\left(\frac{2}{\sqrt{3}}\right). \]